Noncommutative spaces and flat descent.

Maxim Kontsevich, Alexander Rosenberg

Abstract. This paper is dedicated to a prevailing during the last fifteen years approach to noncommutative algebraic geometry in which ’spaces’ are identified with categories thought as categories of quasi-coherent or coherent sheaves. We introduce a universal for this viewpoint category of ’spaces’ and develop basics of algebraic geometry inside of this category; i.e. we define affine objects, classes of morphisms which have geometric meaning, and glue locally affine spaces and schemes. One of the main tools for studying and describing thus defined locally affine spaces and morphisms between them is provided by a noncommutative version of flat descent.

Introduction

Spaces considered in commutative geometry, are either locally ringed topological spaces (otherwise called geometric spaces), or sheaves of sets on the site CAff of commutative affine schemes. Although these two categories of spaces have little in common, the categories of schemes obtained by glueing affine objects are naturally equivalent.

Noncommutative algebraic geometry is developing under influence of examples of interest which are felt to be of geometric nature, but appear in different surroundings. These examples suggest the choices of the category of ’spaces’. Thus, noncommutative analogue of P roj is defined as a category [M1], [V1], or a category with a fixed object O (– a structure sheaf) [AZ]. Projective spaces introduced in [KR1] appear as functors from the category Alg_{k} of associative unital k-algebras to Sets.

We studied ’spaces’ represented by sheaves of sets in [KR2] and noncommutative stacks in [KR3]. Here we undertake an elementary study of ’spaces’ represented by categories regarded as categories of quasi-coherent or coherent sheaves on (maybe ’virtual’) spaces. Similarly to [KR2], one of the purposes of this work is to provide a detailed background for the construction of non-affine spaces sketched in [KR1]. The paper is turning around several examples and establishes (a part of) a dictionary which extends geometric notions to ’abstract nonsense’ environment. This explains the length of the manuscript.

It is important to realize that the setting considered in this work is one of the faces of noncommutative geometry. There are several others. Among them, ’spaces’ represented by triangulated categories, or DG-, or A_{∞}-categories are of increasing importance and require a thorough investigation. We do not discuss them in this paper.

The category of ’spaces’. The minimal starting data for a geometric theory consists of: a category, 𝔅, of ’spaces’, a category A of ’local’ (or ’affine’) objects, a functor from the category A to the category 𝔅 which assigns to each affine object a ’space’. Given a set of covers for all objects of 𝔅, we define the notion of a locally affine ’space’ (details of this formalism can be found in sections 9.1 – 9.5).

We add to this minimal data one more ingredient: a map which assigns to each ’space’ X a category, C_{X}, regarded as the category of quasi-coherent modules on X.

This assignment is supposed to be a contravariant pseudo-functor from the category 𝔅 of ’spaces’ to the category Cat of categories (which belong to a given universum). In other words, we have a fibered category over the category 𝔅. There is a universal (in a sense which is made precise in the text) for this setting category of ’spaces’, |Cat|^{o}. It has same objects as the category Cat^{op} opposite to Cat, morphisms are isomorphism classes of 1-morphisms of Cat^{op}. The universal fibered category of ’spaces’ is given by the functor from Cat^{op} to |Cat|^{o} which is identical on objects and assigns to each 1-morphism of Cat^{op} its isomorphism class. We interpret 1-morphisms of Cat^{op} as inverse image functors of the corresponding morphisms of |Cat|^{o} (i.e morphisms of ’spaces’).

This is the starting point of the work. The immediate purpose is to define and study geometry (or geometries) inside of the category |Cat|^{o}, like commutative scheme theory is defined and studied inside of the category of locally ringed spaces.

What we really study and use is not so much ’spaces’, but certain classes of morphisms between ’spaces’. The most important among them are continuous’, flat, and affine morphisms introduced in [R]. A morphism is continuous if its inverse image functor has a right adjoint (called a direct image functor), and flat if, in addition, the inverse image functor is left exact (i.e. preserves finite limits). A continuous morphism is called affine if its direct image functor is conservative (i.e. reflects isomorphisms) and has a right adjoint.

Affine and locally affine ’spaces’. Given an object S of the category |Cat|^{o}, we define the category Aff_{S} of affine S-spaces as the full subcategory of |Cat|^{o}/S whose objects are affine morphisms to S. The functor from Aff_{S} to |Cat|^{o} is the composition of the inclusion functor Aff_{S} ↪ |Cat|^{o}/S and the canonical functor |Cat|^{o}/S ⟶ |Cat|^{o}. The choice of the object S influences drasticly the rest of the story. Thus, if S = Spℤ (i.e. C_{S} is the category of abelian groups), then Aff_{S} is equivalent to the category opposite to the category 𝔄ss which is defined as follows: objects of 𝔄ss are associative unital rings and morphisms are conjugation classes of unital ring morphisms.

If C_{S} is the category Sets, then the category Aff_{S} is equivalent to the category { \atop 𝔐on}^{op}, where objects of { \atop 𝔐on} are monoids and morphisms are conjugation classes of monoid homomorphisms.

Locally affine objects are defined in a straightforward way, once a notion of a cover (a quasi-pretopology) is fixed. We introduce several canonical quasi-pretopologies on the category |Cat|^{o}. Their common feature is the following: if a set of morphisms to X is a cover, then the set of their inverse image functors is conservative and all inverse image functors are exact in a certain mild way. If, in addition, morphisms of covers are continuous, X has a finite affine cover, and the category C_{S} has finite limits, this requirement suffices to recover the object X from the covering data uniquely up to isomorphism (i.e. the category C_{X} is recovered uniquely up to equivalence) via ’flat descent’.

Semiseparated covers and ’spaces’. We call a cover ’semiseparated’ if it is finite and consists of affine morphisms. Semiseparated covers form a quasi-pretopology on |Cat|^{o}. We call a ’space’ X ’semiseparated’ if it has a semiseparated cover by ’affine spaces’ (i.e. images of objects of Aff_{S}). If the base ’space’ S is Sp(R) for some associative unital ring R (that is C_{S} coincides with the category R − mod of left R-modules), then semiseparated ’spaces’ is the answer to the question ’how much of geometry linear algebra can incorporate?’. Explicitly, if X is semiseparated over Sp(R), then there exists a ring T and a coalgebra H in the category of R-bimodules such that the category C_{X} is equivalent to the category of H-comodules, i.e. left R-modules with a coaction of H. This fact is a consequence of flat descent which is one of the topics and tools of this work.

Recall that a scheme X is called semiseparated if it has an affine cover {U_{i} ↪ X | i ∈ J} such that each morphism U_{i} ↪ X is affine. Any separated scheme is semiseparated; in particular, all classical varieties are semiseparated schemes. One can show that a space X (geometric or functorial) is a semiseparated scheme iff its image in |Cat|^{o} (determined by pseudo-functor which assigns to a space X the category Qcoh_{X} of quasi-coherent sheaves on X) is a semiseparated scheme over Spℤ in the meaning above.

Locally affine morphisms. Even if we consider only semiseparated spaces and schemes over ℤ (i.e. S = Spℤ), the geometry we obtain via the functor Aff_{S} ⟶ |Cat|^{o} is unusual: our locally affine spaces do not have, in general, a global sections functor. Global sections functor appeares if instead of |Cat|^{o}, we take the full subcategory |Cat|_{S}^{o} of the category |Cat|^{o}/S of S-’spaces’ whose objects are pairs (X, f), where f is a continuous morphism X ⟶ S. The choice of the category of local objects is the same as above, i.e. Aff_{S}, and the functor from Aff_{S} to |Cat|_{S}^{o} is the inclusion. It follows from our results that this inclusion is a full (hence a fully faithful) functor. Quasi-pretopologies we introduced on |Cat|^{o} induce quasi-pretopologies on |Cat|_{S}^{o}. Applying general formalism, we define locally affine ’spaces’; using flat descent (in the case of covers formed by continuous morphisms), we obtain their description via local data.

We apply flat descent consideration to describing morphisms between locally affine ’spaces’ in terms of given affine covers of these ’spaces’. A demand to obtain such description immerged in our work [KR1]. Notice that due to the fact that flat covers do not form a pretopology (the base change invariance fails), the standard way to describe morphisms via covers is not available.

Besides of these conceptual facts, the work contains several other things. For instance, we give a useful description of continuous morphisms to the categoric spectrum of a ring. We introduce and study (in ’Complementary facts’) noncommutative quasi-affine ’spaces’ (spectra of non-unital associative rings) illustrated by some examples coming from representation theory.

The paper is organized as follows.

In Section 1, we introduce first notions of categoric geometry (’spaces’ represented by categories, morphisms represented by their inverse image functors, continuous, flat and affine morphisms) and sketch several examples of noncommutative spaces which are among illustrations and motivations of constructions of this work.

In Section 2, we start to view the category Cat^{op} opposite to the category of categories (which belong to a universum 𝔘) as the bicategory of ’spaces’: objects are regarded as ’spaces’, the corresponding categories as categories of quasi-coherent sheaves, and functors as inverse image functors of morphisms they define. We introduce the category of spaces, |Cat|^{o}, by identifying isomorphic inverse image functors. We observe that the natural functor Cat^{op} ⟶ |Cat|^{o} (identical on objects) is a universal object in the category of fibered categories of certain type.

In Section 3, we introduce (or ruther recall the definition of) different classes of morphisms, starting with continuous, flat, and affine morphisms. Their meaning is illustrated by examples. In particular, we consider a functor from the category Aff of affine (noncommutative) schemes defined as the category opposite to the category Rings of associative rings to the category |Cat|^{o} of ’spaces’ which assigns to each ring R its categoric spectrum, Sp(R) (corresponding to the category of left R-modules) and to each ring morphism an affine morphism between the respective categoric spectra.

In Section 4, we describe continuous morphisms from an arbitrary ”space” X to the categoric spectrum, Sp(R), of a ring R. We argue that continuous morphisms X ⟶ Sp(R) are in one-to-one correspondence with isomorphism classes of right R-modules O in the category C_{X} of quasi-coherent modules on X (i.e. R-modules in the opposite category C_{X}^{op}). In the case X = Sp(T) for some ring T, this correspondence expresses the classical fact (a theorem by Eilenberg and Moore) that inverse image functors of continuous morphisms Sp(T) ⟶ Sp(R) are given by (T, R)-bimodules.

In Section 5, we start to study continuous morphisms via monads and comonads associated with them, using as a main tool the Beck’s theorem characterizing the so called monadic and comonadic morphisms. For a monad F on a ”space” X (i.e. on the category C_{X}), we define the spectrum of F as the ”space” Sp(F/X) corresponding to the category of F-modules. The spectrum of a monad is a natural generalization of the categoric spectrum of a ring. Dually, for any comonad G on X, we define its cospectrum, Sp^{o}(X\G) as the space corresponding to the category G − Comod of G-comodules.

In Section 6, we exploit the fact that an affine morphism to X is isomorphic to the canonical morphism Sp(F/X) ⟶ X for a continuous monad F = (F, μ). Here ’continuous’ means that the functor F has a right adjoint. A consequence of this is that any affine morphism Y ⟶ Sp(R) is equivalent to the morphism Sp(T) ⟶ Sp(R) corresponding to a ring morphism R ⟶ T. In particular, a direct image functor of any affine morphism Sp(S) ⟶ Sp(R) is a composition of a Morita equivalence and the ”restriction of scalars” (pull-back) functor corresponding to a ring morphism.

In Section 7, we study affine flat descent. If U ⟶^{f} X is a flat conservative affine morphism (’conservative’ means that f^{∗} reflects isomorphisms), then it follows from Beck’s theorem that X is isomorphic to Sp^{o}(U\G_{f}), where G_{f} = (G_{f}, δ_{f}) is a continuous comonad. ’Continuous’ means that the functor G_{f} has a right adjoint. In the case U = Sp(R) for some ring R, continuous comonads are given by coalgebras in the category of Rbimodules. The main commutative example is an arbitrary semiseparated quasi-compact scheme. Recall that a scheme X is semiseparated iff it has an affine cover {U_{i} ↪ X | i ∈ J} such that all embeddings U_{i} ↪ X are affine morphisms.

Section 8 is dedicated to finiteness conditions (locally finitely presentable morphisms and objects) and smooth and étale morphisms. We start with a simple general formalism and then specialize it in the case of ’spaces’.

In Section 9, we introduce a general formalism of glueing locally affine objects and apply it to define locally affine spaces and schemes in the category |Cat|^{o} and in the categories |Cat|_{S}^{o} of S-’spaces’. Here, we define locally affine spaces and schemes in full generality. Note that noncommutative locally affine spaces and schemes defined in earlier works on the subject [R1], [KR] belong to the class of semiseparated spaces which does not include already most of quasi-separated quasi-compact commutative schemes.

Section 10 is dedicated to the flat descent of morphisms.

The content of the second part of the paper, Complementary facts, reflects its title. In Section C1, we give a description of continuous morphisms to the categoric spectrum

of a monoid which is analogous to the description of continuous morphisms to the categoric spectrum of a ring obtained in Section 4.

In Section C2, we discuss compatibility of monads and continuous morphisms with localizations. The facts obtained here are used in the next 3 sections.

Section C3 complements and clarifies Example 1.6. Here we introduce and study the cone of a non-unital monad and, as a special case, the cone of a non-unital associative ring; the latter is regarded as a noncommutative quasi-affine ’space’. Motivated by important constructions of representation theory of classical and quantum groups and enveloping algebras, we consider Hopf actions on non-unital rings and induced actions on the corresponding quasi-affine ’spaces’.

In Section C4, we construct the Proj of graded non-unital monad. As an application, we recover the Proj of a graded associative ring introduced in Example 1.7.

In Section C5, we resume the study of Hopf actions started in C3. Applying general facts to the action of enveloping algebra of a semisimple (or reductive) Lie algebra we realize the category of D-modules on the base affine space and the flag variety as categories of quasi-coherent sheaves on resp. the cone and the Proj of a graded ring naturally associated with the Lie algebra.

This setting is extended to actions of the quantized enveloping algebra of a semisimple Lie algebra on the quantum base affine ’space’ and the quantum flag variety. Altogeher is a complement to 1.10.

In Section C6 (which is a complement ot Sections 7 and 9), we study affine flat covers. In Section C7 (complementing Section 10), we study descent of flat morphisms. In Sections C8 and C9, we study connection between flat descent and the descent

procedure using relations.

Section C10 gathers miscellaneous facts, some of them suggestive.

We introduce connections and interpret comonads as integrable connections. We introduce weakly quasi-affine morphisms and relatively ample morphisms. We discuss continuous, flat, and affine morphisms in the fibered category of modules

on affine (noncommutative) schemes.

Finally, we complement duality of Section 6 (based on the connection between continuous monads and affine morphisms) by producing conditions which garantee the existence of a canonical functor which assigns to each (additive, or arbitrary) monad a continuous monad.

A part of this paper was written while the second author was visiting Max-Planck Institute für Mathematik in Bonn during the summer of 2001 and provided topics for several talks given at the seminar on noncommutative geometry at MPI. Special thanks to Yu. I. Manin for useful conversations, for proposing these lectures, and for being a stimulating listener. It is a pleasure to acknowledge excellent working conditions provided by the Institute.

The work of the second author on this paper was partially supported by the NSF grant DMS-0070921.

1. ’Spaces’ and categories. Examples.

1.1. Categories and ’spaces’. As usual, Cat, or Cat_{𝔘}, denotes the bicategory of categories which belong to a fixed universum 𝔘. We call objects of Cat^{op} ’spaces’. For any ’space’ X, the corresponding category C_{X} is regarded as the category of quasi-coherent sheaves on X. We denote by f^{∗} the functor C_{Y} ⟶ C_{X} corresponding to a 1-morphism X ⟶^{f} Y and call it the inverse image functor of f. For any 𝔘-category A, we denote by |A| the corresponding object of Cat^{op} (the underlying ’space’) defined by C_{|A|} = A.

We denote by |Cat|^{o} the category having same objects as Cat^{op}. Morphisms from X to Y are isomorphism classes of functors C_{Y} ⟶ C_{X}. For a morphism X ⟶^{f} Y, we denote by f^{∗} any functor C_{Y} ⟶ C_{X} representing f and call it an inverse image functor of the morphism f. We shall write f = [F] to indicate that f is a morphism having an inverse image functor F. The composition of morphisms X ⟶^{f} Y and Y ⟶^{g} Z is defined by g ∘ f = [f^{∗} ∘ g^{∗}]. Thus, the map which assigns to each functor C_{Y} ⟶^{F} C_{X} the morphism

X ⟶^{[F]} Y is a functor Cat^{op} ⟶ |Cat|^{o} which turns Cat^{op} into a fibred category over |Cat|^{o}.

1.2. Localizations and conservative morphisms. Let Y be an object of |Cat|^{o} and Σ a class of arrows of the category C_{Y}. We denote by Σ^{−1}Y the object of |Cat|^{o} such that the corresponding category coincides with (the standard realization of) the quotient of the category C_{Y} by Σ (cf. [GZ, 1.1]): C_{Σ^{−1}Y} = Σ^{−1}C_{Y}. The canonical localization functor

C_{Y} ⟶^{p_{Σ}^{∗}} Σ^{−1}C_{Y} is regarded as an inverse image functor of a morphism, Σ^{−1}Y ⟶^{p_{Σ}} Y.

For any morphism X ⟶^{f} Y in |Cat|^{o}, we denote by Σ_{f} the family of all arrows s of the category C_{Y} such that f^{∗}(s) is invertible (notice that Σ_{f} does not depend on the choice of an inverse image functor f^{∗}). Thanks to the universal property of localizations, f^{∗} is represented as the composition of the localization functor p_{f}^{∗} = p∗_{Σ_{f}} : C_{Y} ⟶ Σ_{f}^{−1}C_{Y}

and a uniquely determined functor Σ^{−1}C_{Y} ⟶^{f_{𝔠}^{∗}} C_{X}. In other words, f = p_{f} ∘ f_{𝔠} for a

uniquely determined morphism X ⟶^{f_{𝔠}} Σ_{f}^{−1}Y.

A morphism X ⟶^{f} Y is called conservative if Σ_{f} consists of isomorphisms only, or, equivalently, p_{f} is an isomorphism.

A morphism X ⟶^{f} Y is called a localization if f_{𝔠} is an isomorphism, i.e. the functor f_{𝔠}^{∗} is an equivalence of categories.

Thus, f = p_{f} ∘ f_{𝔠} is a unique decomposition of a morphism f into a localization and a conservative morphism.

1.3. Continuous, flat, and affine morphisms. A morphism is called continuous if its inverse image functor has a right adjoint (called a direct image functor), and flat if, in addition, the inverse image functor is left exact (i.e. preserves finite limits). A continuous morphism is called ’affine’ if its direct image functor is conservative (i.e. reflects isomorphisms) and has a right adjoint.

1.4. Categoric spectrum of a unital ring. For an associative unital ring R, we define the categoric spectrum of R as the object Sp(R) of |Cat|^{o} such that C_{Sp(R)} = R − mod.

Let R ⟶^{ϕ} S be a unital ring morphism and R−mod ⟶^{ϕ̄^{∗}} S −mod the functor S ⊗_{R} −. The canonical right adjoint to ϕ̄^{∗} is the pull-back functor by the ring morphism ϕ. A right adjoint to ϕ_{∗} is given by

ϕ^{!} : S − mod ⟶ R − mod, L ⟼ Hom_{R}(ϕ_{∗}(S), L).

The map

(R ⟶^{ϕ} S) ⟼ (Sp(S) ⟶^{ϕ̄} Sp(R))

is a functor

Sp : Rings^{op} ⟶ |Cat|^{o}

which takes values in the subcategory formed by affine morphisms.

The image Sp(R) ⟶^{ϕ̄} Sp(T) of a ring morphism T ⟶^{ϕ} R is flat (resp. faithful) iff ϕ turns R into a flat (resp. faithful) right T -module.

1.4.1. Continuous, flat, and affine morphisms from Sp(S) to Sp(R). Let R and S be associative unital rings. A morphism f : Sp(S) ⟶ Sp(R) with an inverse image functor f^{∗} is continuous iff

f^{∗} ≃ M⊗_{R} : L ⟼ M ⊗_{R} L  (1)

for an (S, R)-bimodule M defined uniquely up to isomorphism. The functor

f_{∗} = Hom_{S}(M, −) : N ⟼ Hom_{S}(M, N)  (2)

is a direct image of f.

The morphism f with an inverse image functor (1) is conservative iff M is faithful as a right R-module, i.e. the functor M ⊗_{R} − is faithful.

The direct image functor (2) is conservative iff M is a cogenerator in the category of left S-modules, i.e. for any nonzero S-module N, there exists a nonzero S-module morphism M ⟶ N.

The morphism f is flat iff M is flat as a right R-module.

The functor (2) has a right adjoint, f^{!}, iff f_{∗} is isomorphic to the tensoring (over S) by a bimodule. This happens iff M is a projective S-module of finite type. The latter is equivalent to the condition: the natural functor morphism M_{S}^{∗} ⊗_{S} − ⟶ Hom_{S}(M, −) is an isomorphism. Here M_{S}^{∗} = Hom_{S}(M, S). In this case, f^{!} ≃ Hom_{R}(M_{S}^{∗}, −).

1.5. The graded version. Let G be a monoid and R a G-graded unital ring. We define the ’space’ Sp_{G}(R) by taking as C_{Sp_{G}(R)} the category gr_{G}R − mod of left G-graded

R-modules. There is a natural functor gr_{G}R − mod ⟶^{ϕ_{∗}} R_{0} − mod which assigns to each graded R-module its zero component (’zero’ is the unit element of the monoid G). The functor ϕ_{∗} has a left adjoint, ϕ^{∗}, which maps every R_{0}-module M to the graded R-module R ⊗_{R_{0}} M. The adjunction arrow Id_{R_{0}−mod} ⟶ ϕ_{∗}ϕ^{∗} is an isomorphism. This means that the functor ϕ^{∗} is fully faithful, or, equivalently, the functor ϕ_{∗} is a localization.

The functors ϕ_{∗} and ϕ^{∗} are regarded as respectively a direct and an inverse image functor of a morphism Sp_{G}(R) ⟶^{ϕ} Sp(R_{0}). It follows from the above that the morphism ϕ is affine iff ϕ is an isomorphism (i.e. ϕ^{∗} is an equivalence of categories).

In fact, if ϕ is affine, the functor ϕ_{∗} should be conservative. Since ϕ_{∗} is a localization, this means, precisely, that ϕ_{∗} is an equivalence of categories.

1.6. The cone of a non-unital ring. Let R_{0} be a unital associative ring, and let R_{+} be an associative ring, non-unital in general, in the category of R_{0}-bimodules; i.e. R_{+} is endowed with an R_{0}-bimodule morphism R_{+} ⊗_{R_{0}} R_{+} ⟶^{m} R_{+} satisfying the associativity condition. Let R = R_{0} ⊕ R_{+} denote the augmented ring described by this data. Let T_{R_{+}} denote the full subcategory of the category R − mod whose objects are all R-modules annihilated by R_{+}. Let T_{R_{+}}^{−} be the Serre subcategory (that is a full subcategory closed by taking subquotients, extensions, and arbitrary direct sums) of the category R − mod spanned by T_{R_{+}}.

We define the ’space’ cone of R_{+} by taking as C_{Cone(R_{+})} the quotient category R −

mod/T_{R_{+}}^{−}. The localization functor R −mod ⟶^{u^{∗}} R −mod/T_{R_{+}}^{−} is an inverse image functor of a morphism of ’spaces’ Cone(R_{+}) ⟶^{u} Sp(R). The functor u^{∗} has a (necessarily fully faithful) right adjoint, i.e. the morphism u is continuous. If R_{+} is a unital ring, then u is an isomorphism (see C3.2.1). The composition of the morphism u with the canonical affine morphism Sp(R) ⟶ Sp(R_{0}) is a continuous morphism Cone(R_{+}) ⟶ Sp(R_{0}). Its direct image functor is (regarded as) the global sections functor.

1.7. The graded version: Proj_{G}. Let G be a monoid and R = R_{0} ⊕R_{+} a G-graded ring with zero component R_{0}. Then we have the category gr_{G}R − mod of G-graded Rmodules and its full subcategory gr_{G}T_{R_{+}} = T_{R_{+}} ∩ gr_{G}R − mod whose objects are graded modules annihilated by the ideal R_{+}. We define the ’space’ Proj_{G}(R) by setting

C_{Proj_{G}(R)} = gr_{G}R − mod/gr_{G}T_{R_{+}}^{−}.

Here gr_{G}T_{R_{+}}^{−} is the Serre subcategory of the category gr_{G}R−modspanned by gr_{G}T_{R_{+}}. One can show that gr_{G}T_{R_{+}}^{−} = gr_{G}R − mod ∩ T_{R_{+}}^{−}. Therefore, we have a canonical projection

Cone(R_{+}) ⟶^{𝔭} Proj_{G}(R).

The localization functor gr_{G}R − mod ⟶ C_{Proj_{G}(R_{+})} is an inverse image functor of a continuous morphism Proj_{G}(R) ⟶^{𝔳} Sp_{G}(R). The composition Proj_{G}(R) ⟶^{𝔳} Sp(R_{0}) of

the morphism 𝔳 with the canonical morphism Sp_{G}(R) ⟶^{ϕ} Sp(R_{0}) defines Proj_{G}(R) as a ’space’ over Sp(R_{0}). Its direct image functor is called the global sections functor.

1.7.1. Example: cone and Proj of a ℤ_{+}-graded ring. Let R = ⊕_{n≥0}R_{n} be a ℤ_{+}- graded ring, R_{+} = ⊕_{n≥1}R_{n} its ’irrelevant’ ideal. Thus, we have the cone of R_{+}, Cone(R_{+}), and Proj(R) = Proj_{ℤ}(R), and a canonical morphism Cone(R_{+}) ⟶ Proj(R).

1.8. Example: skew cones and skew projective ’spaces’. Let A be an arbitrary associative k-algebra. And let q denote a matrix [q_{ij}]_{i,j∈J} with entrees in k such that q_{ij}q_{ji} = 1 for all i, j ∈ J and q_{ii} = 1 for all i ∈ J. Let R = A_{q}[x] denote a skew polynomial algebra corresponding to this data. Here x = (x_{i} | i ∈ J) is a set of indeterminates satisfying the relations

x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i, j ∈ J,  (1)
x_{i}r = rx_{i} for all i ∈ J and r ∈ R  (2)

For any i ∈ J, set S_{i} = {x_{i}^{n}|n ≥ 1}. Each of S_{i} is a left and right Ore set, and the Serre subcategory T_{R_{+}}^{−} is generated by R-modules whose elements are annihilated by some elements of ⋃_{i∈J} S_{i}. This implies that the localizations R − mod ⟶ S_{i}^{−1}R − mod factor through the localization R − mod ⟶ C_{Cone(R_{+})}, and the induced

localizations C_{Cone(R_{+})} ⟶^{u_{i}^{∗}} S_{i}^{−1}R − mod form a conservative family. The conservative family {Sp(S_{i}^{−1}R) ⟶ Cone(R_{+}) | i ∈ J} is regarded as an affine cover of the cone Cone(R_{+}). It follows that the algebra S_{i}^{−1}R is isomorphic to A_{q}[x, x_{i}^{−1}].

Let G = ℤ^{J}; and let γ_{i}, i ∈ J, denote the canonical generators of the group G. Assigning to each x_{i} the parity γ_{i}, we turn the skew polynomial algebra R = A_{q}[x] into a G-graded algebra with R_{0} = A. The localizations R − mod ⟶ S_{i}^{−1}R − mod induce localizations gr_{G}R − mod ⟶ gr_{G}S_{i}^{−1}R − mod which factor through the localization

C_{Proj(R)} ⟶^{𝔳_{i}^{∗}} gr_{G}S_{i}^{−1}R − mod = gr_{G}A_{q}[x, x_{i}^{−1}] − mod.  (3)

Let G_{i} denote the quotient group G/ℤγ_{i}. The category gr_{G}A_{q}[x, x_{i}^{−1}] − mod in (3) is naturally equivalent to the category gr_{G_{i}}A_{q_{i}}[x/x_{i}] − mod of left G_{i}-graded modules over the skew polynomial algebra A_{q_{i}}[x/x_{i}]. Here x/x_{i} denotes {x_{j}/x_{i}|j ∈ J, j ≠ i}, and q_{i} denotes the matrix [q_{ni}q_{nm}q_{mi}^{−1}]_{n,m∈J−{i}} (cf. [R, I.7.2.2.4]). Note that A_{q_{i}}[x/x_{i}] is the G_{i}-component of the algebra A_{q}[x, x_{i}^{−1}] of the ’functions’ on Cone(R)/|𝕊′_{i}|.

Let ’spaces’ U_{i} are defined by C_{U_{i}} = gr_{G_{i}}A_{q_{i}}[x/x_{i}] − mod. Note that if the cardinality of J is greater than one, then the natural morphisms U_{i} ⟶^{u_{i}} Proj_{G}(R) do not form an affine cover of Proj_{G}(R) over Sp(A), because the composition of 𝔳_{i∗} with the direct image of the projection Proj_{G}(R) ⟶^{π} Sp(A) is isomorphic to the functor gr_{G_{i}}A_{q_{i}}[x/x_{i}] − mod ⟶ A − mod which assigns to each G_{i}-graded module (resp. G_{i}-graded module morphism) its zero component. If the group G_{i} is non-trivial, this functor is not faithful, hence the morphism π ∘ u_{i} is not affine.

1.8.1. The projective q-’space’ P_{q}^{r}. Let again R = A_{q}[x], x = (x_{0}, x_{1}, …, x_{r}). But, we take G = ℤ with the natural order; and set the parity of each x_{i} equal to 1. One can repeat with Cone_{ℤ}(R) and P_{q}^{r} = Proj_{ℤ}(R) the same pattern as with Cone(R) and P_{G}^{r} = Proj_{G}(R). Only this time the quotient groups G_{i} will be trivial, and we obtain a picture very similar to the classical one: P_{q}^{r} is a ℤ-scheme covered by r + 1 affine spaces A_{q_{i}}[x/x_{i}] − mod, i = 0, 1, …, r.

1.9. The base affine ’space’ and the flag variety of a reductive Lie algebra. Let 𝔤 be a reductive Lie algebra over ℂ and U(𝔤) the enveloping algebra of 𝔤. Let G be the group of integral weights of 𝔤 and G_{+} the semigroup of nonnegative integral weights. Let R = ⊕_{λ∈G_{+}}R_{λ}, where R_{λ} is the vector space of the (canonical) irreducible finite dimensional representation with the highest weight λ. The module R is a G-graded algebra with the multiplication determined by the projections R_{λ} ⊗R_{ν} ⟶ R_{λ+ν}, for all λ, ν ∈ G_{+}. It is well known that the algebra R is isomorphic to the algebra of regular functions on the base affine space of 𝔤. Recall that G/U, where G is a connected simply connected algebraic group with the Lie algebra 𝔤, and U is its maximal unipotent subgroup.

The category C_{Cone(R)} is equivalent to the category of quasi-coherent sheaves on the base affine space Y of the Lie algebra 𝔤. The category P roj_{G}(R) is equivalent to the category of quasi-coherent sheaves on the flag variety of 𝔤.

1.10. The quantized base affine ’space’ and quantized flag variety of a semisimple Lie algebra. Let now 𝔤 be a semisimple Lie algebra over a field k of zero characteristic, and let U_{q}(𝔤) be the quantized enveloping algebra of 𝔤. Define the G-graded algebra R = ⊕_{λ∈G_{+}}R_{λ} the same way as above. This time, however, the algebra R is not commutative. Following the classical example (and identifying spaces with categories of quasi-coherent sheaves on them), we call Cone(R) the quantum base affine ’space’ and Proj_{G}(R) the quantum flag variety of 𝔤.

1.10.1. Canonical affine covers of the base affine ’space’ and the flag variety. Let W be the Weyl group of the Lie algebra 𝔤. Fix a w ∈ W. For any λ ∈ G_{+}, choose a nonzero w-extremal vector e_{wλ}^{λ} generating the one dimensional vector subspace of R_{λ} formed by the vectors of the weight wλ. Set S_{w} = {k^{∗}e_{wλ}^{λ}|λ ∈ G_{+}}. It follows from the Weyl character formula that e_{wλ}^{λ}e_{wμ}^{μ} ∈ k^{∗}e_{w(λ+μ)}^{λ+μ}. Hence S_{w} is a multiplicative set. It was proved by Joseph [Jo] that S_{w} is a left and right Ore subset in R. The Ore sets {S_{w}|w ∈ W } determine a conservative family of affine localizations

Sp(S_{w}^{−1}R) ⟶ Cone(R), w ∈ W,  (4)

of the quantum base affine ’space’ and a conservative family of affine localizations

Sp_{G}(S_{w}^{−1}R) ⟶ Proj_{G}(R), w ∈ W,

of the quantum flag variety. We claim that the category gr_{G}S_{w}^{−1}R − mod is naturally equivalent to (S_{w}^{−1}R)_{0} − mod. By 1.5, it suffices to verify that the canonical functor gr_{G}S_{w}^{−1}R − mod ⟶ S_{w}^{−1}R)_{0} − mod which assigns to every graded S_{w}^{−1}R-module its zero component is faithful; i.e. the zero component of every nonzero G-graded S_{w}^{−1}R-module is nonzero. This is, really, the case, because if z is a nonzero element of λ-component of a G-graded S_{w}^{−1}R-module, then (e_{wλ}^{λ})^{−1}z is a nonzero element of the zero component of this module.

1.11. Noncommutative Grassmannians and projective spaces. Fix an associative unital k-algebra R. Let R\Alg_{k} be the category of associative k-algebras over R (i.e. pairs (S, R → S), where S is a k-algebra and R → S a k-algebra morphism). We call them for convenience R-rings. We denote by R^{e} the k-algebra R ⊗_{k} R^{o}. Here R^{o} is the algebra opposite to R.

1.11.1. The functor Gr_{M,V}. Let M, V be left R-modules. Consider the functor, Gr_{M,V} : R\Alg_{k} ⟶ Sets, which assigns to any R-ring (S, R →^{s} S) the set of isomorphism classes of epimorphisms s^{∗}(M) ⟶ s^{∗}(V) (here s^{∗}(M) = S ⊗_{R} M) and to any R-ring

morphism (S, R →^{s} S) ⟶^{ϕ} (T, R →^{t} T) the map Gr_{M,V} (S, s) ⟶ Gr_{M,V} (T, t) induced by

the inverse image functor S − mod ⟶^{ϕ^{∗}} T − mod, N ⟼ T ⊗_{S} N.

1.11.2. The functor G_{M,V}. Denote by G_{M,V} the functor R\Alg_{k} ⟶ Sets which assigns to any R-ring (S, R →^{s} S) the set of pairs of morphisms s^{∗}(V) →^{v} s^{∗}(M) →^{u} s^{∗}(V) such that u∘v = id_{s^{∗}(V)} and acts naturally on morphisms. Since V is a projective module, the map

π = π_{M,V} : G_{M,V} ⟶ Gr_{M,V} , (v, u) ⟼ [u],  (1)

is a (strict) functor epimorphism.

1.11.3. Relations. Denote by ℜ_{M,V} the ”functor of relations” G_{M,V} ×_{Gr_{M,V}} G_{M,V}.

By definition, ℜ_{M,V} is a subfunctor of G_{M,V} × G_{M,V} which assigns to each R-ring, (S, R →^{s} S), the set of all 4-tuples (u_{1}, v_{1}; u_{2}, v_{2}) ∈ G_{M,V} × G_{M,V} such that the epimorphisms u_{1}, u_{2} are equivalent. The latter means that there exists an isomorphism s^{∗}(V) ⟶^{φ} s^{∗}(V) such that u_{2} = φ ∘ u_{1}, or, equivalently, φ^{−1} ∘ u_{2} = u_{1}. Since u_{i} ∘ v_{i} = id, i = 1, 2, these equalities imply that φ = u_{2} ∘ v_{1} and φ^{−1} = u_{1} ∘ v_{2}. Thus, ℜ_{M,V} (S, s) is a subset of all (u_{1}, v_{1}; u_{2}, v_{2}) ∈ G_{M,V} (S, s) × G_{M,V} (S, s) satisfying the following relations:

u_{2} = (u_{2} ∘ v_{1}) ∘ u_{1}, u_{1} = (u_{1} ∘ v_{2}) ∘ u_{2}  (2)

in addition to the relations describing G_{M,V} (S, s) × G_{M,V} (S, s):

u_{1} ∘ v_{1} = id_{S⊗_{R}V} = u_{2} ∘ v_{2}  (3)

Denote by p_{1}, p_{2} the canonical projections ℜ_{M,V} \overrightarrow{⟶} G_{M,V}. It follows from the surjectivity of G_{M,V} ⟶ Gr_{M,V} that the diagram

ℜ_{M,V} \overrightarrow{⟶}^{p_{1}}_{p_{2}} G_{M,V} ⟶^{π} Gr_{M,V}  (4)

is exact.

1.11.4. Proposition. If both M and V are projective modules of a finite type, then the functors G_{M,V} and ℜ_{M,V} are corepresentable.

Proof. See [KR2, 10.4.3].

1.11.5. Quasi-coherent presheaves on Gr_{M,V}. Suppose that M and V are projective modules of a finite type, hence the functors G_{M,V} and ℜ_{M,V} are corepresentable by R-rings resp. (𝔊_{M,V} , R → 𝔊_{M,V}) and (R_{M,V} , R → R_{M,V}). Then the category Qcoh(G_{M,V}) (resp. Qcoh(ℜ_{M,V})) is equivalent to 𝔊_{M,V} − mod (resp. R_{M,V} − mod), and the category Qcoh(Gr_{M,V}) of quasi-coherent presheaves on Gr_{M,V} is equivalent to the kernel of the diagram

Qcoh(G_{M,V}) \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} Qcoh(ℜ_{M,V})  (5)

This means that, after identifying categories of quasi-coherent presheaves in (5) with corresponding categories of modules, quasi-coherent presheaves on Gr_{M,V} can be realized as pairs (L, ϕ), where L is a 𝔊_{M,V} -module and ϕ is an isomorphism p_{1}^{∗}(L) ⟶^{∼} p_{2}^{∗}(L). Morphisms (L, ϕ) ⟶ (N, ψ) are given by morphisms L ⟶^{g} N such that the diagram

p_{1}^{∗}(L) ⟶^{p_{1}^{∗}(g)} p_{1}^{∗}(N)
ϕ ↓≀  ≀↓ ψ
p_{2}^{∗}(L) ⟶^{p_{2}^{∗}(g)} p_{2}^{∗}(N)

commutes. The functor

Qcoh(Gr_{M,V}) ⟶^{π^{∗}} Qcoh(G_{M,V}), (L, ϕ) ⟼ L,

is an inverse image functor of the projection G_{M,V} ⟶^{π} Gr_{M,V} (see 1.11.3(4)).

1.11.6. Noncommutative Grassmannians. Let 𝔗 be a quasi-topology on the category (R\Alg_{k})^{op} of affine k-schemes over R. Let Gr𝔗_{M,V} be the 𝔗-Grassmannian corresponding to Gr_{M,V}, i.e. a sheaf of sets (a ’space’) associated to the presheaf Gr_{M,V}. By [KR3, 2.9.1], if 𝔗 is coarser than the quasi-topology of effective descent, then the category Qcoh(Gr_{M,V}) of quasi-coherent modules on Gr_{M,V} is naturally equivalent to the category Qcoh(Gr𝔗_{M,V}) of quasi-coherent modules on the 𝔗-Grassmannian Gr𝔗_{M,V}.

1.11.7. Noncommutative projective space. Let M be the free R-module of the rank n+1, V the free R-module of the rank 1. In this case, we denote the functor Gr_{M,V} by Pn_{R}. If a quasi-topology 𝔗 on the category (R\Alg_{k})^{op} of affine k-schemes over R is coarser than the quasi-topology of effective descent, than the category Qcoh(Pn_{R}) of quasi-coherent modules on Pn_{R} is equivalent to the category of quasi-coherent modules on the (associated) projective space ℙn_{R} = Pn^{𝔗}_{R}.

2. Fibered categories and ’spaces’. Quasi-coherent modules.

2.1. Initial objects of |Cat|^{o}. The category • with one (identical) morphism (in particular with one object) is an initial object of |Cat|^{o}. A morphism f : A ⟶ B in |Cat|^{o} with an inverse image functor f^{∗} is an isomorphism iff f^{∗} is a category equivalence. In particular, X ∈ Ob|Cat|^{o} is an initial object of |Cat|^{o} iff the category C_{X} is a connected groupoid; i.e. HomC_{X} consists of isomorphisms and for any x, y ∈ ObC_{X}, the set Hom_{C_{X}} (x, y) is non-empty.

Notice that for any object X of |Cat|^{o}, the set |Cat|^{o}(X, •) of morphisms X → • is isomorphic to the set |X| of isomorphism classes of objects of the category C_{X}.

The category |Cat|^{o} has no ”real” final objects: the empty category is its unique final object.

2.2. Proposition. The category |Cat|^{o} has small limits and colimits.

Proof. (a) Let {X_{i} | i ∈ J} be a set of objects of |Cat|^{o}. Then ∏_{i∈J} X_{i} and ∐_{i∈J} X_{i} are defined by

C_{∏_{i∈J} X_{i}} = ∐_{i∈J} C_{X_{i}} and C_{∐_{i∈J} X_{i}} = ∏_{i∈J} C_{X_{i}}.

(b) Every pair of arrows, X \overrightarrow{⟶}^{f}_{g} Y, in |Cat|^{o} has a cokernel.

Let f^{∗}, g^{∗} : C_{Y} \overrightarrow{⟶} C_{X} be inverse image functors of resp. f and g. Let C_{Z} denote the category whose objects are pairs (x, ϕ), where x ∈ ObC_{Y} and ϕ is an isomorphism f^{∗}(x) →^{∼} g^{∗}(x). A morphism from (x, ϕ) to (y, ψ) is a morphism ξ : x → y such that the diagram

f^{∗}(x) ⟶^{f^{∗}(ξ)} f^{∗}(y)
ϕ ↓  ↓ ψ
f^{∗}(x) ⟶^{f^{∗}(ξ)} f^{∗}(y)

commutes. Denote by 𝔥^{∗} the forgetful functor C_{Z} ⟶ C_{Y} , (x, ϕ) ⟼ x. Let w : Y ⟶ W be a morphism in |Cat|^{o} with an inverse image functor w^{∗} such that w ∘ f = w ∘ g. This means that there exists an isomorphism f^{∗} ∘ w^{∗} ⟶^{ψ} g^{∗} ∘ w^{∗}. The pair (w^{∗}, ψ) defines a functor γ_{w^{∗},ψ}^{∗} : C_{W} ⟶ C_{Z}, b ⟼ (w^{∗}(b), ψ(b)). A different choice, w_{1}^{∗}, of the inverse image functor of w and an isomorphism ψ_{1} : w_{1}^{∗} ∘ f^{∗} ⟶^{∼} w_{1}^{∗} ∘ g^{∗} produces a functor γ_{w_{1}^{∗},ψ_{1}}^{∗} isomorphic to γ_{w^{∗},ψ}^{∗}. This shows that the morphism Y ⟶ Z having the inverse image 𝔥^{∗} is the cokernel of the pair (f, g). The existence of cokernels and (small) coproducts is equivalent to the existence of arbitrary (small) colimits.

(c) Every pair of arrows, X \overrightarrow{⟶}^{f}_{g} Y, in |Cat|^{o} has a kernel.

Let f^{∗}, g^{∗} : C_{Y} \overrightarrow{⟶} C_{X} be inverse image functors of resp. f and g. Denote by 𝔇_{f^{∗},g^{∗}} the diagram scheme defined as follows:

Ob𝔇_{f^{∗},g^{∗}} = ObC_{Y} ∐ ObC_{X} and Hom𝔇_{f^{∗},g^{∗}} = HomC_{X} ∐ Σ_{f^{∗},g^{∗}},

where

Σ_{f^{∗},g^{∗}} = {f^{∗}(x) →^{s_{x}} x, x →^{t_{x}} g^{∗}(x) | x ∈ ObC_{Y} }.

Consider the category Pa𝔇_{f^{∗},g^{∗}} of paths of the diagram 𝔇_{f^{∗},g^{∗}} together with the natural embeddings HomC_{X} ⟶^{τ} HomPa𝔇_{f^{∗},g^{∗}} ⟵ Σ_{f^{∗},g^{∗}} which define the corresponding diagrams. We denote by P𝔇_{f^{∗},g^{∗}} the quotient of the category Pa𝔇_{f^{∗},g^{∗}} by the minimal equivalence relation such that

τ(α ∘ β) ∼ τ(α) ∘ τ(β) and τ(id_{x}) ∼ id_{τ(x)}

for all composable arrows α, β and for all x ∈ ObC_{X}.

Finally, we denote by C_{W} the quotient category Σ_{f^{∗},g^{∗}}^{−1}P𝔇_{f^{∗},g^{∗}}. It follows from the construction that the object W of the category |Cat|^{o} defined this way is the kernel of the pair (f, g). Details are left to the reader.

2.3. Two fibered categories associated with Cat.

2.3.1. The fibered category (Cat^{o}, π) over Cat^{op}. Consider the category Cat^{o} whose objects are pairs (X, M), where X ∈ ObCat^{op} and M ∈ ObC_{X}. Morphisms from (X, M) to (Y, L) are pairs (f, ξ), where f is a morphism X ⟶ Y and ξ is a morphism f^{∗}(L) ⟶ M. The pair (Cat^{o}, π), where π : Cat^{o} ⟶ Cat^{op} is a functor which assigns to each object (X, M) of Cat^{o} the object X and to every morphism (f, ξ) : (X, M) ⟶ (Y, L) the corresponding morphism X ⟶^{f} Y in Cat^{op} is a fibered category with the fiber at X ∈ ObCat^{op} equal to C_{X}^{op}. This fibered category corresponds to the identical functor Cat = (Cat^{op})^{op} ⟶ Cat.

2.3.2. The fibered category ℭat_{𝔘}^{o}. Let |π| denote the composition of the functor π : Cat^{o}_{𝔘} ⟶ Cat_{𝔘}^{op} and the canonical functor Cat_{𝔘}^{op} ⟶ |Cat|_{𝔘}^{o}. The pair (Cat_{𝔘}^{o}, |π|) is a fibered category which we denote by ℭat_{𝔘}^{o}.

2.4. The universal property of the fibered category ℭat_{𝔘}^{o}. Fix two universums, 𝔘 and 𝔙 such that 𝔘 ∈ 𝔙. Denote by F ib_{𝔘,𝔙} the 2-category of fibered categories 𝔉 → E such that the ’base’ E belongs to 𝔙 and all fibers, 𝔉_{x}, x ∈ ObE, belong to 𝔘.

Let 𝔐F ib_{𝔘,𝔙} denote the 2-subcategory of the 2-category F ib_{𝔘,𝔙} formed by cartesian functors (1-morphisms) which induce category equivalence of fibers.

2.4.1. Proposition. (a) For any fibered category 𝔉 which belongs to F ib_{𝔘,𝔙}, there exists a cartesian functor ψ_{𝔉} : 𝔉 ⟶ ℭat_{𝔘}^{o} such that for all cartesian functors F : 𝔉 ⟶ 𝔊 which belong to 𝔐F ib_{𝔘,𝔙}, the diagram

𝔉 ⟶^{F} 𝔊

ψ_{𝔉} ↘ ↙ ψ_{𝔊} ℭat_{𝔘}^{o}

commutes in the 2-category sense.

(b) The union of images of the functors ψ_{𝔉} is ℭat_{𝔘}^{o}.

(c) In particular, the fibered category ℭat_{𝔘}^{o} is a final object of 𝔐F ib_{𝔘,𝔙} iff |Cat|_{𝔘}^{o} belongs to 𝔙.

Proof. Fix a pseudo-functor β : |Cat|_{𝔘} = (|Cat|_{𝔘}^{o})^{op} ⟶ Cat_{𝔘} quasi-inverse to the projection Cat^{o}_{𝔘} ⟶^{π} |Cat|_{𝔘}^{o}; i.e. β assigns to each object X of |Cat| the category C_{X}^{op} and

to any morphism X ⟶^{f} Y the functor f^{∗}^{op} : C_{Y}^{op} ⟶ C_{X}^{op} opposite to a (chosen) inverse image functor of f. This pseudo-functor determines a fibered category ℭat_{β} equivalent to ℭat_{𝔘}^{o}. Recall that objects of ℭat_{β} are pairs (X, x), where X ∈ ObB, x ∈ ObC_{X}, and morphisms (X, x) ⟶ (Y, y) are pairs (f, ξ), where f is a morphism X ⟶ Y and ξ is a morphism x ⟶ f^{∗}(y). Projection p_{β} : F_{β} ⟶ B maps (X, x) to X and (f, ξ) to f.

For any fibered category 𝔉 = (F ⟶^{p} B), there exists a natural functor Ψ_{𝔉} : B ⟶ |Cat|^{o} which sends any object X of B to the class, |F_{X}^{op}|, of the fiber F_{X} over X and any

morphism X ⟶^{f} Y to the morphism [f^{∗}^{op}] : |F_{X}^{op}| ⟶ |F_{Y}^{op}| of |Cat|^{o}. The composition of Ψ_{𝔉}^{op} : B^{op} ⟶ |Cat|_{𝔘} with the pseudo-functor β : |Cat|_{𝔘} ⟶ Cat_{𝔘} is a pseudo-functor, β_{𝔉}, quasi-inverse to p : F ⟶ B. The fibered category 𝔉 is equivalent to the fibered category 𝔉_{β_{𝔉}} = (F_{β_{𝔉}} ⟶^{p_{β_{𝔉}}} B). It follows from the construction that the functor Ψ_{𝔉} : B ⟶ |Cat|^{o} defined above gives rise to a functor Ψ̄_{𝔉} : F_{β_{𝔉}} ⟶ Cat_{𝔘} such that the pair (Ψ̄_{𝔉}, Ψ_{𝔉}) is a cartesian functor from F_{β} ⟶^{p_{β}} B to ℭat_{𝔘}^{o}. Taking the composition with the equivalence 𝔉 ⟶ 𝔉_{β}, we obtain a cartesian functor ψ_{𝔉} : 𝔉 ⟶ ℭat_{𝔘}.

Let 𝔊 = (G ⟶^{q} D) be another fibered category. It follows from the construction of functors ψ_{𝔉} that for any cartesian functor F : 𝔉 ⟶ 𝔊, the diagram

𝔉 ⟶^{F} 𝔊

ψ_{𝔉} ↘ ↙ ψ_{𝔊}

Cat_{2}

which commutes in the 2-category sense.

2.5. The universal property of the fibered category (Cat^{o}, π). As in 2.4, fix universums 𝔘 and 𝔙 such that 𝔘 ∈ 𝔙. Recall that a fibered category over E is split (scindée) if it corresponds to a functor E^{op} ⟶ Cat called splitting (scindage). Consider the category Ssin_{𝔘,𝔙} formed by split fibered categories F ⇄ E such that E belongs to the universum 𝔙 with all fibers from the universum 𝔘). Morphisms are functors preserving splittings as 1-morphisms. The split category Cat_{𝔘} is a final object of the 2-category Scin_{𝔘,𝔙} the same sense as Cat_{2,𝔘} is a final object of the 2-category F ib_{𝔘,𝔙} (see 2.4.1).

2.6. Quasi-coherent modules. The material of this subsection is borrowed from [KR3]. We refer to [KR3] for more detail and proofs.

2.6.1. Modules and quasi-coherent modules on a category over a category. Let E be a category (which belongs to some universum 𝔘) and 𝔉 = (F, F →^{π} E) a category over E. Denote by Mod(𝔉) the category opposite to the category of all sections of 𝔉. We shall call objects of Mod(𝔉) modules on 𝔉.

We denote by Qcoh(𝔉) the category opposite to the category Cart_{E}(E, 𝔉) of cartesian sections of 𝔉. In other words, Qcoh(𝔉) = (Lim𝔉)^{op} (cf. [KR3], A3.5.5). Objects of Qcoh(𝔉) will be called quasi-coherent modules on 𝔉.

Any morphism 𝔉 ⟶ 𝔊 of E-categories induces a functor Mod(𝔉) ⟶ Mod(𝔊). Thus, we have a functor Mod : Cat/E ⟶ Cat from the category of E-categories to the category of categories.

Similarly, the map 𝔉 ⟼ Qcoh(𝔉) extends to a functor Qcoh : Cart_{E} ⟶ Cat from the category of cartesian functors over E to Cat.

Let 𝔉 be a fibered category corresponding to a pseudo-functor E^{op} ⟶ Cat,

ObE ∋ X ⟼ F_{X}, HomE ∋ f ⟼ f^{∗}, HomE ×_{ObE} HomE ∋ (f, g) ⟼ c_{f,g} (1)

(cf. [KR3], A3.7, A3.7.1). Then the category Mod(𝔉) can be described as follows. An object of Mod(𝔉) is a function which assigns to each T ∈ ObE an object M(T) of the fiber F_{T} and to each morphism f : T ⟶ T′ a morphism ξ_{f} : f^{∗}(M(T′)) ⟶ M(T) such that ξ_{gf} ∘ c_{f,g} = ξ_{f} ∘ f^{∗}(ξ_{g}). Morphisms are defined in a natural way.

An object (M, ξ) of Mod(𝔉) belongs to the subcategory Qcoh(𝔉) iff ξ_{f} is an isomorphism for all f ∈ HomE.

2.6.2. The pseudo-functor Qcoh. Let 𝔘, 𝔙 be two universums such that 𝔘 ∈ 𝔙. Denote by Cart_{𝔘,𝔙} the 2-category whose objects are categories over categories 𝔉 = (F →^{π} E) such that the base E belongs to 𝔙 and each fiber belongs to 𝔘. Denote by 𝔐Cart_{𝔘,𝔙} all cartesian functors (1-morphisms of Cart_{𝔘,𝔙})

\begin{matrix}F′ & ⟶^{u} & F \\ π′↓ & & ↓ π \\ E′ & ⟶^{v} & E \end{matrix}  (1)

such that the functors induced on fibers are category equivalences. It follows that 𝔐Cart_{𝔘,𝔙} is a 2-subcategory of the 2-category Cart_{𝔘,𝔙} (and the 2-category F ib_{𝔘,𝔙} introduced in 2.4 is a full 2-subcategory of 𝔐Cart_{𝔘,𝔙}).

2.6.2.1. Proposition. The map 𝔉 ⟼ Qcoh(𝔉) extends to a pseudo-functor

Qcoh : 𝔐Cart_{𝔘,𝔙}^{op} ⟶ Cat_{𝔙}.

Proof. See [KR3, 11.1.4.2].

The pseudo-functor Qcoh gives rise to a functor 𝔖𝔭 : 𝔐Cart_{𝔘,𝔙} ⟶ |Cat|_{𝔙}^{o}. Thus by definition C_{𝔖𝔭(𝔉)} = Qcoh(𝔉) for any category 𝔉 over E. We call 𝔖𝔭(𝔉) the categoric spectrum of 𝔉.

2.6.3. Quasi-coherent modules on presheaves of sets. Let X be a presheaf of sets on the base E, i.e. a functor E^{op} ⟶ Sets. Then we have a functor E/X ⟶ E and the category 𝔉/X = 𝔉 ×_{E} E/X over E/X obtained via a base change (as usual, we identify E with a full subcategory of the category E^{∧} of presheaves of sets on E formed by representable presheaves). Notice that any morphism of the category E/X over E is cartesian. Therefore, by [KR3, 11.1.4.1], the category Qcoh(𝔉/X) is equivalent to the category Cart_{E}(E/X, 𝔉)^{op} opposite to the category of cartesian functors E/X ⟶ 𝔉.

3. Certain classes of morphisms in |Cat|^{o} and Cat^{op}.

3.1. Left exact, right exact, and exact morphisms. A morphism X ⟶^{f} Y is called right exact (resp. left exact, resp. exact), if its inverse image functor preserves colimits (resp. limits, resp. both limits and colimits) of arbitrary finite diagrams.

The following assertion is a reformulation of Proposition 1.1.4 in [GZ].

3.1.1. Proposition. Let f = p_{f} ∘ f_{𝔠} be the canonical decomposition of a morphism

X ⟶^{f} Y into a conservative morphism X ⟶^{f_{𝔠}} Σ_{f}^{−1}Y and a localization Σ_{f}^{−1}Y ⟶^{p_{f}} Y. Suppose C_{Y} has finite limits (resp. finite colimits). Then f is left exact (resp. right exact) iff the class of arrows Σ_{f} satisfies left (resp. right) Ore conditions. In this case both the localization p_{f} and the conservative morphism f_{𝔠} are left (resp. right) exact.

In particular, if the category C_{Y} has limits and colimits of finite diagrams, then f is exact iff both the localization p_{f} and the conservative component f_{𝔠} are exact. The exactness of p_{f} is equivalent to that Σ_{f} satisfies left and right Ore conditions.

3.2. Dualization functor and dual notions. The dualization functor

o : |Cat|^{o} ⟶ |Cat|^{o}

assigns to each object Y of |Cat|^{o}the object Y^{o} defined by C_{Y^{o}} = C_{Y}^{op}, and to each morphism f with an inverse image functor f^{∗}, the morphism f^{o} having (f^{∗})^{op} as an inverse image functor. It follows that the dialization functors is an automorphism of the category |Cat|^{o} and its square is the identical functor.

The dualization functor maps left (resp. right) exact morphisms to right (resp. left) exact morphisms. Conservative morphisms and localizations are stable under the dualization. In particular, the dualization functor preserves the canonical decomposition: f^{o} = p_{f}^{o} ∘ f_{𝔠}^{o} and p_{f}^{o} = p_{f^{o}}, f_{𝔠}^{o} = (f^{o})_{𝔠}.

3.3. Continuous and cocontinuous morphisms. Duality. A morphism X ⟶^{f} Y in |Cat|^{o} is called cocontinuous if f^{o} is continuous. In other words, f is cocontinuous iff its inverse image functor, f^{∗}, has a left adjoint, f_{!}.

Denote by |Cat|_{𝔠}^{o} the subcategory of |Cat|^{o} formed by continuous morphisms and by |Cat|_{𝔠𝔬𝔠}^{o} the subcategory of |Cat|^{o} formed by cocontinuous morphisms. The dualization functor induces isomorphisms of categories |Cat|_{𝔠}^{o} ⟶^{∼} |Cat|_{𝔠𝔬𝔠}^{o} ⟶^{∼} |Cat|_{𝔠}^{o}.

The map which assigns to every continuous morphism X ⟶^{f} Y with a direct image

functor f_{∗} the morphism X ⟶^{f^{∧}} Y having f_{∗} as an inverse image functor, defines an isomorphism of categories

∧ : |Cat|_{𝔠} = (|Cat|_{𝔠}^{o})^{op} ⟶^{∼} |Cat|_{𝔠𝔬𝔠}^{o}.

The inverse isomorphism,

∨ : |Cat|_{𝔠𝔬𝔠} = (|Cat|_{𝔠𝔬𝔠}^{o})^{op} ⟶^{∼} |Cat|_{𝔠}^{o},

assigns to any cocontinuous morphism X ⟶^{g} Y with an inverse image functor g^{∗} the

morphism Y ⟶^{g^{∨}} X having as an inverse image functor a left adjoint, g_{!}, to g^{∗}.

The composition

𝔇^{∧} : |Cat|_{𝔠} ⟶^{∼} |Cat|_{𝔠}^{o},  (1)

of the functor ^{∧} with the dualization functor ^{o} is a duality on |Cat|_{𝔠}^{o}, i.e. a contravariant functor whose square is the identical functor.

Similarly, the composition

𝔇^{∨} : |Cat|_{𝔠𝔬𝔠} ⟶^{∼} |Cat|_{𝔠𝔬𝔠}^{o},  (2)

of the functor ^{∨} with the dualization functor ^{o} is a duality on |Cat|_{𝔠𝔬𝔠}^{o}.

3.4. Flat and coflat morphisms. We call a morphism X ⟶^{f} Y flat if it is continuous and its inverse image functor is exact. Since f^{∗} preserves colimits of arbitrary small diagrams, the exactness requirement means that f^{∗} preserves limits of finite diagrams, i.e. f is left exact (see (ii) above).

By 3.1.1, a continuous morphism f is flat iff both the localization p_{f} and the conservative morphism f_{𝔠} in the decomposition f = p_{f} ∘ f_{𝔠} are exact. One can show that both p_{f} and f_{𝔠} are continuous, hence flat (see [R3]).

3.4.1. Coflat morphisms. A continuous morphism f is called coflat if the dual morphism 𝔇^{∧}(f) is flat; i.e. f is coflat iff its direct image, f_{∗}, is exact, or, equivalently, f_{∗} preserves colimits of finite diagrams.

3.4.2. Weakly flat and coflat morphisms. A pair of arrows g_{1}, g_{2} : M ⟶ L is called coreflexive, if there exists a morphism h : L ⟶ M such that h ∘ g_{1} = id_{M} = h ∘ g_{2}.

Dually, a pair of arrows g_{1}, g_{2} : M ⟶ L is called reflexive, if there exists a morphism h : L ⟶ M such that g_{1} ∘ h = id_{M} = g_{2} ∘ h.

We call a continuous morphism X ⟶^{f} Y in |Cat|^{o} weakly flat if its inverse image functor preserves kernels of coreflexive pairs of arrows.

Dually, f is called weakly coflat if its direct image functor preserves cokernels of reflexive pairs of arrows.

3.5. Affine morphisms. A continuous morphism X ⟶^{f} Y is called affine if its direct image functor f_{∗} has a right adjoint, f^{!}, and is conservative.

Applying the duality 𝔇^{∧}, one can identify an affine morphism X ⟶^{f} Y with a conservative morphism Y ⟶ X with an inverse image functor f_{∗} : X ⟶ Y which is both continuous and cocontinuous.

3.5.1. Coaffine morphisms. We call a continuous morphism X ⟶^{f} Y coaffine iff the dual morphism 𝔇^{∧}(f) is affine. In other words, f is coaffine iff its inverse image functor f^{∗} is conservative and has a left adjoint, f_{!}.

3.6. Classes of morphisms of Cat^{op}. Given a class 𝔐 of morphisms of |Cat|^{o}, we say that a 1-morphism in Cat^{op} belongs to 𝔐 if its image by the canonical functor Cat^{op} ⟶ |Cat|^{o} belongs to 𝔐. Thus we have continuous, flat, affine etc. 1-morphisms of the category Cat^{op}.

3.7. Example: canonical morphisms of the category |Cat|^{o}. Let • be the initial object of the category |Cat|^{o} such that C_{•} is the category with one (hence identical) morphism. For any object X of |Cat|^{o}, denote by f_{X} the unique morphism • ⟶ X.

3.7.1. Lemma. The unique inverse image functor f ∗_{X} : C_{X} ⟶ C_{•} has a right adjoint, f_{_{X}∗}, (i.e. f_{X} is continuous) iff the category C_{X} has a final object. In this case, a direct image functor f_{_{X}∗} maps the unique object of the category C_{•} to a final object of the category C_{X}.

Dually, the functor f ∗_{X} has a left adjoint, f_{_{X}!} (i.e. f_{X} is cocontinuous), iff the category C_{X} has an initial object.

Proof is left to the reader.

3.7.2. Futher observations. All functors from C_{•}, in particular f_{_{X}∗}, are conservative. Thus f_{X} : • ⟶ X is affine iff there exists a direct image functor f_{_{X}∗} (i.e. iff the category C_{X} has a final object), and the functor f_{_{X}∗} has a right adjoint, f_{X}∗^{!}.

Since there is only one functor C_{X} ⟶ C_{•}, the functor f_{_{X}∗} has a right adjoint iff it is left adjoint to the functor f_{X}^{∗}. In this case, f_{_{X}∗} is isomorphic to f_{_{X}!}, hence it maps the unique object of C_{•} to an initial object of C_{X}.

Thus, the unique morphism • ⟶ X is affine iff the category C_{X} has a zero object, i.e. final objects in C_{X} are initial too.

If the category C_{X} has an initial object, then there is a continuous morphism ϕ_{X} : X ⟶ • whose inverse image functor maps the unique object of • to an initial object of C_{X}. The direct image functor of ϕ_{X} is the unique functor ϕ_{∗} : C_{X} ⟶ C_{•} (which coincides with the functor f_{X}^{∗} above).

The functor ϕ_{_{X}∗} has a right adjoint, ϕ_{X}^{!}, iff the category C_{X} is marked. Notice, however, that the morphism ϕ_{X} is affine (that is the functor ϕ_{_{X}∗} is conservative) iff all arrows of C_{X} are invertible, i.e. C_{X} is a connected groupoid, or, what is the same, ϕ_{X}^{∗} is a category equivalence, i.e. X ≃ •.

3.8. Example: cocontinuous and coaffine morphisms between spectra of rings. Let R and S be associative unital rings. A morphism Sp(S) ⟶^{f} Sp(R) is cocontinuous iff its inverse image functor is isomorphic to

Hom_{R}(N , −) : R − mod ⟶ S − mod, L ⟼ Hom_{R}(N , L).  (3)

for some (R, S)-bimodule N.

The functor (1) has a left adjoint, f_{!}, iff it is isomorphic to the functor (3) for some (R, S)-bimodule N; or, equivalently, the canonical functorial morphism

M ⊗_{R} L ⟶ Hom_{R}(M_{R}^{∗}, L)  (4)

is an isomorphism for all L (equivalently, (4) is an isomorphism for L = M_{R}^{∗}). Here M_{R}^{∗} = Hom_{R}(M, R). This happens iff M is a projective R-module of a finite type.

Thus, affine morphisms Sp(R) ⟶ Sp(S) are in bijective correspondence with isomorphism classes of (S, R)-bimodules M which are strictly projective as left S-modules. Recall that strictly projective means projective cogenerator of finite type.

Coaffine morphisms Sp(R) ⟶ Sp(S) are in bijective correspondence with isomorphism classes of (S, R)-bimodules M which are strictly projective as right R-modules.

4. Continuous morphisms to the categoric spectrum of a ring and ’structure sheaves’. Let R be an associative unital ring. For a morphism X ⟶^{f} Sp(R) with an inverse image functor f^{∗}, we denote by O the object f^{∗}(R). It follows that the object O is determined by the pair (X, f) uniquely up to isomorphism. The functor f^{∗} defines a monoid morphism End_{R}(R) ⟶ C_{X}(O, O) the whose composition with the canonical ring isomorphism R^{o} ⟶^{∼} End_{R}(R) gives a monoid morphism ϕ_{f} : R ⟶ C_{X}(O, O)^{o}. Here C_{X}(O, O)^{o} denotes the monoid opposite to C_{X}(O, O). If the category C_{X} is preadditive and the functor f^{∗} is additive, the morphism ϕ_{f} is a unital ring morphism.

In general, the object O does not determine the morphism f. It does, however, if f is continuous:

4.1. Proposition. Let X ⟶^{f} Sp(R) be a continuous morphism. Then

(a) The morphism f is determined by O = f^{∗}(R) uniquely up to isomorphism.

(b) There exists a coproduct of any small set of copies of O.

(c) The object O has a structure of an R-module in the category C_{X}^{op}. In particular, O is an abelian cogroup in the category C_{X} (i.e. an abelian group in C_{X}^{op}) and the canonical map ϕ_{f} : R ⟶ C_{X}(O, O)^{o} is a ring morphism.

Proof. (a) Let f_{∗} be a direct image functor of f (i.e. a right adjoint to f^{∗}). we have functorial isomorphisms C_{X}(f^{∗}(R), M) ≃ Hom_{R}(R, f_{∗}(M)) ≃ f_{∗}(M) which shows that the direct image functor f_{∗} of f is naturally isomorphic to the functor M ⟼ C_{X}(f^{∗}(R), M). Therefore the inverse image functor f^{∗} of f is defined uniqely up to isomorphism (being a left adjoint to f_{∗}) by the object f^{∗}(R). Since f^{∗} preserves colimits, there exists a coproduct of any set of copies of O(X, f) = f^{∗}(R).

(b) Since f^{∗} preserves colimits, there exists a coproduct of any set of copies of O(X, f) = f^{∗}(R).

(c) The assertion follows from the isomorphism f_{∗} ≃ C_{X}(O, −) and the fact that f_{∗} takes values in the category of R-modules.

4.1.1. Global sections functor. The object O is viewed as the ’structure sheaf’ on X. We denote C_{X}(O, O)^{o} by Γ_{X}O. The functor

f_{_{O}∗} : C_{X} ⟶ Γ_{X}O − mod, M ⟼ C_{X}(O, M)

will be called the ’global sections functor’ on (X, O). In particularly, Γ_{X}O = f_{_{O}∗}(O) is the ring of global sections of the structure sheaf O.

4.2. Right R-modules. We call R-modules in C_{X}^{op} right R-modules in C_{X}, or right R-modules on X. Right R-modules on X form a category which we denote by R^{o} − Mod_{X}. A morphism from an R^{o}-module (M, ϕ) to an R^{o}-module (M′, ϕ′) is a morphism h : M ⟶ M′ of (abelian) groups in the category C_{X} such that the diagram

\begin{matrix}C_{X}(M, M) & ⟶^{C_{X}(M,h)} & C_{X}(M, M′) \\ ϕ ↑ & & ↑ C_{X}(h, M) \\ R & ⟶^{ϕ′} & C_{X}(M′, M′) \end{matrix}  (1)

commutes. The composition is defined in an obvious way.

4.3. Proposition. For every continuous morphism X ⟶^{f} Y, its inverse image functor induces a functor R^{o} − Mod_{Y} ⟶ R^{o} − Mod_{X}.

Proof. In fact, the functorial isomorphism C_{X}(f^{∗}(M), T) ≃ C_{Y} (M, f_{∗}(T)) implies that if M has a structure of an abelian group in the category C_{Y}^{op} = C_{Y^{o}} (i.e. the functor C_{Y} (M, −) : C_{Y}^{op} ⟶ Sets, has a lifting to a functor C_{Y}^{op} ⟶ ℤ − mod), then f^{∗}(M) has a structure of an abelian group in the category C_{X}^{op} = C_{X^{o}}. In particular, for any abelian group M in C_{Y}^{op} and for any object T in C_{Y}, the morphism f_{M,T}^{∗} : C_{Y} (M, T) ⟶ C_{X}(f^{∗}(M), f^{∗}(T)) ≃ C_{Y} (M, f_{∗}f^{∗}(T)) is an abelian group morphism. It follows that f_{M,M}^{∗} : C_{Y} (M, M) ⟶ C_{X}(f^{∗}(M), f^{∗}(M)) is a ring morphism. Thus if R^{o} ⟶ C_{Y} (M, M) is a ring morphism (i.e. a right R-module structure on M), its composition with f_{M,M}^{∗} is a right R-module structure on f^{∗}(M). This extends to a functor R^{o}−Mod_{Y} ⟶ R^{o}−Mod_{X}, hence the assertion.

Thus, for any associative ring R, we have a pseudo-functor

β_{R} : |Cat|_{𝔠} = (|Cat|_{𝔠}^{o})^{op} ⟶ Cat

which defines the fibered category (R^{o} − Mod ⟶ |Cat|_{𝔠}^{o}). Here |Cat|_{𝔠}^{o} is the subcategory of the category |Cat|^{o} formed by the continuous morphisms.

4.4. Right R-modules and continuous morphisms to Sp(R): functorial picture. We denote by |Cat|^{o}(R) the category of right R-modules. Its objects are triples (X, O_{X}, ϕ), where X is an object of |Cat|^{o}, and (O_{X}, ϕ) is a right R-module in the category C_{X} (cf. 4.2). Morphisms from (X, O_{X}, ϕ) to (Y, O_{Y} , ψ) are morphisms X ⟶^{f} Y such that there exists an isomorphism λ : f^{∗}(O_{Y}) ⟶^{∼} O_{X} making the diagram

\begin{matrix}R^{o} & ⟶^{ϕ} & C_{X}(O_{X}, O_{X}) \\ ψ ↓ & & ↓ 𝔠_{λ} \\ C_{Y} (O_{Y} , O_{Y}) & ⟶^{f_{O_{Y} ,O_{Y}}^{∗}} & C_{X}(f^{∗}(O_{Y}), f^{∗}(O_{Y})) \end{matrix}  (2)

commute. Here 𝔠_{λ} denotes the conjugation by λ, h ⟼ λ^{−1} ∘ h ∘ λ.

Let Cat^{op}(R) denote the category whose objects are same as objects of |Cat|^{o}(R). Morphisms from (X, O_{X}, ϕ) to (Y, O_{Y} , ψ) are given by pairs (f^{∗}, λ), where f^{∗} is a functor C_{Y} ⟶ C_{X}, λ an isomorphism f^{∗}(O_{Y}) ⟶^{∼} O_{X} such that the diagram (2) commutes. Composition is defined in an obvious way. The canonical functor Cat^{op}(R) ⟶ |Cat|^{o}(R) is a fibered category.

4.4.1. Note. Morphisms of the categories Cat^{op}(R) and |Cat|^{o}(R) (more precisely, the meaning of the component λ of a morphism in Cat^{op}(R), see above) might be viewed as follows. For any object X of |Cat|^{o}, let Cat^{op}(R)_{X} denote the fiber of the category Cat^{op}(R) over X; its objects are triples (X, O_{X}, ϕ), where ϕ is a right action of the ring R on O_{X}, morphisms are morphisms of actions. An inverse image functor, f^{∗}, of a morphism X ⟶^{f} Y induces a functor

f_{R}^{∗} : Cat^{op}(R)_{Y} ⟶ Cat^{op}(R)_{X}  (3)

which maps an object (Y, O_{Y} , ψ) to the object (X, f^{∗}(O_{Y}), ψ_{f^{∗}}), where the action ψ_{f^{∗}} is the composition of ψ : R^{o} ⟶ C_{Y} (O_{Y} , O_{Y}) and the ring morphism

f_{O_{Y} ,O_{Y}}^{∗} : C_{Y} (O_{Y} , O_{Y}) ⟶ C_{X}(f^{∗}(O_{Y}), f^{∗}(O_{Y})).

The commutativity of the diagram (2) means exactly that λ : f^{∗}(O_{Y}) ⟶ O_{X} is an isomorphism f_{R}^{∗}(Y, O_{Y} , ψ) ⟶ (X, O_{X}, ϕ).

We denote by |Cat|^{o}(R)_{𝔠} the full subcategory of the category |Cat|^{o}(R) whose objects are triples (X, O_{X}, ϕ) such that the following conditions hold:

(a) (O_{X}, ϕ) is a right R-module in the category C_{X}, i.e. O_{X} is an abelian group in C_{X}^{op} and ϕ a ring morphism R^{o} ⟶ C_{X}(O_{X}, O_{X}) (cf. 4.1(c) and 4.2).

(b) There exists a coproduct of any set of copies of O_{X}.

(c) Let Φ_{ϕ} denote the functor from the subcategory 𝔏_{R} of free R-modules to C_{X} which is uniquely defined by the action ϕ (thanks to (b) above). The image by Φ_{ϕ} of any pair of arrows X_{1} ⇉ X_{0} has a cokernel.

Let Cat^{op}(R)_{𝔠} denote the preimage of |Cat|^{o}(R)_{𝔠} in Cat^{op}(R). On the other hand, let (|Cat|^{o}/Sp(R))_{𝔠} denote the full subcategory of the category |Cat|^{o}/Sp(R) whose objects are continuous morphisms to Sp(R), and let (Cat^{op}/Sp(R))_{𝔠} denote its preimage in Cat^{op}/Sp(R).

4.4.2. Proposition. The functor (3) induces an equivalence of the fibered categories

( (Cat^{op}/Sp(R))_{𝔠} ; ↓ ; (|Cat|^{o}/Sp(R))_{𝔠} ) ⟶ ( (Cat^{op}(R)_{𝔠} ; ↓ ; |Cat|^{o}(R)_{𝔠} ) .  (4)

Proof. By 4.1, the cartesian morphism (3) induces a cartesian morphism (4).

Thanks to 4.1, in order to prove that (4) is an equivalence, it suffices to show that the functor f^{∗} can be reconstructed (uniquely up to isomorphism) from the right R-module

(O, R →^{ϕ_{f}} C_{X}(O, O)^{o}) associated with f. In fact, the right R-module (O, ϕ_{f}) gives raise to a functor, Φ_{f}, from the category 𝔏_{R} of free left R-modules (a full subcategory of R − mod) to C_{X} which is isomorphic to the restriction of f^{∗} to 𝔏_{R}: direct sums of copies of R are maped to direct sums of copies of O and ϕ_{f} (regarded as End_{R}(R) ⟶ C_{X}(O, O)) determines map on morphisms. Every R-module M is a cokernel of a pair of morphisms L_{1} ⇉ L_{0}, where L_{1}, L_{0} are free R-modules. Since the functor f^{∗} preserves cokernels of pairs of arrows, f^{∗}(M) is isomorphic to the cokernel of the pair Φ_{f}(L_{1}) ⇉ Φ_{f}(L_{0}).

4.5. ℤ-’spaces’. Let X ⟶^{f} Sp(R) be a continuous morphism with an inverse image functor f^{∗}, and let O = f^{∗}(R). It follows from 4.1(c) that the functor f_{∗} = C_{X}(O, −) is naturally decomposed into

\begin{matrix}C_{X} ⟶^{f_{_{O} ∗}} Γ_{X}O − mod \\ f_{∗} ↘  ↙ ϕ̄_{_{f} ∗} \\ R − mod \end{matrix}  (5)

where ϕ̄_{_{f} ∗} is the pull-back by the ring morphism ϕ_{f} : R ⟶ Γ_{X}O defining a right R-module structure on O.

4.5.1. Lemma. The global sections functor f_{_{O}∗} is a direct image functor of a continuous morphism, f_{O} : X ⟶ Sp(Γ_{X}O), iff any pair of arrows O^{⊕I} \overrightarrow{⟶} O^{⊕J} between coproducts of copies of O has a cokernel in C_{X}.

Proof. The fact follows from (the argument of) 4.4.2 applied to the case when R = Γ_{X}O: the inverse image functor f ∗_{O} assigns to a free Γ_{X}O-module Γ_{X}O^{⊕J} the coproduct O^{⊕J} of J copies of the object O.

4.5.2. The category of ℤ-spaces. Denote by |Cat|o_{ℤ} the category whose objects are all pairs (X, O), where X is an object of |Cat|^{o} and O is an abelian group in C_{X}^{op} such that there exist coproducts of small sets of copies of O and any pair of arrows O^{⊕I} \overrightarrow{⟶} O^{⊕J} between coproducts of copies of O has a cokernel in C_{X}. Morphisms from (X, O) to (X′, O′) are morphisms X ⟶^{f} X′ such that there exists an isomorphism f^{∗}(O′) ⟶^{∼} O. Composition is defined in an obvious way. Objects of the category |Cat|o_{ℤ} will be called ℤ-’spaces’.

4.5.2.1. A reformulation. By 4.5.1, ℤ-spaces are pairs (X, O) such that O is an abelian group in the category C_{X}^{op} and the canonical functor

f_{_{O}∗} : C_{X} ⟶ Γ_{X}O − mod, M ⟼ C_{X}(O, M),  (6)

has a left adjoint; or, equivalently, f_{_{O}∗} is a direct image functor of a continuous morphism.

4.5.2.2. Example. If C_{X} is an additive category with small coproducts and cokernels, then (X, O) is a ℤ-space for any O ∈ ObC_{X}.

4.5.3. Affine ℤ-spaces. We call a ℤ-space (X, O) affine if the canonical morphism f_{O} : X ⟶ Sp(Γ_{X}O) is an isomorphism; i.e. the functor f_{_{O}∗} (see (6)) is a category equivalence. By a Mitchel’s theorem, affine ℤ-spaces are pairs (X, O), where C_{X} is an abelian category with small coproducts, and O is a projective cogenerator of finite type. We denote by Aff_{ℤ} the full subcategory of the category |Cat|o_{ℤ} formed by affine ℤ-spaces.

The functor Sp : Rings^{op} ⟶ |Cat|^{o}, R ⟼ Sp(R), gives rise to the functor

Sp_{ℤ} : Rings^{op} ⟶ |Cat|o_{ℤ}, R ⟼ (Sp(R), R)

which takes values in the subcategory Aff_{ℤ}. We denote the image of the functor Sp_{ℤ} by 𝔄𝔣𝔣_{ℤ}. Thus, objects of the category 𝔄𝔣𝔣_{ℤ} are pairs (Sp(R), R) and morphisms from (Sp(R), R) ⟶ (Sp(T), T) are morphisms Sp(R) ⟶ Sp(T) corresponding to unital ring morphisms T ⟶ R. The functor Sp_{ℤ} induces an inclusion functor γ_{∗} : 𝔄𝔣𝔣_{ℤ} ⟶ |Cat|o_{ℤ} which takes values in the subcategory of affine ℤ-spaces.

4.5.4. Proposition. The functor γ_{∗} : 𝔄𝔣𝔣_{ℤ} ⟶ |Cat|o_{ℤ} is fully faithful and has a left adjoint. In particular, the functor γ_{∗} induces an equivalence of 𝔄𝔣𝔣_{ℤ} and the category of affine ℤ-spaces.

Proof. Let f be a morphism (X, O) ⟶ (X′, O′). A choice of an inverse image functor, f^{∗}, of f and an isomorphism λ : f^{∗}(O′) ⟶^{∼} O determines a ring morphism ψ_{f^{∗},λ} : Γ_{X′}O′ ⟶ Γ_{X}O. One can check that the corresponding morphism of categoric spectra, Sp(Γ_{X}O) ⟶ Sp(Γ_{X′}O′), does not depend on choices. Thus we have a functor γ^{∗} : |Cat|o_{ℤ} ⟶ 𝔄𝔣𝔣_{ℤ}. By 4.5.1, we have a natural morphism η_{γ} : Id_{|Cat|_{ℤ}^{o}} ⟶ γ_{∗}γ^{∗}. And there is an isomorphism ϵ_{γ} : γ^{∗}γ_{∗} ⟶^{∼} Id_{𝔄𝔣𝔣_{ℤ}}. These are adjunction morphisms. Since ϵ_{γ} is an isomorphism, the functor γ_{∗} is fully faithful.

4.5.5. A description of the category 𝔄𝔣𝔣_{ℤ}. Let 𝔄ss denote the category whose objects are associative rings; and morphisms from a ring R to a ring S are equivalence classes of ring morphisms R ⟶ S by the following equivalence relation: two ring morphisms f, g : R ⟶ S are equivalent if they are conjugated, i.e. g(−) = tf(−)t^{−1} for an invertible element t of S.

4.5.5.1. Proposition. Two ring morphisms, R \overrightarrow{⟶}^{ϕ}_{ψ} S, are conjugate iff the corre-

sponding inverse image functors, R − mod \overrightarrow{⟶}^{ϕ^{∗}}_{ψ^{∗}} S − mod, are isomorphic.

Proof. (a) Suppose that ψ and ϕ are conjugate, i.e. there exists an invertible element, t, of S such that ψ(r) = tϕ(r)t^{−1} for all r ∈ R. For any R-module M = (M, m), we have a commutative diagram

\begin{matrix}S ⊗ M & ⟶^{·t} & S ⊗ M \\ γ_{ψ} ↓ & & ↓ γ_{ϕ} \\ S ⊗_{R,ψ} M & ⟶^{λ_{t}} & S ⊗_{R,ϕ} M \end{matrix}  (1)

Here ·t denotes the S-module morphism s ⊗ z ⟼ st ⊗ z for all s ∈ S, z ∈ M; γ_{ψ}, γ_{ϕ} are canonical epimorphisms.

In fact, for any s ∈ S, r ∈ R, z ∈ M, γ_{ψ}(sψ(r) ⊗ z) = γ_{ψ}(s ⊗ r · z), and ·t(s ⊗ r · z) = st ⊗ r · z.

On the other hand, ·t(sψ(r) ⊗ z) = sψ(r)t ⊗ z = stϕ(r) ⊗ z, and γ_{ϕ}(stϕ(r) ⊗ z) = γ_{ϕ}(st ⊗ r · z). Since γ_{ψ} is by definition the cokernel of two maps

S ⊗_{k} R ⊗_{k} M \overrightarrow{⟶}^{ψ_{l}}_{ψr} S ⊗_{k} M, s ⊗ r ⊗ z ⟼^{ψ_{l}} sψ(r) ⊗ z, and s ⊗ r ⊗ z ⟼^{ψ_{r}} s ⊗ r · z,

it follows the existence of a (necessarily unique) morphism λ_{t} : S ⊗_{R,ψ} M ⟶ S ⊗_{R,ϕ} M such that the diagram (1) commutes; i.e. λ_{t} is given by γ_{ψ}(s ⊗ z) ⟼ γ_{ϕ}(st ⊗ z).

(b) Conversely, suppose ϕ, ψ are unital ring morphisms such that there is a functorial isomorphism u : ψ^{∗} ⟶^{∼} ϕ^{∗}. Identifying both ϕ^{∗}(R) and ψ^{∗}(R) with the left S-module S, we obtain, in particular, an S-module morphism u(R) : S ⟶ S. Since S is a ring with unit, u(R) equals to ·t : s ⟼ st for some t ∈ S. Since u is a functor morphism, for any r ∈ R, u(R) ∘ ψ^{∗}(·r) = ϕ^{∗}(·r) ∘ u(R). This means that for any s ∈ S, sψ(r)t = stϕ(r), hence ψ(r) = tϕ(r)t^{−1}.

4.5.5.2. Corollary. The functor Sp_{ℤ} : Rings^{op} ⟶ |Cat|o_{ℤ}, R ⟼ (Sp(R), R) induces an isomorphism of categories 𝔄ss^{op} ⟶^{∼} 𝔄𝔣𝔣_{ℤ}.

4.5.5.3. Corollary. The functor Sp_{ℤ} : Rings^{op} ⟶ |Cat|o_{ℤ} induces an equivalence of categories 𝔄ss^{op} ⟶ Aff_{ℤ}.

Proof. This follows from 4.5.4 and 4.5.5.2.

4.5.6. Remark. If S is a commutative ring, then for any ring R, the surjection Rings(R, S) ⟶ 𝔄ss(R, S) is a bijective map. In particular, the full subcategory of 𝔄ss formed by commutative rings is isomorphic to the category CRings of commutative rings. Thus, the equivalence of categories 𝔄ss^{op} ⟶ Aff_{ℤ} induces an equivalence between the category CRings^{op} opposite to the category of commutative unital rings and the full subcategory CAff_{ℤ} formed by affine ℤ-spaces (X, O) such that the global sections ring Γ_{X}O is commutative. This shows by passing that the category CAff_{ℤ} of commutative affine ℤ-spaces is equivalent to the category of commutative affine schemes in the usual sense.

4.6. Proposition. Let X ∈ Ob|Cat|^{o} be such that the category C_{X} has cokernels of pairs of morphisms. Then (X, O) is a ℤ-space for any object O of C_{X} such that there exists a coproduct of any small set of copies of O. In particular, continuous morphisms X ⟶ Sp(R) are in one-to-one correspondence with isomorphism classes of right Rmodules (O, ϕ) in C_{X} such that (X, O) is a ℤ-space.

In particular, if C_{X} is an abelian category with small coproducts, then morphisms X ⟶ Sp(R) are in one-to-one correspondence with isomorphism classes of right Rmodules in C_{X}.

Proof. The assertion is a corollary of 4.5.4.

4.6.1. Example. Let X = Sp(S) for some associative ring S, i.e. C_{X} is the category S − mod of left S-modules. By 4.6, continuous morphisms from Sp(S) ⟶ Sp(R) are in one-to-one correspondence with isomorphism classes of right R-modules in the category S − mod. Notice that the category of right R-modules in S − mod is isomorphic to the category of (S, R)-bimodules. If O is a (S, R)-bimodule corresponding to a morphism f : Sp(S) ⟶ Sp(R), then L ⟼ Hom_{S}(O, L) is a direct image functor of f. Therefore N ⟼ O ⊗_{R} N is an inverse image functor of f. Thus we have recovered a classical fact already mentioned in 3.8.1.

4.7. Additive morphisms and continuous morphisms. For any objects X, Y of Cat^{op}, denote by Cat_{c}^{op}(X, Y) the full subcategory of the category Cat^{op}(X, Y) whose objects are continuous morphisms. If C_{X}, C_{Y} are (pre)additive categories, we call a morphism X ⟶^{f} Y additive if its inverse image functor is additive. We denote by Cat_{a}^{op}(X, Y) the full subcategory of the category Cat^{op}(X, Y) whose objects are additive morphisms. Continuous morphisms are additive, i.e. Cat_{c}^{op}(X, Y) is a (full) subcategory of Cat_{a}^{op}(X, Y).

4.7.1. Proposition. Let X be an object of Cat^{op} such that C_{X} is an abelian category with small coproducts. Then for any unital associative ring R, the inclusion functor Cat_{c}^{op}(X, Sp(R)) ⟶ Cat_{a}^{op}(X, Sp(R)) has a right adjoint.

Proof. Since C_{X} is an abelian category with small coproducts (or, what is the same, colimits of small diagrams), it follows from 4.5.1 that for any object O ∈ ObC_{X}, the functor

f_{_{O}∗} = C_{X}(O, −) : C_{X} ⟶ Γ_{X}O − mod  (6)

is a direct image functor of a continuous morphism f_{O} : X ⟶ Sp(Γ_{X}O). To every morphism X ⟶^{f} Sp(R), we assign the composition of the morphism f_{O}, where O = f^{∗}(R), and the morphism Sp(Γ_{X}O) ⟶ Sp(R) corresponding to the right R-module structure ϕ_{f} : R ⟶ Γ_{X}O. The assertion follows from 4.6. Details are left to the reader.

4.7.2. Corollary. Let R, S be associative unital rings. The functor

(S, R) − bimod ⟶ Cat^{op}(Sp(S), Sp(R))  (7)

which assigns to each (S, R)-bimodule M the morphism f_{M} with the inverse image functor M ⊗_{R} − : R − mod ⟶ S − mod has a right adjoint.

Proof. The assertion follows from 4.7.1 and the fact that the functor (7) establishes an equivalence between the category (S, R)-bimodules and the category of continuous morphisms from (Sp(S) to Sp(R)) (see 4.6.1).

4.8. Grothendieck categories and flat localizations of spectra of rings. Recall that a Grothendieck category is an abelian category with small coproducts, exact inductive limits, and cogenerators.

Let C_{X} be a Grothendieck category. By 4.6, every object O of C_{X} defines a continuous morphism f_{O} : X ⟶ Sp(Γ_{X}O) with a direct image functor C_{X}(O, −). Here Γ_{X}O denotes the ring C_{X}(O, O)^{o}.

4.8.1. Proposition. Let C_{X} be a Grothendieck category. A continuous morphism X ⟶^{f} Sp(R) is a flat localization iff the corresponding right R-module (O, ϕ_{f}) (cf. 4.6) has the following properties:

(a) O is a cogenerator of the category C_{X};

(b) the ring morphism ϕ_{f} : R ⟶ Γ_{X}O is flat (i.e. it makes Γ_{X}O ≃ f_{∗}f^{∗}(R) a flat right R-module), and the morphism Γ_{X}O ⊗_{R} Γ_{X}O ⟶ Γ_{X}O induced by the multiplication on Γ_{X}O is an isomorphism.

Proof. 1) Suppose the conditions (a) and (b) hold.

(a) If C_{X} is a Grothendieck category, then the morphism f_{O} : X ⟶ Sp(Γ_{X}O) with the direct image functor given by C_{X}(O, −) is a localization iff O is a cogenerator of the category C_{X}.

This assertion follows from the argument of the Theorem by Gabriel and Popescu which describes Grothendieck categories as quotient categories (localizations) of categories of modules. See [BD, 6.25].

(b) The condition (b) means exactly that the morphism SpΓ_{X}O ⟶ Sp(R) corresponding to the ring morphism ϕ_{f} is a localization.

Indeed, the (right) flatness of ϕ_{f} : R ⟶ Γ_{X}O means that the corresponding inverse image functor, ϕ_{f}^{∗} = Γ_{X}O ⊗_{R} −, is exact. The condition that Γ_{X}O ⊗_{R} Γ_{X}O ⟶ Γ_{X}O is an isomorphism implies that the adjunction morphism ϕ∗_{f}ϕ_{_{f} ∗} ⟶ Id_{Γ_{X}O−mod} is an isomorphism. The latter is equivalent to that ϕ_{_{f} ∗} is fully faithful, i.e. ϕ∗_{f} is a localization.

Thus, f, being a composition of flat localizations is a flat localization.

2) Suppose X ⟶^{f} Sp(R) is a localization.

(i) For any continuous morphism X ⟶^{f} Sp(R), the object O = f^{∗}(R) is a cogenerator iff f_{∗} is a faithful functor.

In fact, by definition, the object O is a cogenerator iff the functor C_{X}(O, −) is faithful. But, this functor is isomorphic to f_{∗}.

A continuous morphism f is a localization iff its direct image functor, f_{∗}, is fully faithful; in particular, O is a cogenerator.

(ii) Since O is a cogenerator, the morphism f_{O} : X ⟶ Sp(Γ_{X}O) is a flat localization. Let ϕ̄_{f} denote the morphism Sp(Γ_{X}O) ⟶ Sp(R) corresponding to the ring morphism ϕ_{f}. Since the composition f = ϕ̄_{f} ∘ f_{O} is a localization and f_{O} is a localization, it follows from the universal property of localizations that ϕ̄_{f} is a localization. The latter means that the adjunction morphism ϕ̄^{∗}ϕ̄_{∗}(M) = Γ_{X}O ⊗_{R} ϕ_{∗}(M) ⟶ M is an isomorphism for any Γ_{X}Omodule M. Taking M = Γ_{X}O, we obtain that the natural morphism Γ_{X}O ⊗_{R} Γ_{X}O ⟶ Γ_{X}O is an isomorphism.

(iii) It remains to verify that Γ_{X}O is flat as a right R-module. This is a consequence of the following general fact:

4.8.1.1. Lemma. Let X ⟶^{g} Y ⟶^{h} Z be continuous localizations. Suppose that the category C_{Z} has finite limits and that inverse image functors of g and h ∘ g are left exact. Then h^{∗} is left exact.

Let D : 𝔇 ⟶ C_{Y} be a finite diagram. Since h^{∗} is a localization, the functor h_{∗} is fully faithful. By [GZ, I.1.4], lim D exists if lim h_{∗}D exists, and the natural morphism h^{∗}(lim h_{∗}D) ⟶ lim h^{∗}h_{∗}D ≃ lim D is an isomorphism. Therefore

g^{∗}(lim D) ≃ g^{∗}h^{∗}(lim h_{∗}D) ≃ (hg)^{∗}(lim h_{∗}D).

Since the functor (hg)^{∗} is exact,

(hg)^{∗}(lim h_{∗}D) ≃ lim((hg)^{∗}h_{∗}D) ≃ lim(g^{∗}h^{∗}h_{∗}D) ≃ lim(g^{∗}D)

(the last isomorphism is induced by the adjunction isomorphism h^{∗}h_{∗} ⟶ Id_{C_{Y}}). Hence the assertion.

4.8.1.2. Note. The requirement in 4.8.1.1 that localizations are continuous can be dropped. The argument in this case is a little bit more involved. It is based on the fact that if X ⟶^{f} Y is a localization and the category C_{Y} has finite limits, then the functor f^{∗} is exact iff the class of morphisms Σ_{f} = {s ∈ HomC_{Y} | f^{∗}(s) is an isomorphism} satisfies the right Ore conditions (see 3.1.1, or [GZ, I.3.1 and 1.3.4]).

4.8.2. Corollary. Let C_{X} be a Grothendieck category. Any continuous morphism X ⟶^{f} Sp(R) such that f_{∗} is a faithful functor is uniquely represented as the composition

X ⟶^{ψ_{f}} Sp(R′) ⟶^{ϕ̄_{f}} Sp(R),  (8)

where ϕ̄_{f} is an affine morphism corresponding to a ring morphism ϕ_{f} : R ⟶ R′, and ψ_{f} is a flat localization such that the adjunction morphism R′ ⟶ ψ_{f∗}ψ_{f}^{∗}(R′) is an isomorphism.

The morphism f is a localization (resp. a flat localization) iff ϕ̄_{f} is a localization (resp. a flat localization).

Proof. Let O = f^{∗}(R). The functor f_{∗} is faithful iff the object O is a cogenerator in C_{X}. By (the argument of) Gabriel-Popescu Theorem, this implies that the canonical morphism f_{O} : X ⟶ Sp(Γ_{X}O) is a flat localization. We set ψ_{f} = f_{O} and R′ = Γ_{X}O = C_{X}(O, O)^{o}. The morphism ϕ_{f} : R ⟶ R′ is the right R-module structure on O (which is the same as the adjunction morphism R ⟶ f_{∗}f^{∗}(R) if the ring C_{X}(O, O)^{o} is identified with f_{∗}f^{∗}(R)). It follows from 4.8.1 that if f is a localization (i.e. f_{∗} is fully faithful), then ϕ̄_{f} is a localization.

The condition R′ ≃ ψ_{f∗}ψ_{f}^{∗}(R′) implies that the ring R′ is isomorphic to Γ_{X}O, where O = f^{∗}(R). This implies uniqueness (up to isomorphism) of the decomposition (8).

5. Monads, comonads, and continuous morphisms.

5.1. Monads and their categoric spectrum. Let Y be an object of |Cat|^{o}. A monad on Y is by definition a monad on the category C_{Y}, i.e. a pair (F, μ), where F is a functor C_{Y} ⟶ C_{Y} and μ a morphism F^{2} ⟶ F (multiplication) such that μ ∘ F μ = μ ∘ μF and there exists a morphism η : Id_{C_{Y}} ⟶ F uniquely determined by the equalities μ ∘ F η = id_{F} = μ ∘ ηF (a unit).

A morphism from a monad F = (F, μ) to a monad F′ = (F′, μ′) is given by a functor morphism F ⟶ F′ such that φ∘μ = μ′ ∘φ⊙φ and φ∘η = η′. Here φ⊙φ = F′φ∘φF, and η, η′ are units of the monads resp. F and F′. The composition of morphisms is defined naturally, so that the map 𝔐𝔬𝔫_{Y} ⟶ End(C_{Y}) forgetting monad structure, i.e. sending a monad morphism (F, μ) ⟶^{φ} (F′, μ′) to the natural transformation F ⟶^{φ} F′, is a functor.

For an object Y of |Cat|^{o}, we denote by 𝔐𝔬𝔫_{Y} the category of monads on Y.

Given a monad F = (F, μ) on Y, we denote by (F/Y) − mod, or simply by F − mod, the category of (F/Y)-modules. Its objects are pairs (M, ξ), where M ∈ ObC_{Y} and ξ is a morphism F (M) ⟶ M such that ξ ∘ F ξ = ξ ∘ μ(M) and ξ ∘ η(M) = id_{M}. Morphisms from (M, ξ) to (M′, ξ′) are given by morphisms g : M ⟶ M′ such that ξ′ ∘ F g = g ∘ ξ.

We denote by Sp(F/Y) the object of |Cat|^{o} such that the corresponding category is the category (F/Y) − mod of (F/Y)-modules. The object Sp(F/Y) is regarded as the spectrum of the monad F in |Cat|^{o}.

The forgetful functor

(F/Y) − mod ⟶^{f_{∗}} C_{Y} , (M, ξ) ⟼ M,

is a right adjoint to the functor

C_{Y} ⟶^{f^{∗}} (F/Y) − mod, L ⟼ (F (L), μ(L)), (L →^{g} N) ⟼ (f^{∗}(L) ⟶^{F (g)} f^{∗}(L′)).

In other words, we have a canonical continuous morphism Sp(F/Y) ⟶^{f} Y.

5.1.1. Example. Let R, S be unital associative rings. Any unital ring morphism φ : S ⟶ R defines a monad, R_{φ}^{∼} = (R_{φ}, μ_{φ}), on Y = Sp(S). Here the functor R_{φ} is M ⟼ R⊗_{S}M, and the multiplication is induced by the multiplication on R. The canonical morphism Sp(R/Sp(S)) ⟶ Sp(S) has the pull-back functor ϕ_{∗} : R − mod ⟶ S − mod as a direct image functor. Notice that the category (R_{φ}^{∼}/Sp(S))-modules is isomorphic to the category R − mod of R-modules; in particular, Sp(R_{φ}^{∼}/Sp(S)) ≃ Sp(R).

If S = ℤ, i.e. C_{Y} = ℤ−mod, the category (R/Spℤ)−mod coincides with the category R−mod of left R-modules. In this case, consistantly with our previous notations, we write Sp(R) instead of Sp(R/Spℤ).

5.1.2. Example. Any monoid morphism M ⟶^{ϕ} cN defines a monad, F = (F, μ), on Y = Sp(M/E), where the functor F is M ⊠_{N} −. It maps any left M-set (L, ξ) to the cokernel of the pair of morphisms N × M × L \overrightarrow{⟶} N × L, where one arrow is N × ξ

and another is the composition of N × M × L ⟶^{N ×ϕ×L} N × N × L ⟶^{ν×L} N × L. Here N × N ⟶^{ν} N is the multiplication on N. The multiplication F^{2} ⟶^{μ} F is induced by the multiplication on N. The canonical morphism Sp(F/Sp(M/E)) ⟶ Sp(M/E) has the pull-back functor N − sets ⟶^{ϕ_{∗}} M − sets as a direct image functor.

5.1.3. Example: localizations of modules. Let R be an associative unital ring and 𝔉 a set of left ideals in R. Denote by R − mod_{𝔉} the full subcategory of R − mod whose objects are R-modules M such that the canonical morphism

M ⟶ Hom_{R}(m, M), z ⟼ (r ↦ r · z) for all r ∈ m and z ∈ M,  (1)

is an isomorphism for all m ∈ 𝔉. The inclusion functor

R − mod_{𝔉} ⟶^{𝔧_{_{𝔉}∗}} R − mod  (2)

preserves limits, hence it has a left adjoint, 𝔧∗_{𝔉}. Since 𝔧_{_{𝔉}∗} is fully faithful, 𝔧∗_{𝔉} is a localization. The R-module R_{𝔉} = 𝔧_{_{𝔉}∗}𝔧∗_{𝔉}(R) has a structure of a ring uniquely determined by the fact

that the adjunction arrow, R ⟶^{η_{𝔉}} R_{𝔉} is a ring morphism. There is a canonical functor morphism

R_{𝔉} ⊗_{R} − ⟶^{τ_{𝔉}} 𝔧_{_{𝔉}∗}𝔧∗_{𝔉}.  (3)

Suppose all ideals in 𝔉 are projective modules. Then the inclusion functor (2) is exact. This implies that the morphism

R_{𝔉} ⊗_{R} M ⟶^{τ_{𝔉} (M)} 𝔧_{_{𝔉}∗}𝔧∗_{𝔉}(M)

is an isomorphism for any module M of finite type.

If 𝔉 consists of projective ideals of finite type, then (2) is strictly exact, i.e. it has a right adjoint. In this case, the localization R−mod ⟶ R−mod_{𝔉} is an affine morphism, or, equivalently, the functor morphism (3) is an isomorphism. Thus the category R−mod_{𝔉} is equivalent to the category R_{𝔉} − mod of left R_{𝔉}-modules.

5.1.3.1. Note. In general, the localization 𝔧_{𝔉} is not flat, i.e. the functor 𝔧∗_{𝔉} is not exact. Denote by 𝔉^{−} the set of all left ideals of the ring R such that the canonical morphism M ⟶ Hom_{R}(m, M) is an isomorphism for all M ∈ ObR − mod_{𝔉}. Clearly R − mod_{𝔉^{−}} = R − mod_{𝔉}. It follows from results of Gabriel (cf. [Gab], or [BD, Ch. 6]) that the localization 𝔧_{𝔉} is flat iff 𝔉^{−} is a radical filter; i.e. with any left ideal m, the set 𝔉^{−} contains left ideals (m : r) = {a ∈ R | ar ∈ m} for all r ∈ R and all left ideals n in R such that (n : r) ∈ 𝔉^{−} for all r ∈ m. These conditions are equivalent to that the full subcategory T_{𝔉^{−}} of R − mod whose objects are all R-modules M such that every element of M is annihilated by some ideal m ∈ 𝔉^{−}, is a Serre subcategory.

5.1.4. Curves. Let R be a ring of the homological dimension one, or, equivalently, every left ideal in R is projective. Then for any set of left ideals 𝔉, the inclusion functor (2) is exact. If, in addition, R is left noetherian, the functor (2) is strictly exact.

5.2. Morphisms of monads and morphisms of their categoric spectra. Let Y be an object of |Cat|^{o} and F, F′ monads on Y. Any monad morphism F ⟶^{φ} F′ induces the ’pull-back’ functor

(F′/Y) − mod ⟶^{φ_{∗}} (F/Y) − mod, (M, ξ) ⟼ (M, ξ ∘ φ(M)).

This correspondence defines a functor 𝔐𝔬𝔫_{Y}^{op} ⟶ Cat/C_{Y} which takes values in the full subcategory of Cat/C_{Y} objects of which are functors C_{Z} ⟶ C_{Y} having a left adjoint.

5.2.1. Reflexive pairs of arrows and weakly continuous functors and mon-

ads. Recall that a pair of arrows M \overrightarrow{⟶}^{g_{1}}_{g_{2}} L in C_{Y} is called reflexive, if there exists a

morphism L ⟶^{h} M such that g_{1} ∘ h = id_{M} = g_{2} ∘ h.

We call a functor C_{Y} ⟶ C_{Z} weakly continous if it preserves cokernels of reflexive pairs of arrows.

We call a monad F = (F, μ) on Y weakly continuous if the functor C_{Y} ⟶^{F} C_{Y} is weakly continuous. We denote by 𝔐𝔬𝔫_{Y}^{w} the full subcategory of the category 𝔐𝔬𝔫_{Y} whose objects are weakly continuous monads on Y.

5.2.2. Lemma. Let F, F′ be monads on Y and φ a monad morphism F ⟶ F′. Suppose the category C_{Y} has cokernels of reflexive pairs of morphisms and the monad F′ is weakly continuous. Then the functor φ_{∗} has a left adjoint.

In particular, the map (F/Y) ⟼ Sp(F/Y), φ ⟼ [φ^{∗}] is a functor,

Sp_{Y} : 𝔐𝔬𝔫_{Y}^{w} ⟶ |Cat|^{o},  (1)

which takes values in the subcategory |Cat|_{cont}^{o} of |Cat|^{o} formed by continuous morphisms.

Proof. The left adjoint, (F/Y) − mod ⟶^{φ^{∗}} (F′/Y) − mod assigns to each (F/Y)- module (M, F (M) →^{ξ} M) the cokernel of the pair of arrows

F′F (M) \overrightarrow{⟶}^{μ′∘F′φ}_{F′ξ} F′(M).  (1)

Since by hypothesis F′ preserves cokernels of reflexive pairs and both arrows (1) are F′- module morphisms, there exists a unique F′-module structure on the cokernel of (1). Details are left to the reader.

5.2.3. Note. Suppose that the category C_{X} has colimits of certain type 𝔇, and let F = (F, μ) be a monad on X such that the functor F preserves colimits of this type. Then the category (F/X) − mod has colimits of this type.

In fact, for a diagram 𝔇 ⟶^{D} (F/X)−mod, the colimit of the composition f_{∗}∘D (where f_{∗} is the forgetful functor (F/X) − mod ⟶ C_{X}) has a unique F-module structure, ξ_{D}. The F-module (colim(f_{∗} ∘ D), ξ_{D}) is a colimit of the diagram D.

In particular, if F = (F, μ) is a weakly continuous monad on X, and the category C_{X} has cokernels of reflexive pairs of arrows, then the category (F/X) − mod has cokernels of reflexive pairs of arrows.

5.3. Comonads and their cospectrum. A comonad on Y is a monad on the dual object (’space’) Y^{o} defined by C_{Y^{o}} = C_{Y}^{op}. In other words, a comonad on Y is a pair (G, δ), where G is a functor C_{Y} ⟶ C_{Y} and δ a functor morphism G ⟶ G^{2} (a comultiplication) such that Gδ ∘ δ = δG ∘ δ and Gϵ ∘ δ = id_{G} = ϵG ∘ δ for a uniquely determined morphism

G ⟶^{ϵ} Id_{C_{Y}} (a counit).

We denote the category of comonads on Y by ℭ𝔐𝔬𝔫_{Y}. It is defined by the formula ℭ𝔐𝔬𝔫_{Y} = 𝔐𝔬𝔫_{Y^{o}}.

Comodules over a comonad G = (G, δ) are just modules over the dual monad on Y^{o}. In terms of Y, a G-comodule is a pair (M, ξ), where M ∈ ObC_{Y} and ξ a morphism M ⟶ G(M) such that δ(M) ∘ ξ = Gξ ∘ ξ and ϵ(M) ∘ ξ = id_{M}. We denote the category of comodules over G by (Y \G) − Comod, or simply by G − Comod.

We denote by Sp^{o}(Y \G) the object of |Cat|^{o} (or Cat^{op}) such that the corresponding category is (Y \G) − Comod. This definition can be rephrased as follows:

Sp^{o}(Y \G) = Sp(G^{o}/Y^{o})^{o}.  (1)

Here G^{o} is the monad (G^{o}, δ^{o}) on Y^{o} dual to the comonad G.

We call Sp^{o}(Y \G) the cospectrum of the comonad G in |Cat|^{o}.

By duality, there is a canonical continuous morphism Y ⟶^{g} Sp^{o}(Y \G) with an inverse image functor

(Y \G) − Comod ⟶^{g^{∗}} C_{Y} , (M, ξ) ⟼ M,  (2)

and having a direct image functor

C_{Y} ⟶^{g_{∗}} (Y \G) − Comod, L ⟼ (G(L), δ(L)).  (3)

5.3.1. Example. Let R be an associative unital ring and H = (H, δ) a coalgebra in the monoidal category of R-bimodules. This means that H is an R-bimodule, δ an R-bimodule morphism H ⟶ H ⊗_{R} H such that δ ⊗_{R} id_{H} ∘ δ = id_{H} ⊗_{R} δ ∘ δ, and there exists a (necessarily unique) R-bimodule morphism ϵ : H ⊗_{R} H ⟶ R such that λ_{r}(H) ∘ ϵ ⊗_{R} id_{H} ∘ δ = id_{H} = λ_{l}(H) ∘ id_{H} ⊗_{R} ϵ ∘ δ. Here λ_{l}(H) : R ⊗_{R} H ⟶ H and λ_{r}(H) : H⊗_{R} ⟶ H are canonical isomorphisms. The coalgebra H induces a comonad on the category R−mod of left R-modules tensoring by H over R, L ⟼ H ⊗_{R} L, as a functor and with comultiplication H ⊗_{R} − ⟶ H ⊗_{R} H ⊗_{R} − induced by the comultiplication δ.

The canonical morphism Sp(R) ⟶ Sp^{o}(Sp(R)\H) has the forgetful functor (Sp(R)\H)− Comod ⟶ R − mod as an inverse image functor.

5.3.2. Functoriality of the cospectrum. Let G = (G, δ) and G′ = (G′, δ′) be comonads on Y and ψ a comonad morphism G ⟶ G′; i..e. ψ is a morphism of functors G ⟶ G′ such that δ′ ∘ ψ = ψ ⊙ ψ ∘ δ and ϵ′ ∘ ψ = ϵ. Here ψ ⊙ ψ = G′ψ ∘ ψG, and ϵ, ϵ′ are counits of the comonads resp. G and G′. The morphism ψ induces the ’pull-back’ functor

(Y \G) − Comod ⟶^{ψ^{∗}} (Y \G′) − Comod, (M, ξ) ⟼ (M, ψ(M) ∘ ξ),  (4)

which is regarded as an inverse image functor of a morphism

Sp^{o}(ψ) : Sp^{o}(Y \G′) ⟶ Sp^{o}(Y \G)  (5)

The map (4) defines a functor

S̃p_{Y}^{o} : ℭ𝔐𝔬𝔫_{Y}^{op} ⟶ (C_{Y} \Cat^{op})_{𝔠},  (6)

where (C_{Y} \Cat^{op})_{𝔠} denotes the full subcategory of the category C_{Y} \Cat^{op} whose objects are continuous morphisms.

5.3.2.1. Proposition. The functor (6) is fully faithful and has a right adjoint.

Proof. Let

\begin{matrix}X ⟶^{h} Z \\ f ↖ ↗ g \\ Y \end{matrix}

be a morphism in (C_{Y} \Cat^{op})_{𝔠} given by the commutative diagram

C_{Z} ⟶^{h^{∗}} C_{X}

f^{∗} ↘ ↙ g^{∗}

C_{Y}

of functors. Fix direct image functors of f and g and the corresponding adjunction arrows. Set

φ_{h} = f^{∗}(f_{∗}ϵ_{g} ∘ η_{f}h^{∗}g_{∗}) : g^{∗}g_{∗} ⟶ f^{∗}f_{∗}.  (7)

One can check that ϕ_{h} is a monad morphism G_{g} ⟶ G_{f} and the map Γ̃_{Y} : h ⟼ φ_{h} is functorial. The composition Γ̃_{Y} ∘ S̃p_{Y}^{o} is the identical functor which provides one of the adjunction arrows and shows that the functor S̃p_{Y}^{o} is fully faithful. We leave the other adjunction arrow to the reader (it is defined in 5.4).

Recall that pair of arrows M \overrightarrow{⟶}^{g_{1}}_{g_{2}} L is called coreflexive, if there exists a morphism

L ⟶^{h} M such that h ∘ g_{1} = id_{M} = h ∘ g_{2}.

5.3.2.2. Lemma. Suppose that the category C_{Y} has kernels of coreflexive pairs of morphisms and the functor G preserves these kernels. Then the functor ψ^{∗} has a right adjoint, i.e. the morphism (4) is continuous.

Proof. The assertion is the dual version of 5.2.2.

5.4. Beck’s theorem. Let X ⟶^{f} Y be a continuous morphism in |Cat|^{o} with inverse image functor f^{∗}, direct image functor f_{∗}, and adjunction morphisms

Id_{C_{Y}} ⟶^{η_{f}} f_{∗}f^{∗} and f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{X}} .

Let G_{f} denote the comonad (G_{f}, δ_{f}), where G_{f} = f^{∗}f_{∗} and δ_{f} = f^{∗}η_{f}f_{∗}. There is a commutative diagram

\begin{matrix}C_{Y} ⟶^{f̃^{∗}} (X\G_{f}) − Comod \\ _{f^{∗}} ↘  ↙f̆^{∗} \\ C_{X} \end{matrix}  (1^{o})

Here f̃^{∗} is the canonical functor C_{Y} ⟶ (X\G_{f}) − Comod, M ⟼ (f^{∗}(M), f^{∗}η_{f}(M)), and f̆^{∗} is the forgetful functor (X\G_{f}) − Comod ⟶ C_{X}. The diagram (1^{o}) is regarded as the diagram of inverse image functors of the commutative diagram

\begin{matrix}X ⟶^{f̆} Sp^{o}(X\G_{f}) \\ _{f} ↘  ↙f̃ \\ Y \end{matrix}  (2^{o})

in |Cat|^{o}. The following statement is one of the versions of the Beck’s theorem.

5.4.1. Theorem. Let X ⟶^{f} Y be a continuous morphism.

(a) If the category C_{X} has kernels of coreflexive pairs of arrows, then the functor f̃^{∗}

has a right adjoint, f̃_{∗}, i.e. Sp^{o}(X\G_{f}) ⟶^{f̃} Y is a continuous morphism.

(b) If, in addition, f is weakly flat, i.e. the functor f^{∗} preserves kernels of coreflexive pairs, then the adjunction arrow f̃^{∗}f̃_{∗} ⟶^{ϵf̃} Id_{(X\G_{f})−Comod} is an isomorphism, i.e. f̂_{∗} is a fully faithful functor, or, equivalently, f̃^{∗} is a localization.

(c) If, in addition to (a) and (b), f^{∗} reflects isomorphisms, then the adjunction arrow Id_{C_{Y}} ⟶^{ηf̃} f̃_{∗}f̃^{∗} is an isomorphism too, i.e. f̃ is an isomorphism.

Proof. See [MLM], IV.4.2, or [ML], VI.7.

We need also the dual version of the theorem 5.4.1. Let F_{f} denote the monad (F_{f}, μ_{f}), where F_{f} = f_{∗}f^{∗} and μ_{f} = f_{∗}ϵ_{f}f^{∗}. There is a commutative diagram

\begin{matrix}C_{X} ⟶^{f̄_{∗}} (F_{f}/Y) − mod \\ f_{∗} ↘  ↙ f̂_{∗} \\ C_{Y} \end{matrix}  (1)

Here f̄_{∗} is the canonical functor

C_{X} ⟶ (F_{f}/Y) − mod, M ⟼ (f_{∗}(M), f_{∗}ϵ_{f}(M)),

f̂_{∗} the forgetful functor (F_{f}/Y) − mod ⟶ C_{Y}.

5.4.2. Theorem. Let X ⟶^{f} Y be a continuous morphism.

(a) If the category C_{Y} has cokernels of reflexive pairs of arrows, then the functor f̄_{∗} has a left adjoint, f̄^{∗}; hence f̄_{∗} is a direct image functor of a continuous morphism

X̄ ⟶^{f} Sp(F_{f}/Y).

(b) If, in addition, the functor f_{∗} preserves cokernels of reflexive pairs, then the adjunction arrow f̄^{∗}f̄_{∗} ⟶ Id_{C_{X}} is an isomorphism, i.e. f̄_{∗} is a localization.

(c) If, in addition to (a) and (b), the functor f_{∗} is conservative, then f̄_{∗} is a category equivalence.

Proof. The theorem is dual (hence equivalent) to the theorem 5.4.1.

If the condition (a) in 5.4.2 holds, then to the diagram (1), there corresponds a commutative diagram

\begin{matrix}X ⟶^{f̄} Sp(F_{f}/Y) \\ _{f} ↘  ↙f̂ \\ Y \end{matrix}  (2)

in |Cat|^{o}. If the condition (c) in 5.4 holds, the morphism f̄ in (2) is an isomorphism.

Thus, given a continuous morphism X ⟶^{f} Y such that the category C_{Y} has cokernels of reflexive pairs of arrows, we have a commutative diagram

\begin{matrix}X ⟶^{f̄} Sp(F_{f}/Y) \\ f̆ ↙  ↘^{f} ↙ f̂ \\ Sp^{o}(X\G) ⟶^{f̃} Y \end{matrix}  (3)

Notice that the diagrams (1) and (1^{o}) are uniquely defined by the data (f^{∗}, f_{∗}, ϵ_{f}, η_{f}), since the monad F_{f}, the comonad G_{f}, and the functors f̄_{∗} : C_{X} ⟶ (F_{f}, Y) − mod in (1) and f̃^{∗} : C_{Y} ⟶ (X\G_{f}) − Comod are defined in terms of this data. Given the functor f^{∗} (resp. f_{∗}), the rest of the data, f_{∗}, ϵ_{f}, η_{f} (resp. f^{∗}, ϵ_{f}, η_{f}), is determined uniquely up to isomorphism. Thus, the monad F_{f} and the comonad G_{f} in the diagrams (1) and (1^{o}) are determined by f^{∗} uniquely up to isomorphism.

5.5. Weakly flat and weakly affine morphisms. We call a functor weakly continuous (resp. weakly flat) if it preserves cokernels of reflexive pairs of arrows (resp. kernels of coreflexive pairs of arrows).

We call a monad (F, μ) weakly continuous if the functor F is weakly continuous, and a comonad (G, δ) weakly flat if the functor G is weakly flat.

Let 𝔐𝔬𝔫_{X}^{w} denote the full subcategory of the category 𝔐𝔬𝔫_{X} of monads on X spanned by weakly continuous monads, and let ℭ𝔐𝔬𝔫_{X}^{w} be the full subcategory of the category ℭ𝔐𝔬𝔫_{X} of comonads on X spanned by weakly flat comonads on X.

We call a continuous morphism X ⟶^{f} Y weakly affine if its direct image functor is weakly continuous and the category C_{X} has cokernels of reflexive pairs of arrows. Let Aff_{Y}^{w} denote the full subcategory of the category |Cat|^{o}/Y whose objects are weakly affine morphisms to Y.

We call a continuous morphism X ⟶^{f} Y in weakly flat if its inverse image functor is weakly flat and the category C_{Y} has kernels of coreflexive pairs of arrows. We denote by F lat_{X}^{w} the full subcategory of the category X\|Cat|^{o} whose objects are weakly flat morphisms from X.

5.5.1. Proposition. (a) Suppose the category C_{Y} has cokernels of reflexive pairs of arrows. The map (F/Y) ⟼ (Sp(F/Y) → Y) defines a full functor

Sp_{Y}^{w} : (𝔐𝔬𝔫_{Y}^{w})^{op} ⟶ Aff_{Y}^{w}.

(b) Dually, if the category C_{Y} has kernels of coreflexive pairs of arrows, then the map (X\G) ⟼ (X → Sp^{o}(X\G)) defines a full functor

Sp^{o}_{X}^{w} : (ℭ𝔐𝔬𝔫_{X}^{w})^{op} ⟶ F lat_{X}^{w}.

Proof. These facts follow from Beck’s Theorem and the following

5.5.2. Lemma. Let X ⟶^{f} Y be a continuous morphism with a direct image functor f_{∗} and an inverse image functor f^{∗}.

(a) Suppose the morphism f is monadic and the category C_{Y} has colimits of a type 𝔖. Then f_{∗} preserves colimits of the type 𝔖 iff the functor F_{f} = f_{∗}f^{∗} has this property.

(b) Dually, if the morphism f is comonadic and the category C_{X} has limits of a certain type, then f^{∗} preserves these limits iff the functor G_{f} = f^{∗}f_{∗} does the same.

Recall that a continuous morphism X ⟶^{f} Y is called comonadic if the induced morphism f̃: Sp^{o}(X\G_{f}) ⟶ Y is an isomorphism.

Dually, a continuous morphisms X ⟶^{f} Y (in |Cat|^{o}, or in Cat^{op}) is called monadic if the associated morphism X̄ ⟶^{f} Sp(F_{f}/Y) is an isomorphism.

The proof of the lemma and details of the proof of 5.5.1 are left to the reader.

5.5.3. Proposition. Let

\begin{matrix}X ⟶^{h} Y \\ f ↘ ↙ g \\ Z \end{matrix}

be a commutative diagram in |Cat|^{o}. Suppose C_{Z} has colimits of reflexive pairs of arrows. If f and g are weakly affine, then h is weakly affine.

Dually, if

\begin{matrix}X ⟶^{h} Y \\ f ↖ ↗ g \\ Z \end{matrix}

is a commutative diagram in |Cat|^{o} such that C_{Z} has kernels of coreflexive pairs of arrows and the morphisms f and g are weakly flat, then h is weakly flat.

Proof. Fix inverse and direct image functors of f and g together with adjunction morphisms. By hypothesis, the canonical morphisms C_{X} ⟶ (F_{f}/Z) − mod and C_{Y} ⟶ (F_{g}/Z) − mod are category equivalences. Here F_{f} = (f_{∗}f^{∗}, μ_{f}) and F_{g} = (g_{∗}g^{∗}, μ_{g}) are monads associated with resp. f and g. It follows from the dual version of 5.3.2.1 (or the argument of 10.7.1 in [R4]) that a choice of an inverse image functor h^{∗} of the morphism h determines a monad morphism F_{g} ⟶^{ϕ_{h}} F_{f} such that the diagram

C_{Y} ⟶^{∼} (F_{g}/Z) − mod
h^{∗} ↓  ↓ ϕ_{h}^{∗}
C_{X} ⟶^{∼} (F_{f}/Z) − mod

quasi-commutes. Here ϕ_{h}^{∗} is the inverse image functor associated with the monad morphism ϕ_{h} (a left adjoint to the pull-back functor). The pull-back functor, ϕ_{h_{∗}}, is, evidently, conservative and weakly continuous. The latter follows from the fact that the monads F_{f} and F_{g} are weakly continuous.

6. Continuous monads and affine morphisms. Duality. A functor F is called continuous if it has a right adjoint. A monad F = (F, μ) on Y (i.e. on the category C_{Y}) is called continuous if the functor F is continuous.

Dually, a comonad G = (G, δ) on Y is called cocontinuous if the functor G has a left adjoint. In other words, a cocontinuous comonad on Y is the same as a continuous monad on Y^{o} and vice versa.

It follows that a continuous monad is weakly continuous, because the the functor F preserves all colimits. Dually, a cocontinuous monad is weakly flat.

6.1. Duality. Let F = (F, μ) be a continuous monad on Y; i.e. the functor F has a right adjoint, F^{∧}. The multiplication F^{2} ⟶^{μ} F induces a morphism F^{∧} ⟶^{δ} (F^{∧})^{2} which is a comonad structure on F^{∧} with the counit F^{∧} ⟶^{ϵ} Id_{C_{Y}} induced by the unit Id_{C_{Y}} ⟶^{η} F of the monad F. Thus, we have a comonad, F^{∧} = (F^{∧}, μ^{∧}) dual to the monad F. The map which assigns to any morphism F (L) ⟶ L, L ∈ ObC_{Y} , the dual morphism L ⟶ F^{∧}(L) induces an isomorphism of categories

Φ : (F/Y) − mod ⟶^{∼} (Y \F^{∧}) − Comod  (1)

such that the diagram

\begin{matrix}(F/Y) − mod ⟶^{Φ} (Y \F^{∧}) − Comod \\ f̂_{∗} ↘  ↙ f̆^{∗} \\ C_{Y} \end{matrix}  (2)

commutes. Here f̆^{∗} denotes the functor forgetting F^{∧}-comodule structure.

It follows from the construction that F^{∧} is a cocontinuous comonad on Y determined by the monad F uniquely up to isomorphism.

Conversely, to any cocontinuous comonad, G = (G, δ), on Y, there corresponds a continuous monad G^{∨} = (G^{∨}, δ^{∨}), where G^{∨} is a left adjoint to G. The monad G^{∨} is determined by G uniquely up to isomorphism, and we have a comonad and monad isomorphisms, respectively

G ⟶^{∼} (G^{∨})^{∧} and F ⟶^{∼} (F^{∧})^{∨}.

6.2. Proposition. A monad F = (F, μ) on Y is continuous iff the canonical mor-

phism Sp(F/Y) ⟶^{f̂} Y is affine. Dually, a comonad G = (G, δ) on Y is cocontinuous iff the canonical morphism Y ⟶ Sp^{o}(Y \G) is coaffine.

Proof. A canonical direct image functor of f̂ is the forgetful functor

(F/Y) − mod ⟶^{f̂_{∗}} C_{Y} , (M, ξ) ⟼ M.

Since the functor f̂_{∗} is, evidently, conservative, the morphism f̂ is affine iff f̂_{∗} has a right adjoint.

(a) If f̂^{!} is a right adjoint to f̂_{∗}, then the functor F^{∧} = f̂_{∗}f̂^{!} is a right adjoint to F = f̂_{∗}f̂^{∗}. Here f̂^{∗} denotes the functor L ⟼ (F (L), μ(L)).

(b) Conversely, suppose F = (F, μ) is a continuous monad on Y; i.e. the functor F has a right adjoint, F^{∧}. The functor f̆^{∗} in the diagram (2) has a right adjoint, f̆_{∗}, which maps

every object M of C_{Y} to the (Y \F^{∧})-comodule (F^{∧}(M), M ⟶^{δ(M)} (F^{∧})^{2}(M)). It follows from the commutativity of (2) that the functor f̂^{!} = Φ^{−1} ∘ f̆_{∗} : C_{Y} ⟶ F − mod is a right

adjoint to the forgetful functor F − mod ⟶^{f̂_{∗}} C_{Y}. Since f̂_{∗} is, obviously, conservative, it is a direct image functor of an affine morphism Sp(F/Y) ⟶ Y.

6.2.1. Corollary. Suppose that the category C_{Y} has cokernels of reflexive pairs of arrows. A continuous morphism X ⟶^{f} Y in |Cat|^{o} is affine iff its direct image functor C_{X} ⟶^{f_{∗}} C_{Y} is the composition of a category equivalence

C_{X} ⟶ (F_{f}/Y) − mod

for a continuous monad F_{f} in C_{Y} and the forgetful functor (F_{f}/Y) − mod ⟶ C_{Y}. The monad F_{f} is determined by f uniquely up to isomorphism.

Proof. The conditions of the Beck’s theorem are fullfiled if f is affine, hence f_{∗} is the composition of an equivalence C_{X} ⟶ (F_{f}/Y) − mod for a monad F_{f} = (f_{∗}f^{∗}, μ_{f}) in C_{Y} and the forgetful functor (F_{f}/Y) − mod ⟶ C_{Y} (see (1)). The functor F_{f} = f_{∗}f^{∗} has a right adjoint f_{∗}f^{!}, where f^{!} is a right adjoint to f_{∗}. The rest follows from 6.2.

6.3. Proposition. Suppose X is an object of |Cat|^{o} such that the category C_{X} has kernels of reflexive pairs of arrows. Let F_{f} = (F_{f}, μ_{f}) and F_{g} = (F_{g}, μ_{g}) be continuous monads on X. Then for any monad morphism F_{f} ⟶^{φ} F_{g}, the corresponding morphism

Sp(φ) : Sp(F_{f}/X) ⟶ Sp(F_{g}/X)

is affine.

Proof. The morphism φ induces a dual comonad morphism F_{g}^{∧} ⟶^{φ̂} F_{f}^{∧} such that the diagram

\begin{matrix}(F_{g}/X) − mod & ⟶^{φ_{∗}} & (F_{f}/X) − mod \\ Φ_{F_{g}} ↓ & & ↓ Φ_{F_{f}} \\ (X\F_{g}^{∧}) − Comod & ⟶^{φ̂^{∗}} & (X\F_{f}^{∧}) − Comod \end{matrix}  (3)

commutes. Here Φ_{F_{f}} and Φ_{F_{g}} are the canonical category isomorphisms (cf. 6.1). Since the category C_{X} has kernels of coreflexive pairs of arrows and the functor F_{g}^{∧} preserves limits (in particular, it preserves kernels of pairs of arrows), the functor φ̂^{∗} has a right adjoint (cf. 5.3.2.2), hence φ_{∗} has a right adjoint. Since the functor φ_{∗} is conservative, the morphism Sp(φ) is affine.

6.4. Proposition. Let

\begin{matrix}X ⟶^{h} Y \\ f ↘ ↙ g \\ Z \end{matrix}

be a commutative diagram in |Cat|^{o}. Suppose the category C_{Z} has cokernels of coreflexive pairs of arrows. If f and g are affine, then h is affine.

Proof. Fix inverse and direct image functors of f and g together with adjunction morphisms. By the Beck’s theorem, the canonical morphisms C_{X} ⟶ (F_{f}/Z) − mod and C_{Y} ⟶ (F_{g}/Z)−mod are category equivalences. Here F_{f} = (f_{∗}f^{∗}, μ_{f}) and F_{g} = (g_{∗}g^{∗}, μ_{g}) are monads associated with resp. f and g. By 5.5.3 (or 10.7.1 in [R4]), a choice of an inverse image functor h^{∗} of the morphism h determines a monad morphism F_{g} ⟶^{ϕ_{h}} F_{f} such that the diagram

C_{Y} ⟶^{∼} (F_{g}/Z) − mod
h^{∗} ↓  ↓ ϕ_{h}^{∗}
C_{X} ⟶^{∼} (F_{f}/Z) − mod

quasi-commutes. By 6.3, since the monads F_{g} and F_{f} are continuous, the direct image functor ϕ_{h∗} (the pull-back by the morphism ϕ_{h}) has a right adjoint, ϕ_{h}^{!}.

For Z ∈ Ob|Cat|^{o}, denote by Aff_{Z} the full subcategory of |Cat|^{o}/Z whose objects are affine morphisms. Let |Cat|_{aff}^{o} be the subcategory of |Cat|^{o} formed by affine morphisms.

6.4.1. Proposition. Suppose that the category C_{Z} has cokernels of reflexive pairs of arrows. Then the natural embedding |Cat|_{aff}^{o}/Z ⟶ Aff_{Z} is an isomorphism of categories.

Proof. The assertion is a corollary of 6.4.

6.5. Proposition. Let X ⟶^{f} Y be an affine morphism in |Cat|^{o}. If the category C_{Y} is additive (resp. abelian, resp. abelian with small coproducts, resp. a Grothendieck category), then the category C_{X} has the same property.

Proof. By 6.2, the category C_{X} is equivalent to the category (F_{f}/Y)−mod of (F_{f}/Y)- modules for a continuous monad F = (F_{f}, μ_{f}) on Y. Since the functor F_{f} has a right adjoint and the category C_{Y} is additive, F_{f} is additive and preserves colimits of arbitrary small diagrams. This implies that for any diagram D ⟶^{𝔇} F −mod, the object colim(f_{∗}∘𝔇) (where f_{∗} is the forgetful functor (F/Y)−mod ⟶ C_{Y}) has a unique (F_{f}/Y)-module structure ξ_{𝔇} such that all morphisms f_{∗}𝔇(x) ⟶ colim(f_{∗} ∘ 𝔇) are (F/Y)-module morphisms 𝔇(x) ⟶ (colim(f_{∗} ∘ 𝔇), ξ_{𝔇}). This implies the assertion. Details are left to the reader.

6.6. Affine morphisms to Sp(R).

6.6.1. Proposition. Let R be an associative unital ring. A continuous morphism X ⟶^{f} Sp(R) in |Cat|^{o} is affine iff its direct image functor, C_{X} ⟶^{f_{∗}} R − mod, is the composition of an equivalence of categories C_{X} ⟶ R_{f} −mod for an associative unital ring

R_{f} and the pull-back functor R_{f} − mod ⟶^{ϕ_{∗}} R − mod = C_{Y} for a ring morphism R ⟶^{ϕ} R_{f} determined by f uniquely up to isomorphism.

Proof. (i) If X ⟶^{f} Y is such a morphism in |Cat|^{o} that C_{X} ⟶^{f_{∗}} C_{Y} is an equivalence of categories, then f is, obviously, affine. The morphism S −mod ⟶ R−mod corresponding to a ring morphism R ⟶ S is affine by 3.8. Finally, the composition of affine morphisms is affine.

(ii) Conversely, suppose that X ⟶^{f} Sp(R) is an affine morphism. Then the functor f_{∗}f^{∗} : R − mod ⟶ R − mod has a right adjoint, hence it is isomorphic to the functor R_{f} ⊗_{R} − : L ⟼ R_{f} ⊗_{R} L for some R-bimodule R_{f}. The monad structure on f_{∗}f^{∗} induces an associative ring structure, R_{f} ⊗_{R} R_{f} ⟶^{m_{f}} R_{f}, on R_{f}; and the adjunction morphism

Id_{R−mod} ⟶^{η_{f}} f_{∗}f^{∗} corresponds to a ring morphism R ⟶^{ϕ} R_{f} so that the diagrams of functor morphisms

\begin{aligned}Id_{R−mod} & ⟶^{∼} & R ⊗_{R} − & & (f_{∗}f^{∗})^{2} & ⟶^{∼} & R_{f} ⊗_{R} R_{f} ⊗_{R} − \\ η_{f} ↓ & & ↓ ϕ⊗_{R} & and & μ_{f} ↓ & & ↓ m_{f} \\ f_{∗}f^{∗} & ⟶^{∼} & R_{f} ⊗_{R} − & & f_{∗}f^{∗} & ⟶^{∼} & R_{f} ⊗_{R} − \end{aligned} (2)

commute. Thus we have a commutative diagram

\begin{matrix}(F_{f}/Sp(R)) − mod ⟶^{∼} R_{f} − mod \\ f̂_{∗} ↘  ↙ ϕ_{∗} \\ R − mod \end{matrix}  (3)

in which the horizontal arrow is an isomorphism of categories. Combining with the commutative diagram (1), we obtain a commutative diagram

\begin{matrix}C_{X} ⟶^{∼} R_{f} − mod \\ f_{∗} ↘  ↙ ϕ_{∗} \\ R − mod \end{matrix}  (4)

in which the horizontal arrow is an equivalence of categories.

Notice that R_{f} = f_{∗}f^{∗}(R). Therefore, the ring morphism R ⟶^{ϕ} R_{f} is defined uniquely up to isomorphism by a choice of the functor f^{∗}.

6.6.2. A comparison of two descriptions. Let X ⟶^{f} Sp(R) be an affine morphism. Being continuous, the morphism f is determined uniquely up to isomorphism by the object O = f^{∗}(R), and a right R-module structure R ⟶ Γ_{X}O = C_{X}(O, O)^{o} (cf. 4.1). By 4.5, we have a commutative diagram of direct image functors of continuous morphisms

\begin{matrix}C_{X} ⟶^{f_{_{O} ∗}} Γ_{X}O − mod \\ f_{∗} ↘  ↙ ϕ̄_{_{f} ∗} \\ R − mod \end{matrix}  (1)

Here ϕ̄_{_{f} ∗} is the pull-back by the ring morphism R ⟶^{ϕ_{f}} Γ_{X}O defining a right R-module structure on O. The morphism f_{_{O}∗} has an inverse image functor f ∗_{O} which maps the left module Γ_{X}O to O. The adjunction morphism Γ_{X}O ⟶ f_{_{O}∗}f ∗_{O}(Γ_{X}O) is an isomorphism.

Since morphisms f and SpΓ_{X}O ⟶^{ϕ̄_{f}} Sp(R) are affine, the morphism f_{O} is affine too (cf. 6.3). In particular, f_{_{O}∗}f ∗_{O} has a right adjoint, hence it preserves colimits. Since Γ_{X}O is a generator of the category Γ_{X}O − mod, the isomorphness of the adjunction arrow Γ_{X}O ⟶ f_{_{O}∗}f ∗_{O}(Γ_{X}O) implies that M ⟶ f_{_{O}∗}f ∗_{O}(M) is an isomorphism for any Γ_{X}Omodule M. This means that the functor f ∗_{O} is fully faithful, hence f_{_{O}∗} is a localization. Since by condition f_{_{O}∗} is conservative, it is a category equivalence.

This shows that C_{X} is naturally equivalent to the category of Γ_{X}O-modules. Thus the ring morphism R ⟶ R_{f} in 6.6.1 is isomorphic to the ring morphism R ⟶ Γ_{X}O defining a right R-module structure on the object O.

This observation was made before (in [R], Ch.7) using a slightly different argument.

7. Flat descent.

7.1. Continuous, flat comonads. A comonad G = (G, δ) on X is called

— continuous if the functor G has a right adjoint,

— flat if the functor G preserves finite limits,

— weakly flat if the functor G preserves kernels of coreflexive pairs of arrows,

— conservative if the functor G is conservative.

7.2. Proposition. Let X ⟶^{f} Y be a continuous morphism, and let C_{X} have kernels of coreflexive pairs of morphisms. The morphism X ⟶^{f} Y in |Cat|^{o} is affine, flat (resp.

weakly flat), and conservative iff its inverse image functor C_{Y} ⟶^{f^{∗}} C_{X} is the composition of an equivalence C_{Y} ⟶ (X\G_{f}) − Comod for a continuous flat (resp. weakly flat) conservative comonad G_{f} on X and the forgetful functor (X\G_{f}) − Comod ⟶ C_{X}. The monad G_{f} is determined by f uniquely up to isomorphism.

Proof. The conditions of the Beck’s theorem are fullfiled if f is weakly flat and conservative, hence f^{∗} is the composition of an equivalence C_{Y} ⟶ (X\G_{f}) − Comod for a comonad G_{f} = (f^{∗}f_{∗}, δ_{f}) on X and the forgetful functor (X\G_{f}) − Comod ⟶ C_{X} (see (1)). If f is affine, then the functor G_{f} = f^{∗}f_{∗} has a right adjoint f^{!}f_{∗}, where f^{!} is a right adjoint to f_{∗}.

Let now G = (G, δ) be a continuous comonad on X and f^{∗} the forgetful functor (X\G) − Comod ⟶ C_{X}. The functor f_{∗} which assigns to each object M of C_{X} the G-comodule (G(M), δ(M)) is right adjoint to f^{∗}: the canonical adjunction arrows are

ϵ_{f} : f^{∗}f_{∗} = G ⟶ Id_{C_{X}} and η_{f} : Id_{C_{Y}} ⟶ f_{∗}f^{∗},

where C_{Y} = G − Comod, ϵ_{f} is the counit of the monad G and

η_{f}(M, ξ) = ξ : (M, ξ) ⟶ f_{∗}f^{∗}(M, ξ) = (G(M), δ(M))

for any G-comodule (M, M →^{ξ} G(M)).

Let G^{!} be a right adjoint to G, and let GG^{!} ⟶^{ϵ′} Id_{C_{X}} and Id_{C_{X}} ⟶^{η′} G^{!}G be adjunction arrows. Let f^{!} denote the functor

C_{Y} = (X\G) − Comod ⟶ C_{X}, (M, ξ) ⟼ G^{!}(M).

Since f^{!}f_{∗} = G^{!}G, the adjunction arrow η′ is a morphism Id_{C_{X}} ⟶ f^{!}f_{∗}. The composition f_{∗}f^{!} assigns to each G-comodule (M, ξ) the G-comodule f_{∗}G^{!}(M) = (GG^{!}(M), δG^{!}(M)). One can check that the adjunction arrow ϵ′(M) : GG^{!}(M) ⟶ M is a G-comodule morphism f_{∗}G^{!}(M) ⟶ (M, ξ), i.e. the diagram

\begin{matrix}GG^{!}(M) & ⟶^{ϵ′(M)} & M \\ δG^{!}(M) ↓ & & ↓ ξ \\ G^{2}G^{!}(M) & ⟶^{Gϵ′(M)} & G(M) \end{matrix}  (1)

commutes. This implies that ϵ′f^{∗} and η′ are adjunction morphisms, hence the assertion.

7.2.1. Corollary. Let a morphism X ⟶^{f} Y be affine, weakly flat, and conservative. If the category C_{X} is additive (resp. abelian, resp. abelian with small coproducts, resp. a Grothendieck category), then the category C_{Y} has the same property, and the morphism f is flat.

Proof. Under the hypothesis, the category C_{Y} is equivalent to the category (X\G_{f})-

comodules for a continuous comonad G_{f} = (G_{f}, δ_{f}) on X. Since the functor C_{X} ⟶^{G_{f}} C_{X} has a right adjoint, it is additive and preserves small colimits. Since G_{f} = f^{∗}f_{∗}, the functor f_{∗} preserves all small limits, and the functor f^{∗} preserves kernels of coreflexive pairs of arrows, the functor G_{f} preserves kernels of coreflexive pairs of arrows too. For additive categories (more generally, for categories with coproducts and a zero object) functors which preserve kernels of coreflexive pairs of arrows preserve kernels of any pairs of arrows. Thus G_{f} preserves kernels of any pairs of arrows and, being additive, products (which coincide with coproducts), hence G_{f} reserves limits of any finite diagrams. This implies that the category (X\G_{f}) − Comod has limits of finite diagrams which are preserved (and reflected) by the forgetful functor (X\G_{f}) − Comod ⟶ C_{X}. This implies the additivity of (X\G_{f}) − Comod. The rest follows from the compatibility of G_{f} with arbitrary small colimits (cf. the argument of 6.5).

7.3. Affine, flat morphisms from Sp(R). If R is an associative ring and G a comonad on Sp(R), we shall write for convenience (R\G) instead of (Sp(R)\G).

7.3.1. Proposition. A continuous morphism Sp(R) ⟶^{f} X in |Cat|^{o} is flat, conser-

vative, and affine iff its inverse image functor, C_{X} ⟶^{f^{∗}} R − mod, is the composition of an equivalence of categories C_{X} ⟶ (R\H_{f}) − Comod for a coalgebra H_{f} = (H_{f}, δ_{f}) in the category of R-bimodules such that H_{f} is a flat right R-module, and the forgetful functor (R\H_{f}) − mod ⟶ R − mod.

Proof. Let Sp(R) ⟶^{f} X be a flat, conservative, and affine morphism with an inverse

image functor f^{∗}. By 7.2, the functor C_{X} ⟶^{f^{∗}} R − mod is the composition of a category equivalence C_{X} ⟶ (R\G_{f}) − Comod for a comonad G_{f} = (G_{f}, δ_{f}) on Sp(R) and the forgetful functor (R\G_{f}) − mod ⟶ R − mod. Since the comonad G_{f} is continuous, the functor G_{f} is isomorphic to the functor H_{f} ⊗_{R} − for an R-bimodule H_{f} (equal to G_{f}(R)).

The comultiplication G_{f} ⟶^{δ_{f}} G_{f}^{2} induces a comultiplication H_{f} ⟶ H_{f} ⊗_{R} G_{f} (see the argument of 6.6.1).

Conversely, let H = (H, δ) be a coalgebra in the category of R-bimodules, and let f^{∗} denote the forgetful functor

(R\H) − Comod ⟶ R − mod, (M, M → H(M)) ⟼ M.  (1)

The functor f^{∗} has a right adjoint,

L ⟼ H ⊗_{R} L = (H ⊗_{R} L, δ ⊗_{R} L)  (2)

(see the argument of 7.2). The comonad H⊗_{R} is continuous, since the functor H ⊗_{R} − has a right adjoint, Hom_{R}(H, −).

The functor f^{∗} being flat is equivalent to the flatness of H as a right R-module. The assertion follows now from 7.2.

7.3.2. Corollary. Let (C, O) be a ringed category, and let X ⟶^{f} Y be a morphism

of presheaves of sets on C such that its inverse image functor, Qcoh_{Y} ⟶^{f^{∗}} Qcoh_{X}, is flat, conservative, and its direct image functor is conservative and has a right adjoint. Suppose X is representable. Then there exists a coalgebra H in the category of O(X)-bimodules such that f^{∗} is the composition of a category equivalence Qcoh_{Y} ⟶^{∼} (O(X)\H)−Comod and the forgetful functor (O(X)\H) − Comod ⟶ O(X) − mod.

Proof. The assertion follows from 7.3.1.

7.3.3. Example: semiseparated schemes and algebraic spaces. Let X be a scheme, or an algebraic space. Recall that an affine cover {U_{i} →^{u_{i}} X | i ∈ J} of X is called semiseparated if each morphism U_{i} →^{u_{i}} X is affine. A scheme (or an algebraic space) is called semiseparated if it has a semiseparated cover. Evidently, every separated algebraic space (or scheme) is semiseparated.

If {U_{i} → X | i ∈ J} is a semiseparated cover of X, the corresponding morphism

U = ∐_{i∈J} U_{i} ⟶^{π} X

is affine which implies that the space of relations, R = ∐_{i,j∈J} U_{i} ×_{X} U_{j} ≃ U ×_{X} U is affine too. Since morphisms u_{i} are étale, their inverse image functors, u_{i}^{∗} are flat and the family {u_{i}^{∗}| i ∈ J} is conservative. The latter means exactly that an inverse image functor π^{∗} of the morphism π is flat and conservative. It follows by construction, that the inverse images of projections R ⇉ U are flat and conservative (equivalently, faithfully flat). And they are affine, since both R and U are affine.

8. Generalities on finiteness conditions and smooth and étale morphisms. Fix a functor 𝔄 ⟶^{𝔉} 𝔅. In what follows, 𝔄 is a category of ’local’, or ’affine’, objects, 𝔅 is a category of spaces, and 𝔉 assigns to local objects corresponding spaces. In a standard commutative prototype, 𝔅 is the category of locally ringed topological spaces, otherwise called geometric spaces, 𝔄 is the category opposite to the category CRings of commutative unital rings, and the functor 𝔄 ⟶^{𝔉} 𝔅 assigns to every commutative ring R the affine scheme (SpecR, O_{R}).

8.1. Locally finitely presentable objects and morphisms. Given a functor 𝔄 ⟶^{𝔉} 𝔅, we call an object X of 𝔅 of (𝔄, 𝔉)-finite type (resp. (𝔄, 𝔉)-finitely presentable), if for any filtered projective system D ⟶^{𝔇} 𝔄 such that there exists lim(𝔉∘𝔇, the canonical map

colim 𝔅(𝔉 ∘ 𝔇, X) ⟶ 𝔅(lim(𝔉 ∘ 𝔇), X)  (1)

is injective (resp. bijective).

We call a morphism X ⟶^{f} Y of 𝔅 of (𝔄, 𝔉)-finite type (resp. (𝔄, 𝔉)-finitely presentable), if for any filtered projective system D ⟶^{𝔇} 𝔄/Y, the canonical morphism

colim 𝔅/Y (𝔉_{Y} ∘ 𝔇, (X, f)) ⟶ 𝔅/Y (lim(𝔉_{Y} ∘ 𝔇), (X, f))  (2)

is injective (resp. bijective), provided lim(𝔉_{Y} ∘ 𝔇) exists. Here 𝔉_{Y} denotes the canonical functor 𝔉/Y ⟶ 𝔅/Y.

It follows from these definitions that if the category 𝔅 has a final object, •, then an object X of 𝔅 is of (𝔄, 𝔉)-finite type (resp. 𝔄-finitely presentable) iff the unique morphism X ⟶ • is of (𝔄, 𝔉)-finite type (resp. (𝔄, 𝔉)-finitely presentable).

8.1.1. Proposition. Let Σ_{𝔄}^{1} (resp. Σ_{𝔄}^{0}) denote the class of all (𝔄, 𝔉)-finitely presentable morphisms (resp. morphisms of (𝔄, 𝔉)-finite type) of the category 𝔅.

(a) Both Σ_{𝔄}^{0} and Σ_{𝔄}^{1} are closed under compositions and contain all isomorphisms.

(b) If the morphism f in the cartesian square

X′ ⟶^{g′} X
f′ ↓  ↓ f
Y′ ⟶^{g} Y

belongs to Σ_{𝔄}^{i}, then f′ belongs to Σ_{𝔄}^{i}, i = 0, 1.

(c) Suppose that X ⟶^{f} Y and Z ⟶^{h} W are morphisms over an object S which belong

to Σ_{𝔄}^{i}. If X ×_{S} Z and Y ×_{S} W exist, then the morphism X ×_{S} Z ⟶^{f×_{S}h} Y ×_{S} W belongs to Σ_{𝔄}^{i}, i = 0, 1.

(d) If the composition g ∘ f of two morphisms is (𝔄, 𝔉)-finitely presentable and g is of (𝔄, 𝔉)-finite type, then f is (𝔄, 𝔉)-finitely presentable.

Proof. The argument is similar to that of [KR2, 5.12.2].

8.2. Formally smooth, formally unramified and formally étale morphisms. Fix a functor 𝔄 ⟶^{𝔉} 𝔅 and a family, 𝔐, of morphisms of 𝔄 containing all identical morphisms.

8.2.1. Definitions. (i) We call a morphism X ⟶^{f} Y of 𝔅 formally 𝔐-smooth if any commutative diagram

\begin{matrix}𝔉(T) & ⟶^{g} & X \\ 𝔉(ϕ) ↓ & & ↓ f \\ 𝔉(S) & ⟶^{g′} & Y \end{matrix}  (1)

such that ϕ ∈ 𝔐 extends to a commutative diagram

\begin{matrix}𝔉(T) ⟶^{g′} X \\ 𝔉(ϕ) ↓ ↗_{γ} ↓ f \\ 𝔉(S) ⟶^{g} Y \end{matrix}  (2)

(ii) We call X ⟶^{f} Y formally 𝔐-unramified if for any commutative diagram (1) such that ϕ ∈ 𝔐, there exists at most one morphism S ⟶^{γ} X such that the diagram (2) commutes.

(iii) We call X ⟶^{f} Y formally 𝔐-étale if it is both formally 𝔐-smooth and formally 𝔐-unramified.

We denote by 𝔐_{fsm} (resp. 𝔐_{fnr}, resp. 𝔐_{fet}) the class of all formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) morphisms.

8.2.2. Proposition. (a) Each monomorphism is formally unramified and each isomorphism is formally étale.

(b) Composition of formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) morphisms is formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale).

(c) Let X ⟶^{f} Y, Y ⟶^{h} Z be morphisms of 𝔅.

(i) If h ∘ f is formally 𝔐-unramified, then f is formally 𝔐-unramified.

(ii) Suppose h is formally 𝔐-unramified. If X ⟶^{h∘f} Z is formally 𝔐-smooth (resp. formally 𝔐-étale), then f is formally 𝔐-smooth (resp. formally 𝔐-étale).

(d) Let X ⟶^{ξ} S ⟵^{ξ′} X′ and Y ⟶^{ν} S ⟵^{ν′} Y′ be morphisms such that there exist

X×_{S}X′ and Y ×_{S}Y′. Let (X, ξ) ⟶^{f} (Y, ν) and (X′, ξ′) ⟶^{f′} (Y′, ν) be morphisms of objects over S. The morphisms f, f′ are formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) iff the morphism f ×_{S} f′ : X ×_{S} X′ ⟶ Y ×_{S} Y′ has the respective property.

(e) Let X ⟶^{f} S ⟵^{h} Y be such a diagram that there exists a fiber product X ×_{S} Y. If f is formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally étale), then the

canonical projection X ×_{S} Y ⟶^{f′} Y is formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale).

Proof. See [KR2, 6.6].

8.3. Smooth, unramified, and étale morphisms. We call a morphism X ⟶^{f} Y 𝔐-smooth (resp. 𝔐-étale, resp. 𝔐-unramified) if it is (𝔄, 𝔉)-finitely presentable and formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale).

We denote by 𝔐_{sm} (resp. 𝔐_{nr}, resp. 𝔐_{et}) the family of all 𝔐-smooth (resp. 𝔐unramified, resp. 𝔐-étale) morphisms.

We call a morphism X ⟶^{f} Y 𝔐-open immersion if it is an 𝔐-smooth monomorphism.

8.3.1. Proposition. (a) Each monomorphism is 𝔐-unramified and each isomorphism is 𝔐-open immersion.

(b) Composition of 𝔐-smooth (resp. 𝔐-unramified, resp. 𝔐-étale) morphisms is 𝔐-smooth (resp. 𝔐-unramified, resp. 𝔐-étale).

(c) Let X ⟶^{f} Y, Y ⟶^{h} Z be morphisms of 𝔅.

(i) If g∘f is formally 𝔐-unramified and g is of 𝔐-finite type, then f is 𝔐-unramified.

(ii) Suppose g is 𝔐-unramified. If X ⟶^{g∘f} Z is 𝔐-smooth (resp. 𝔐-étale), then f is 𝔐-smooth (resp. 𝔐-étale).

(d) Let X ⟶^{ξ} S ⟵^{ξ′} X′ and Y ⟶^{ν} S ⟵^{ν′} Y′ be morphisms such that there exist

X ×_{S} X′ and Y ×_{S} Y′. Let (X, ξ) ⟶^{f} (Y, ν) and (X′, ξ′) ⟶^{f′} (Y′, ν) be morphisms of objects over S. The morphisms f, f′ are 𝔐-smooth (resp. 𝔐-unramified, resp. 𝔐-étale) iff f ×_{S} f′ : X ×_{S} X′ ⟶ Y ×_{S} Y′ has the respective property.

(e) Let X ⟶^{f} S ⟵^{h} Y be such a diagram that there exists a fiber product X ×_{S} Y.

If f is 𝔐-smooth (resp. 𝔐-unramified, resp. étale), then the projection X ×_{S} Y ⟶^{f′} Y is 𝔐-smooth (resp. 𝔐-unramified, resp. 𝔐-étale).

Proof. The assertion follows from 8.2.2 and 8.1.1.

8.4. Standard examples.

8.4.1. Let 𝔄 be the category CRings^{op}, as in 8.4.1. Let 𝔅 be the category Esp of spaces in the sense of Grothendieck (and [DG]); i.e. 𝔅 is the category of sheaves of sets on 𝔄 for the flat topology. In other words, objects of 𝔅 are functors CRings ⟶ Sets which preserve finite products, and for any faithfully flat ring morphism R ⟶ T, the diagram

X(R) ⟶ X(T) \overrightarrow{⟶} X(T ⊗_{R} T)  (1)

is exact. The functor 𝔄 ⟶^{𝔉} 𝔅 is the Yoneda functor which maps every object R of 𝔄 to the functor 𝔄(−, R) = CRings(R, −) represented by R (here we identify objects of 𝔄 with the corresponding objects of CRings).

Then (𝔄, 𝔉)-finitely presentable morphisms (resp. morphims of (𝔄, 𝔉)-finite type) are precisely locally finitely presentable morphisms (resp. morphisms of locally finite type).

We take as 𝔐 the family of all morphisms of 𝔄 such that the corresponding ring morphism is an epimorphism with a nilpotent kernel. The formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) morphisms are formally smooth (resp. formally unramified, resp. formally étale) in the usual sense. Therefore, 𝔐-smooth, 𝔐unramified, 𝔐-étale morphisms are resp. smooth, unramified and étale. And 𝔐-open immersions are precisely open immersions in the conventional sense.

8.4.2. Let 𝔄 be the opposite category to the category Alg_{k} of associative unital kalgebras, Let 𝔅 be the category of presheaves of sets on 𝔄, i.e. functors Alg_{k} ⟶ Sets, which are local in the following sense: they preserve finite products, and for any faithfully flat k-algebra morphism R ⟶ T, the diagram

X(R) ⟶ X(T) \overrightarrow{⟶} X(T ⋆_{R} T)  (1)

is exact. Here ⋆_{R} denote the ’star’-product of rings over R (which is a traditional name for a push-forward of associative rings). We denote this category by Esp_{N C} and call its objects ’noncommutative spaces’, or simply ’spaces’. The functor 𝔄 ⟶^{𝔉} 𝔅 is the Yoneda embedding, R ⟼ 𝔄(−, R) = Alg_{k}(R, −) (here we identify objects of 𝔄 with the corresponding k-algebras).

It follows that (𝔄, 𝔉)-finitely presentable morphisms (resp. morphims of (𝔄, 𝔉)-finite type) are precisely locally finitely presentable morphisms (resp. morphisms of locally finite type) in the sense of [KR, 5.12.1].

We take as 𝔐 the family of all morphisms of 𝔄 such that the corresponding k-algebra morphism is an epimorphism with a nilpotent kernel. Then formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) morphisms are formally smooth (resp. formally unramified, resp. formally étale) in the sense of [KR2, 5.8].

8.5. Smooth and étale morphisms of ’spaces’. Open immersions.

8.5.1. Cosubspaces and closed immersions. Let X be a ’space’, that is an object of |Cat|^{o}. We call Y a cosubspace of X if C_{Y} is a full subcategory of the category C_{X} which is closed under finite limits and colimits taken in C_{X} and, in addition, the following condition holds:

If M ⇉ N is a pair of arrows such that M ∈ ObC_{Y} (resp. N ∈ ObC_{Y}) and the kernel (resp. cokernel) of the pair M ⇉ N exists in C_{X}, then this kernel (resp. cokernel) belongs to the subcategory C_{Y}. In particular, C_{Y} is strictly full (i.e. it contains with any of its objects all objects of C_{X} isomorphic to this object).

We call a continuous morphism U ⟶^{u} X a closed immersion if its direct image functor, u_{∗}, induces an equivalence between C_{U} and C_{Y} for some cosubspace Y of X. In particular, u_{∗} is a fully faithful.

8.5.1.1. Topologizing subcategories. Suppose C_{X} is an abelian category, and C_{Y} is a subcategory of C_{X}. Then Y is a cosubspace of X iff C_{Y} is a topologizing subcategory; i.e. C_{Y} is closed by direct sums and subquotients (taken in C_{X}).

8.5.2. Thickennings. We call a closed immersion U ⟶^{u} T a thickenning (and say that T is a thickenning of U), if the smallest saturated multiplicative system in HomC_{T} containing u_{∗}(HomC_{U}) coincides with HomC_{T}. The latter condition means precisely that if Y ⟶^{q} X is an exact localization such that the composition q^{∗}u_{∗} maps all arrows to isomorphisms, then C_{Y} is a groupoid.

8.5.2.1. Abelian case. Suppose that the category C_{T} is abelian. Then a continuous morphism U ⟶^{u} T is a thickenning iff its direct image functor, u_{∗}, induces an equivalence between C_{U} and a topologizing subcategory of C_{T}, and the smallest thick subcategory of C_{T} containing u_{∗}(C_{U}) coincides with C_{T}.

8.5.3. Formally smooth, formally unramified, and formally étale morphisms. Fix a ’space’ S, and consider the category |Cat|_{S}^{o} and its full subcategory Aff_{S}. Recall that |Cat|_{S}^{o} is a full subcategory of |Cat|^{o}/S whose objects are pair (X, f), where

X ⟶^{f} S is a continuous morphism; and Aff_{S} is the full subcategory of |Cat|^{o}/S whose objects are pairs (Y, g), where Y ⟶^{g} S is an affine morphism.

Let X ⟶^{f} Y be a morphism in |Cat|^{o}/S. We call the morphism f formally smooth (resp. formally unramified) if for every commutative diagram

U ⟶^{g′} X
u ↓  ↓ f
T ⟶^{g} Y

of S-’spaces’ such that U and T belong to Aff_{S} and U ⟶^{u} T is a thickening, there exists (resp. at most one) morphism T ⟶^{γ} X such that the diagram

U ⟶^{g′} X

u ↓ ↗_{γ} ↓ f

T ⟶^{g} Y

commutes. The morphism X ⟶^{f} Y is called formally étale if it is both formally smooth and formally unramified.

We call an S-’space’ (X, ξ) formally smooth (resp. formally unramified, resp. formally étale), if the morphism (X, ξ) ⟶^{ξ} (S, id_{S}) is formally smooth (resp. formally unramified, resp. formally étale).

8.5.3.1. Proposition. (a) The class of formally smooth (resp. formally unramified, resp. formally étale) morphisms is closed under composition and contains all isomorphisms.

(b) If h ∘ f is formally unramified, then f is formally unramified.

(c) Suppose a morphism h is formally unramified. If X ⟶^{h∘f} Z is formally smooth (resp. formally étale), then f is formally smooth (resp. formally étale).

(d) Let X ⟶^{ξ} T ⟵^{ξ′} X′ and Y ⟶^{ν} T ⟵^{ν′} Y′ be morphisms such that there exist

X ×_{T} X′ and Y ×_{T} Y′. Let (X, ξ) ⟶^{f} (Y, ν) and (X′, ξ′) ⟶^{f′} (Y′, ν) be morphisms of objects over T. The morphisms f, f′ are formally smooth (resp. formally unramified, resp. formally étale) iff the morphism f ×_{T} f′ : X ×_{T} X′ ⟶ Y ×_{T} Y′ has the respective property.

(e) Let X ⟶^{f} T ⟵^{h} Y be such a diagram that there exists a fiber product X ×_{T} Y. If f is formally smooth (resp. formally unramified, resp. formally étale), then the canonical

projection X ×_{T} Y ⟶^{f′} Y is formally smooth (resp. formally unramified, resp. formally étale).

Proof. The assertion follows from [KR2 6.6].

8.5.5. Locally finitely presentable S-’spaces’ and morphisms of S-’spaces’. Let S be a ’space’. We take as 𝔅 the category |Cat|_{S}^{o}. If 𝔄 is the subcategory Aff_{S} of affine S-’spaces’, then we shall call 𝔄-presentable S-’spaces’ (resp. morphisms of S- ’spaces’) locally presentable. Similarly, we call (morphisms of) S-’spaces’ of 𝔄-finite type locally of finite type.

8.5.6. Smooth, unramified, and étale S-’spaces’ and morphisms of S- ’spaces’. Open immersions. We call a morphism X ⟶^{f} Y of S-’spaces’ smooth (resp. étale, resp. unramified) if it is locally finitely presentable and formally smooth (resp. formally étale, resp. formally unramified).

We call a morphism X ⟶^{f} Y of S-’spaces’ an open immersion if it is a smooth monomorphism.

An S-’space’ (X, ξ) is called smooth (resp. unramified, resp. étale) iff it is formally smooth (resp. formally unramified, resp. formally étale) and locally finitely presentable.

Since (S, id_{S}) is a final object in |Cat|_{S}^{o}, an S-’space’ (X, ξ) is smooth (resp. unramified, resp. étale) if the unique morphism (X, ξ) ⟶ (S, id_{S}) is smooth (resp. unramified, resp. étale).

9. Locally affine ’spaces’ and schemes.

9.1. Generalities on glueing. Fix a functor 𝔄 ⟶^{𝔉} 𝔅.

We assume that the category 𝔅 is endowed with a quasi-pretopology, τ. The latter is a function which assigns to each object X of 𝔅 a family, τ_{X}, of covers of X. An element of τ_{X} is set of arrows {U_{i} →^{u_{i}} X | i ∈ J}. We assume that any isomorphism forms a cover, and the composition of covers is a cover.

A cover of the form {𝔉(U_{i}) →^{u_{i}} X | i ∈ J} of an object X is called a (𝔄, τ)-cover, or simply 𝔄-cover, if τ is fixed.

An object X of 𝔅 is called locally (𝔄, τ)-affine (or locally 𝔄-affine, if no ambiguity arises) if it has an (𝔄, τ)-cover.

We denote by Sp_{𝔄,τ} the full subcategory of the category 𝔅 whose objects are locally (𝔄, τ)-affine.

9.1.1. Quasi-finite locally 𝔄-affine objects. Given a quasi-pretopology τ on 𝔅, let τ_{𝔣} denote the quasi-pretopology formed by all finite covers of τ. We call an object X of 𝔅 quasi-finite locally (𝔄, τ)-affine if it is locally (𝔄, τ_{𝔣})-affine.

9.1.2. 2-𝔄-covers and 2-locally 𝔄-affine objects. An (𝔄, τ)-cover {𝔉(U_{i}) →^{u_{i}} X | i ∈ J} will be called a 2-(𝔄, τ)-cover (or 2-𝔄-cover), if for any i, j ∈ J, there exists a set of morphisms (U_{i}, u_{i}) ⟵ (U_{ij}^{ν}, u_{ij}^{ν}) ⟶ (U_{j}, u_{j}), ν ∈ J_{ij} in the category 𝔉/X such that the corresponding set of morphisms {𝔉(U_{ij}^{ν}) ⟶ 𝔉(U_{i}) ×_{X} 𝔉(U_{j}) | ν ∈ J_{ij}} is a cover for any i, j ∈ J. We call the diagram

(U_{i}, u_{i}) ⟵ (U_{ij}^{ν}, u_{ij}^{ν}) ⟶ (U_{j}, u_{j}), ν ∈ J_{ij}, i, j ∈ J,  (1)

(in the category 𝔉/X) a diagram of relations of the 2-cover 𝔘 = {𝔉(U_{i}) →^{u_{i}} X | i ∈ J}. We call an object X of 𝔅 2-locally 𝔄-affine if it has a 2-locally 𝔄-affine cover.

9.1.3. Weakly semiseparated covers. We call an 𝔄-cover {𝔉(U_{i}) →^{u_{i}} X | i ∈ J} weakly semiseparated if for any i, j ∈ J, there exists a diagram

(U_{i}, u_{i}) ⟵ (U_{ij}, u_{ij}) ⟶ (U_{j}, u_{j})

in 𝔉/X such that the square

𝔉(U_{ij}) ⟶ 𝔉(U_{j})
↓ ↓
𝔉(U_{i}) ⟶ X

is cartesian; in particular, the object 𝔉(U_{i})×_{X} 𝔉(U_{j}) is isomorphic to an object of the form 𝔉(U_{ij}). It follows that any weakly semiseparated 𝔄-cover is a 2-𝔄-cover.

We say that an object X of 𝔅 is 𝔄-weakly semiseparated if it has a weakly semiseparated 𝔄-cover.

9.2. 𝔄-Representable morphisms and covers, and locally 𝔄-representable objects. If E is a subcategory of 𝔅 such that ObE = Ob𝔅, we denote by τ^{E} the quasipretopology on 𝔅 formed by all covers {U_{i} →^{u_{i}} X | i ∈ J} in τ such that all morphisms u_{i} belong to E. Given a functor 𝔄 ⟶^{𝔉} 𝔅, we have a natural choice of the subcategory E, which is the subcategory of 𝔄-representable morphisms described below.

We call a morphism X ⟶^{f} Y of the category 𝔅 𝔄-representable if for any morphism 𝔉(V) ⟶^{g} Y, there exist morphisms 𝔉(W) ⟶^{g̃} X and W ⟶^{v} V such that

𝔉(W) ⟶^{g̃} X
𝔉(v) ↓  ↓ f
𝔉(V) ⟶^{g} Y

is a cartesian square; in particular, it commutes.

9.2.1. Lemma. (a) Every isomorphism in 𝔅 is 𝔄-representable.

(b) The composition of 𝔄-representable morphisms is 𝔄-representable.

(c) If in a cartesian square

X′ ⟶ X
f′ ↓ ↓ f
Y′ ⟶ Y

the morphism X ⟶^{f} Y is 𝔄-representable, then the morphism X′ ⟶^{f′} Y′ is 𝔄-representable.

Proof. The assertion (a) is obvious. The assertions (b) and (c) follow from the general nonsense fact that the composition

of cartesian squares is a cartesian square: if in the commutative diagram

X′′ ⟶  X′ ⟶ X
f′′ ↓ f′ ↓ ↓ f
Y′′ ⟶  Y′ ⟶ Y

both squares are cartesian, then the square

X′′ ⟶ X
f′′ ↓ ↓ f
Y′′ ⟶ Y

is cartesian. Details are left to the reader.

9.2.2. Representable covers. We call a cover {U_{i} ⟶^{u_{i}} X | i ∈ J} 𝔄-representable if each morphism u_{i} of the cover is 𝔄-representable. We denote by τ^{𝔄} the function which assigns to each object X of the category 𝔅 the set, τ_{X}^{𝔄}, of 𝔄-representable covers of X.

9.2.2.1. Lemma. The function τ^{𝔄} is a quasi-pretopology on 𝔅. If τ is a pretopology, then τ^{𝔄} is a pretopology.

Proof. The assertion is a corollary of 9.2.1.

9.2.3. Locally 𝔄-representable objects. Evidently, every representable 𝔄-cover is weakly semiseparated (cf. 9.1.3.). In particular, it is a 2-𝔄-cover.

We say that an object X of 𝔅 is locally 𝔄-representable if it has a representable 𝔄-cover. Thus, every locally 𝔄-representable object is locally 𝔄-affine.

9.3. Coinduced pretopology. Let 𝔄 ⟶^{𝔉} 𝔅 be a functor, and let 𝔗 be a quasipretopology on 𝔄. The coinduced quasi-pretopology, 𝔗^{𝔉}, on 𝔅 is defined as follows: a set of arrows {U_{i} ⟶^{u_{i}} X | i ∈ J} is a cover of X iff for any morphism 𝔉(V) ⟶^{g} X, there exists a cover {V_{j} ⟶^{v_{j}} V | j ∈ I} ∈ 𝔗_{V} such that for every j ∈ I, the morphism g ∘ 𝔉(v_{j}) : 𝔉(V_{j}) ⟶ X factors through u_{i} for some i ∈ J (cf. [R4, 4.4]).

9.3.1. Proposition. Suppose 𝔅 is a category with fiber products. Then the coinduced quasi-pretopology 𝔗^{𝔉} on 𝔅 is a pretopology.

Proof. Let {U_{i} ⟶^{u_{i}} X | i ∈ J} be a cover in 𝔗^{𝔉} and Y ⟶^{g} X an arbitrary morphism. The claim (equivalent to the proposition) is that the set of arrows {U_{i} ×_{X} Y ⟶^{ū_{i}} Y | i ∈ J} is a cover in 𝔗^{𝔉}.

In fact, let 𝔉(V) ⟶^{v} Y be an arbitrary morphism. Since {U_{i} ⟶^{u_{i}} X | i ∈ J} is a cover, there exists a cover {V_{j} ⟶^{v_{i}} V | j ∈ I} in 𝔗 such that for any j ∈ I, there exists

i_{j} ∈ J and a morphism 𝔉(V_{j}) ⟶^{v̄_{j}} U_{i_{j}} which make the diagram

\begin{matrix}𝔉(V_{j}) & ⟶^{v̄_{j}} & U_{i_{j}} \\ 𝔉(v_{j}) ↓ & & ↓ u_{i_{j}} \\ 𝔉(V) & ⟶^{g∘v} & X \end{matrix}  (1)

commute. The commutativity of (1) implies the existence if a unique morphism 𝔉(V_{j}) ⟶^{v′_{j}} U_{i_{j}} ×_{X} Y such that the diagram

\begin{matrix}𝔉(V_{j}) & ⟶^{v̄′_{j}} & U_{i_{j}} ×_{X} Y \\ 𝔉(v_{j}) ↓ & & ↓ u_{i_{j}} \\ 𝔉(V) & ⟶^{v} & Y \end{matrix}  (2)

commutes and the morphism 𝔉(V_{j}) ⟶^{v̄_{j}} U_{i_{j}} is the composition of 𝔉(V_{j}) ⟶^{v′_{j}} U_{i_{j}} ×_{X} Y and the canonical projection U_{i_{j}} ×_{X} Y ⟶ U_{i_{j}}. This shows that {U_{i} ×_{X} Y ⟶^{ū_{i}} Y | i ∈ J} is a cover in 𝔗^{𝔉}.

9.3.2. Note. Let 𝔄 be a category with fiber products and 𝔗 a quasi-pretopology on 𝔄. Taking 𝔉 = Id_{𝔄}, we obtain the coinduced pretopology, 𝔗^{g}, on 𝔅 = 𝔄. The pretopology 𝔗^{g} is the finest pretopology among those pretopologies on 𝔄 which are coarser than 𝔗.

9.4. Standard commutative examples.

9.4.1. Geometric spaces and schemes. Let 𝔅 be the category of locally ringed topological spaces which we call otherwise geometric spaces, 𝔄 the category opposite to the category CRings of commutative unital rings, 𝔉 the functor 𝔄 ⟶ 𝔅 which assigns to every commutative ring its spectrum. The pretopology on 𝔅 is the standard Zariski pretopology given by families of open immersions covering the underlying space: a set {(U_{i}, O_{U_{i}}) →^{u_{i}} (X, O_{X}) | i ∈ J} of open immersions is a cover iff ⋃_{i∈J} U_{i} = X.

Then locally 𝔄-affine objects of 𝔅 are arbitrary schemes.

9.4.1.1. Semiseparated schemes. Locally 𝔄-representable objects of 𝔅 are precisely semiseparated schemes. Recall that a scheme X = (X, O) is called semiseparated if it has an affine cover {U_{i} →^{u_{i}} X | i ∈ J} such that each morphism U_{i} →^{u_{i}} X is representable. Clearly, every semiseparated scheme is weakly separated.

9.4.2. Quasi-finite 𝔄-objects. Let 𝔄, 𝔅, and 𝔄 ⟶^{𝔉} 𝔅 are same as in 9.4.1. Then quasi-finite 𝔄-objects (i.e. locally (𝔄, τ_{ℨ_{𝔣}})-affine objects, where τ_{ℨ_{𝔣}} is the subpretopology of τ_{ℨ} formed by finite covers, cf. 9.1.1) are exactly quasi-compact schemes.

Notice that 2-locally (𝔄, τ_{ℨ_{𝔣}})-affine objects are quasi-compact quasi-separated schemes.

9.4.3. Spaces as sheaves of sets. Let 𝔄 be the category CRings^{op}, as in 9.2.1. Let 𝔅 be category Esp of sheaves of sets on CRings^{op} for the fpqc topology, and let 𝔄 ⟶^{𝔉} 𝔅 be the Yoneda embedding: R ⟼ CRings(R, −) (see 8.4.1).

Zariski covers in CRings^{op} = 𝔄 are given by sets of morphisms {R ⟶ R_{i} | i ∈ J} such that R_{i} is a localization of R at an element of R (that is at the multiplicative set generated by this element), and ⋃_{i∈J} Spec(R_{i}) = Spec(R). Zariski covers form a (Zariski) pretopology, 𝔗_{ℨ}. We define Zariski pretopology on 𝔅 = Esp as the pretopology coinduced by 𝔗_{ℨ} (cf. 9.3).

Locally affine 𝔄-objects in this setting are schemes in the sense of [DG], that is schemes realized as functors CRings ⟶ Sets. The functor S which assigns to each geometric space X = (X, O) the functor R ⟼ Hom((SpecR, O_{R}), X) establishes an equivalence between geometric schemes and functorial schemes.

Representable morphisms in 𝔅 are corepresentable functors CRings ⟶ Sets. The functor S induces an equivalence of the category of semiseparated schemes (cf. 9.4.1.1) and the category of locally 𝔄-representable objects of 𝔅.

Replacing the Zariski pretopology 𝔗_{ℨ} by its finite version, 𝔗_{ℨ_{𝔣}}, we obtain a full subcategory of the category of locally 𝔄-affine objects formed by quasi-finite locally 𝔄-affine objects. The functor S induces an equivalence of this category and the category of quasicompact geometric schemes. The functor S induces an equivalence of the category of 2-locally (𝔄, 𝔗_{ℨ_{𝔣}})-affine objects and the category of quasi-compact, quasi-separated geometric schemes.

9.5. Standard noncommutative examples. We take as 𝔄 the category Aff_{k} = Alg_{k}^{op} opposite to the category of associative unital k-algebras, together with one of the canonical quasi-pretopologies defined below.

9.5.1. Canonical quasi-pretopologies on Aff_{k} = Alg_{k}^{op}. We call the image in Aff_{k} of a set of k-algebra morphisms {R ⟶ R_{i} | i ∈ J} an fpqc cover if all morphisms R ⟶ R_{i} are flat (i.e. R_{i} is a flat right R-module), and there is a finite subset I of J such that the family of functors {R_{i} ⊗_{R} | i ∈ I} is conservative. The composition of fpqc covers is an fpqc cover, and any faithfully flat ring morphism (in particular, any ring isomorphism) R ⟶ S forms an fpqc cover. Thus, fpqc covers form a quasi-pretopology which we denote by τ_{fpqc}.

We call an fpqc cover {SpecR_{i} ⟶ SpecR | i ∈ J} an lqc cover if the corresponding ring morphisms R ⟶ R_{i} are localizations, or, equivalently, the corresponding ’restriction of scalars’ functor R_{i} − mod ⟶ R − mod is full (hence fully faithful). It follows from [R4, 2.6.3.1] that lqc covers form pretopology on Aff which we denote by τ_{lqc}.

We call an fpqc cover {R ⟶ R_{i} | i ∈ J} an fppf cover if it consists of finitely presentable morphisms. We denote the fppf quasi-pretopology by τ_{fppf} .

A set of algebra morphisms {R ⟶ R_{i} | i ∈ J} defines a Zariski cover if it consists of finitely presentable localizations and the family of functors {R_{i} ⊗_{R} | i ∈ J} is conservative. Zariski covers form a pretopology which we denote by τ_{ℨ} and call it the Zariski pretopology.

9.5.2. Noncommutative schemes as presheaves of sets. Let 𝔅 be the category Esp_{N C} of sheaves of sets on Aff_{k} for fpqc quasi-pretopology. In other words, objects of 𝔅 are functors Rings ⟶ Sets which preserve finite products, and for any faithfully flat ring morphism R ⟶ T, the diagram

X(R) ⟶ X(T) \overrightarrow{⟶} X(T ⋆_{R} T)  (1)

is exact. The functor 𝔄 ⟶^{𝔉} 𝔅 is the Yoneda embedding, R ⟼ Alg_{k}(R, −) (see 8.4.2).

Let τ o_{ℨ} denote the pretopology on 𝔅 coinduced by the Zariski pretopology τ_{ℨ} via the functor 𝔉. We define schemes as locally (𝔄, τ o_{ℨ})-affine objects of 𝔅.

9.5.3. Remark on fpqc-locally affine spaces. Let the category 𝔅 and the functor 𝔄 ⟶^{𝔉} 𝔅 be the same as in 9.5.2. But, we take the fpqc quasi-pretopology on 𝔄 = Aff_{k} instead of the Zariski pretopology. Let τ o_{fpqc} denote the pretopology coinduced on 𝔅 by the fpqc quasi-pretopology on 𝔄 = A^{op}. Applying the formalism of 9.1, we obtain locally (𝔄, τ o_{fpqc})-affine spaces.

This approach, however, is less satisfactory in the case of general fpqc covers, than in the case of Zariski covers. The reason is that fpqc covers do not form a pretopology; hence the operation of coinduction decimates the original quasi-pretopology on Aff_{k}. Fortunately, this inconvenience is easily avoided by defining fpqc quasi-pretopology directly on the category 𝔅.

9.6. Flat quasi-pretopologies in |Cat|^{o}. Let 𝔅 = |Cat|^{o}. We call a set of morphisms {U_{i} →^{u_{i}} X | i ∈ J} in |Cat|^{o} a weakly flat cover if all u_{i} are weakly flat and the set of their inverse image functors, {u_{i}^{∗}| i ∈ J}, is conservative. This defines a weakly flat quasi-pretopology, τ^{w}, on the category |Cat|^{o}.

9.6.1. Finiteness conditions. We call a weakly flat cover an fpqc cover, if it contains a finite subcover. We denote the corresponding quasi-pretopology by τ_{fpqc}.

Let 𝔈 be a set of types of diagrams. We denote by τ 𝔈_{fpqc} the quasi-pretopology defined as follows: {U_{i} ⟶^{u_{i}} X | i ∈ J} belongs to τ 𝔈_{fpqc} iff it is a weakly flat fpqc cover such that all direct image functors, u_{i∗} preserve colimits from 𝔈.

We denote by τ af_{fpqc} the quasi-pretopology formed by weakly flat fpqc covers which consist of affine morphisms. We denote by τ_{1}^{w} the quasi-pretopology generated by weakly flat covers which consist of one morphism.

Finally, we denote by τ_{1}^{af} the quasi-pretopology generated by weakly flat covers which consist of one affine morphism.

9.6.2. Semiseparated covers. We call a weakly flat cover {U_{i} →^{u_{i}} X | i ∈ J} semiseparated if all morphisms u_{i} are affine. We denote the corresponding quasipretopology by τ^{af}.

9.6.3. Proposition. Let 𝔘 = {U_{i} →^{u_{i}} X | i ∈ J} be a weakly flat cover and U = ∐_{i∈J} U_{i} ⟶^{u} X the canonical morphism corresponding to the cover 𝔘.

(a) If the category C_{X} has products of J objects, then the morphism U ⟶^{u} X is weakly flat and conservative.

(b) If the category C_{X} is additive and the cover 𝔘 is finite and semiseparated (i.e. every morphism u_{i} is affine), then the morphism U ⟶^{u} X is affine.

Proof. The family of inverse image functors 𝔘 = {C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}} | i ∈ J} is conservative iff the corresponding functor

C_{X} ⟶^{u^{∗}} ∏_{i∈J} C_{U_{i}} = C_{U}, x ⟼ (u_{i}^{∗}(x)|i ∈ J),

is conservative. Similarly, the functors u_{i}^{∗} preserve kernels of coreflexive pairs of arrows for all i ∈ J iff the functor u^{∗} has the same property.

(a) Suppose the category C_{X} has products of J objects. Then the functor u_{∗} : (a_{i}|i ∈ J) ⟼ ∏_{i∈J} u_{i∗}(a_{i}) is a right adjoint to the functor u^{∗}.

(b) If every direct image functor u_{i∗} is conservative, then the functor u_{∗} is conservative. If the category C_{X} is additive and the cover 𝔘 is finite (i.e. J is finite), then u_{∗}(a_{i}|i ∈ J) = ∐_{i∈J} u_{i∗}(a_{i}) for any object (a_{i}|i ∈ J) of the category C_{U}, and for any x ∈ ObC_{X}, we have:

C_{X}(u_{∗}(a_{i}|i ∈ J), x) = C_{X}( ∐_{i∈J} u_{i∗}(a_{i}), x) ≃ ∏_{i∈J} C_{X}(u_{i∗}(a_{i}), x) ≃

∏_{i∈J} C_{X}(a_{i}, u_{i}^{!}(x)) = C_{U}((a_{i}|i ∈ J), (u_{i}^{!}(x)|i ∈ J)).

Here u_{i}^{!} is a right adjoint to the direct image functor u_{i∗}. This shows that the functor

C_{X} ⟶^{u^{!}} C_{U}, x ⟼ (u_{i}^{!}(x)|i ∈ J),

is a right adjoint to the functor u_{∗}.

9.7. Locally affine morphisms. Relative schemes. Fix a ’space’ S, and consider the category 𝔅 = |Cat|_{S}^{o}. Recall that |Cat|_{S}^{o} is a full subcategory of |Cat|^{o}/S whose

objects are pairs (X, f), where X ⟶^{f} S is a continuous morphism.

We assume that the category C_{S} has cokernels of reflexive pairs of arrows. There are two extreme choices of the category of ’local’ (or ’affine’) objects 𝔄. The largest (in a certain sense) choice is the category 𝔄 = Aff_{S}^{w} of weakly affine morphisms to S. The other extremity is the category Aff_{S} of affine morphisms to S (as in 8.5.3). Intermediate choices are categories Aff_{S}^{𝔖}, where 𝔖 is a set of types of diagrams: objects of Aff_{S}^{𝔖} are pairs (X, f), where f is a continuous morphism X ⟶ S such that f_{∗} is weakly affine and preserves colimits of diagrams of 𝔖.

In each of these cases, the functor 𝔄 ⟶^{𝔉} 𝔅 is the inclusion functor.

Any quasi-pretopology on |Cat|^{o} induces a quasi–pretopology on |Cat|^{o}/S. In particular, each of the canonical quasi-pretopologies on |Cat|^{o} defined in 9.7 induces a canonical quasi-pretopology on 𝔅 = |Cat|_{S}^{o}. Thus, we have the quasi-pretopology τ_{fpqc} given by finite weakly flat covers and its versions, τ 𝔈_{fpqc} and τ af_{fpqc} (cf. 9.6.1).

9.7.1. Locally affine S-’spaces’. We call an object (X, X →^{f} S) of the category 𝔅 = |Cat|_{S}^{o} a locally affine S-’space’, if it is a locally 𝔄-affine object of |Cat|_{S}^{o} for the quasi-pretopology τ_{fpqc}, and 𝔄 = Aff_{S}.

An object (X, X →^{f} S) of |Cat|_{S}^{o} is called a semiseparated locally affine S-’space’, if it is a locally 𝔄-affine object of |Cat|_{S}^{o} with respect to the quasi-pretopology τ af_{fpqc} and

𝔄 = Aff_{S}. In other words, (X, X →^{f} S) has a finite weakly flat Aff_{S}-affine cover which consists of affine morphisms.

Restricting to fpqc covers {U_{i} ⟶^{u_{i}} X | i ∈ J} such that all morphisms u_{i} are flat localizations, we obtain quasi-pretopologies τ_{lpqc}, τ 𝔈_{lpqc}, and τ af_{lpqc}.

9.7.2. Relative schemes. We call a conservative set {U_{i} ⟶^{u_{i}} X | i ∈ J} of morphisms of |Cat|_{S}^{o} a Zariski cover, if all morphisms u_{i} are locally finitely presentable localizations. Zariski covers form a quasi-pretopology, τ_{ℨ}.

We call an object (X, X →^{f} S) of the category |Cat|_{S}^{o} an S-scheme, if it is a locally 𝔄-affine object of |Cat|_{S}^{o} with respect to the quasi-pretopology τ_{ℨ} and 𝔄 = Aff_{S}.

Taking Zariski covers which are fpqc covers, we obtain the corresponding versions of Zariski quasi-pretopology: τ_{ℨ}, τ_{ℨ}^{𝔈}, and τ_{ℨ}^{af}.

We call an object (X, X →^{f} S) of |Cat|^{o}/S a semiseparated S-scheme, if it is a locally 𝔄-affine object of |Cat|^{o}/S with respect to the quasi-pretopology τ af_{ℨ} and 𝔄 = Aff_{S}. In

other words, (X, X →^{f} S) has a finite weakly flat Aff_{S}-affine Zariski cover which consists of affine localizations.

9.8. Locally S-affine ’spaces’ and schemes. Fix an object S of the category 𝔅 = |Cat|^{o}. One can take as 𝔄 the category Aff_{S}^{w}, or Aff_{S}, or Aff_{S}^{𝔖} for some set 𝔖 of types of diagrams. This time, 𝔄 ⟶^{𝔉} 𝔅 is the natural functor which maps an object (X, X → S) of the category 𝔄 to X. We call an object X of 𝔅 S-locally affine ’space’ if it is a locally (𝔄, 𝔉)-affine object with respect to the quasi-pretopology τ_{fpqc}.

We call a conservative set {𝔉(U_{i}) ⟶^{u_{i}} X | i ∈ J} of morphisms of |Cat|_{S}^{o} a Zariski cover, if all morphisms u_{i} are (𝔄, 𝔉)-finitely presentable localizations. Zariski covers form a quasi-pretopology which we denote by τ_{ℨ}, as in the relative case sketched in 9.7.2.

We call an object (X, X →^{f} S) of the category |Cat|_{S}^{o} an S-scheme, if it is a locally 𝔄-affine object of |Cat|_{S}^{o} with respect to the quasi-pretopology τ_{ℨ} and 𝔄 = Aff_{S}.

9.8.1. Locally affine ℤ-spaces and ℤ-schemes. Let S = Spℤ (i.e. C_{S} is the category of abelian groups). We call an object X of 𝔅 locally affine ℤ-space (resp. a ℤscheme) if it is a locally 𝔄-affine object with respect to the quasi-pretopology τ_{fpqc} (resp. the Zariski quasi-pretopology τ_{ℨ}).

9.8.2. Note. Let 𝔄 be the category Rings^{op}, and let 𝔉 be the functor Sp which assigns to each associative unital ring its categoric spectrum. Then the category of locally (𝔄, 𝔉)-affine ’spaces’ is isomorphic to the category of locally affine ℤ-’spaces’ defined in 9.8.1. Similarly, the category of (𝔄, 𝔉)-schemes (defined in an obvious way) is isomorphic to the category of ℤ-schemes.

9.9. The structure of locally affine ’spaces’. Fix a ’space’ S. Let 𝔄 be the category Aff_{S} of affine morphisms to S, or the category Aff_{S}^{𝔖} for some set of diagram types 𝔖, and let 𝔄 ⟶^{𝔉} 𝔅 = |Cat|^{o} be the forgerful functor (X, X → S) ⟼ X.

Suppose that C_{S} has finite products and cokernels of reflexive pairs of arrows. Let

{𝔉(U_{i}, ū_{i}) = U_{i} ⟶^{u_{i}} X | i ∈ J} be a finite 𝔄-cover of X, and let U = ∐_{i∈J} U_{i} ⟶^{u} X be

the corresponding morphism.

If C_{X} has finite products, then u is a continuous morphism (see 9.6.3), hence u is weakly flat and conservative. By Beck’s theorem, X is isomorphic to Sp^{o}(U\G_{u}); i.e. the category C_{X} is equivalent to the category of (U\G_{u})-comodules.

10. ’Spaces’ of morphisms and flat descent.

10.1. ’Spaces’ of morphisms. For any two objects X, Y of the category |Cat|^{o}, we denote by Hom(X, Y) the object of |Cat|^{o} such that C_{Hom(X,Y)} is the category Cat^{op}(X, Y) of (inverse image) functors from C_{Y} to C_{X}.

Let • be the standard initial object of the category |Cat|^{o}, i.e. C_{•} is the category with one morphism. For any object X of |Cat|^{o}, there are natural isomorphisms:

Hom(•, X) ≃ • and Hom(X, •) ≃ X.

As it is observed in 2.2, |Cat|^{o}(X, •) ≃ |X|, where |X| is the set of isomorphism classes of objects of C_{X}. In particular, |Cat|^{o}(Hom(X, Y), •) ≃ |Cat|^{o}(X, Y).

Let X, Y, Z be objects of |Cat|^{o}. The composition of functors (1-morphisms of the category Cat)

Cat(C_{X}, C_{Y}) × Cat(C_{Y} , C_{Z}) ⟶ Cat(C_{X}, C_{Z})

can be rewritten as

C_{Hom(Y,X)} × C_{Hom(Y,X)} = C_{Hom(Y,X) ∐ Hom(Z,Y)} ⟶ C_{Hom(Z,X)}

and interpreted as a inverse image functor of a morphism

Hom(Z, X) ⟶ Hom(Y, Z) ∐ Hom(X, Y)  (1)

Applying to (1) the functor |Cat|^{o}(−, •), we obtain the composition in |Cat|^{o}:

|Cat|^{o}(Y, X) × |Cat|^{o}(Z, Y) ⟶ |Cat|^{o}(Z, X).

10.1.1. Remark. One can view Cat as a monoidal category with the monoidal structure given by the product of categories. The unit object is the category C_{•}, and for any two objects, C_{X}, C_{Y}, of the category Cat, the category Hom(X, Y) = Hom(C_{Y} , C_{X}) of functors C_{Y} ⟶ C_{X} is the inner hom from C_{Y} to C_{X}. Thus, we have an enriched monoidal category. Formally dualizing this structure, we obtain the one described above, i.e. the category |Cat|^{o} enriched by ’spaces’ of morphisms Hom(X, Y) for any pair of objects X, Y.

10.1.2. Morphisms of S-’spaces’. Fix an object S of the category |Cat|^{o} and consider the category |Cat|^{o}/S of objects over S. For any two objects, X = (X, f), Y = (Y, g) of |Cat|^{o}/S, we denote by Hom_{S}(X , Y) the object of |Cat|^{o} such that C_{Hom_{S}(X ,Y)} is the full subcategory of the category C_{Hom(X,Y)} = Hom(C_{Y} , C_{X}) formed by those functors ϕ^{∗} : C_{Y} ⟶ C_{X} for which the diagram

C_{Y} ⟶^{ϕ^{∗}} C_{X}

g^{∗} ↖ ↗_{f^{∗}}

C_{S}

quasi commutes. If ϕ^{∗} is an object of C_{Hom_{S}(X ,Y)} and ψ^{∗} is an object of C_{Hom_{S}(Y,Z)}, then ϕ^{∗}ψ^{∗} is an object of C_{Hom_{S}(X ,Z)}. Thus, we have the enriched category |Cat|^{o}/S of S-’spaces’ such that the forgetful functor |Cat|^{o}/S ⟶ |Cat|^{o} defines a morphism of enriched categories.

10.2. Certain enriched categories of morphisms. For any two objects X and Y of the category |Cat|^{o}, we denote by Hom^{w}(X, Y), Hom_{w}(X, Y), and Hom^{c}(X, Y) objects of |Cat|^{o} defined as follows:

C_{Hom^{w}(X,Y)} is the full subcategory of C_{Hom(X,Y)} formed by weakly right exact functors C_{Y} ⟶ C_{X}, i.e. functors which preserve cokernels of reflexive pairs of arrows;

C_{Hom_{w}(X,Y)} is the full subcategory of C_{Hom(X,Y)} formed by weakly left exact functors, i.e. functors C_{Y} ⟶ C_{X} which preserve kernels of coreflexive pairs of arrows.

C_{Hom^{c}(X,Y)} is the full subcategory of C_{Hom(X,Y)} whose objects are continuous functors C_{Y} ⟶ C_{X}. In particular, C_{Hom^{c}(X,Y)} is a full subcategory of the category C_{Hom^{w}(X,Y)}.

Continuous morphisms play a special role, as the following example shows.

10.2.1. Example. For any unital associative rings R, S, there is a natural isomorphism Hom^{c}(Sp(R), Sp(S)) ≃ Sp(S ⊗ R^{o}).

In fact, the category of continuous functors from S − mod = C_{Sp(S)} to R − mod = C_{Sp(R)} is equivalent to the category of (R, S)-bimodules, which, in turn, is isomorphic to the category of R ⊗ S^{o}-modules.

10.2.1.1. Note. Suppose that S and R are commutative rings. Then S ⊗ R^{o} = S ⊗ R is a commutative ring. Thus, if U and V are spectra of commutative rings, then Hom^{c}(U, V) is the spectrum of a commutative ring.

10.2.2. Generalizations of Hom^{w} and Hom_{w}. Let 𝔖 and 𝔗 be sets of types of diagrams. For any two objects, X, Y of |Cat|^{o}, we denote by Hom^{𝔗}(X, Y) (resp. by Hom_{𝔖}(X, Y)) the object of |Cat|^{o} such that C_{Hom^{𝔗}(X,Y)} (resp. C_{Hom_{𝔖}(X,Y)}) is a full subcategory of C_{Hom(X,Y)} whose objects are all functors C_{Y} ⟶ C_{X} which preserve colimits of diagrams from 𝔗 (resp. limits of diagrams from 𝔖). Besides Hom^{w} and Hom_{w}, the cases of special interest are when 𝔖 (or 𝔗, or both) consist of all finite diagrams, or 𝔖 consists of countable diagrams.

We denote by Hom_{𝔖}^{𝔗}(X, Y) the intersection (or, ruther, cointersection) of Hom_{𝔖}(X, Y) and Hom^{𝔗}(X, Y) which is the object of |Cat|^{o} defined by

C_{Hom_{𝔖}^{𝔗}(X,Y)} = C_{Hom_{𝔖}(X,Y)} ⋂ C_{Hom^{𝔗}(X,Y)}

i.e. objects of C_{Hom_{𝔖}^{𝔗}(X,Y)} are functors which preserves limits from 𝔖 and colimits from 𝔗. In particular, objects of C_{Hom_{w}^{w}(X,Y)} are functors C_{Y} ⟶ C_{X} which are weakly right and left exact.

10.2.3. Functoriality of these Hom-s. The maps

(X, Y) ⟼ Hom^{c}(X, Y) and (X, Y) ⟼ Hom_{𝔖}^{𝔗}(X, Y),

in particular, the maps (X, Y) ⟼ Hom^{w}(X, Y) and (X, Y) ⟼ Hom_{w}(X, Y), are functorial and stable under the composition (1), hence they define enriched subcategories of the enriched category of morphisms.

10.3. Flat descent of morphisms. Any pair U ⟶^{u} X and V ⟶^{v} Y of morphisms such that v is continuous induces a morphism

(u|v) : Hom(U, V) ⟶ Hom(X, Y)  (1)

having an inverse image functor (u^{∗}|v_{∗}) defined by

C_{Hom(X,Y)} ⟶ C_{Hom(U,V)}, ϕ^{∗} ⟼ u^{∗}ϕ^{∗}v_{∗}.  (2)

10.3.1. Lemma. (a) If U ⟶^{u} X and V ⟶^{v} Y are continuous morphisms, then (u|v) is continuous.

(b) If u^{∗} and v_{∗} are weakly continuous (i.e. they preserve cokernels of reflexive pairs of arrows), then the morphism (u|v) induces a morphism

(u|v)_{w} : Hom^{w}(U, V) ⟶ Hom^{w}(X, Y).

(c) If the morphisms u, v are continuous and v_{∗} has a right adjoint (e.g. v is affine), then the morphism (u|v) induces a morphism

(u|v)_{c} : Hom^{c}(U, V) ⟶ Hom^{c}(X, Y)

which is continuous if u_{∗} has a right adjoint.

Proof. (a) If u has a direct image, u_{∗}, then the functor

(u_{∗}|v^{∗}) : C_{Hom(U,V)} ⟶ C_{Hom(X,Y)}, ψ^{∗} ⟼ u_{∗}ϕ^{∗}v^{∗},  (3)

is a direct image functor of the morphism (u|v).

In fact,

(u_{∗}|v^{∗}) ∘ (u^{∗}|v_{∗}) = (u_{∗}u^{∗}|v_{∗}v^{∗}) : ϕ^{∗} ⟼ u_{∗}u^{∗}ϕ^{∗}v_{∗}v^{∗},

and

(u^{∗}|v_{∗}) ∘ (u_{∗}|v^{∗}) = (u^{∗}u_{∗}|v^{∗}v_{∗}) : ψ^{∗} ⟼ u^{∗}u_{∗}ψ^{∗}v^{∗}v_{∗}.

Adjunction arrow η_{u}, η_{v} (resp. ϵ_{u}, ϵ_{v}) determine adjunction arrows

η_{(u|v)} : Id_{C_{Hom(X,Y )}} ⟶ (u|v)_{∗} ∘ (u|v)^{∗} = (u_{∗}u^{∗}|v_{∗}v^{∗})

and

ϵ_{(u|v)} : (u|v)^{∗}(u|v)_{∗} = (u^{∗}u_{∗}|v^{∗}v_{∗}) ⟶ Id_{C_{Hom(U,V )}}.

(b) If the functors u^{∗}, v_{∗} and ϕ^{∗} : C_{Y} ⟶ C_{X} preserve kernels of coreflexive pairs of arrows, then their composition, u^{∗}ϕ^{∗}v_{∗} = (u|v)^{∗}(ϕ^{∗}) has the same property.

(c) If each of the functors u^{∗}, v_{∗} and ϕ^{∗} has a right adjoint, then the functor (u|v)^{∗}(ϕ^{∗}) = u^{∗}ϕ^{∗}v_{∗} has a right adjoint. Similarly, if the functors u_{∗} and ψ^{∗} have right adjoints, then (u|v)_{∗}(ψ^{∗}) = u_{∗}ψ^{∗}v^{∗} has a right adjoint.

10.3.2. Lemma. Let U ⟶^{u} X be a continuous morphism with an inverse image u^{∗} and a direct image u_{∗}. Suppose the category C_{X} has kernels of coreflexive pairs of arrows and the functor u^{∗} preserves these kernels. Then the morphism u (i.e. its inverse image u^{∗}) is conservative iff the canonical diagram

Id_{C_{X}} ⟶^{η_{u}} F_{u} \overrightarrow{⟶}^{Fuηu}_{ηuFu} F_{u}^{2}  (4)

is exact. Here F_{u} = u_{∗}u^{∗} and η_{u} is an adjunction morphism.

Proof. Since the pair of arrows

F_{u} \overrightarrow{⟶}^{Fuηu}_{ηuFu} F_{u}^{2}  (5)

is coreflexive, and by hypothesis, the category C_{X} has kernels of coreflexive arrows, the pair (5) has a kernel, K_{η}, and the morphism η : Id_{C_{X}} ⟶ F_{u} is uniquely decomposed into Id_{C_{X}} ⟶^{s_{u}} K_{η} ⟶^{κ_{u}} F_{u}, where κ_{u} is the canonical morphism. Notice that the diagram

u^{∗} ⟶^{u^{∗}η_{u}} u^{∗}F_{u} \overrightarrow{⟶}^{u^{∗}Fuηu}_{u^{∗}ηuFu} u^{∗}F_{u}^{2}  (6)

is exact without any additional conditions on u (except for u being continuous). Since u^{∗} preserves kernels of coreflexive pairs of arrows, the exactness of (6) is equivalent to that u^{∗}(s_{u}) is an isomorphism. Thus, if u^{∗} is conservative, then s_{u} is an isomorphism, i.e. the diagram (4) is exact.

Conversely, suppose the diagram (4) is exact. Let s : L ⟶ M be a morphism such that u^{∗}(s) is an isomorphism. Then the vertical arrows F_{u}(s) and F_{u}^{2}(s) in the commutative diagram

\begin{matrix}L & ⟶^{η_{u}(L)} & F_{u}(L) & \overrightarrow{⟶}^{Fuηu(L)}_{ηuFu(L)} & F_{u}^{2}(L) \\ s ↓ & & F_{u}(s) ↓ & & ↓ F_{u}^{2}(s) \\ M & ⟶^{η_{u}(M)} & F_{u}(M) & \overrightarrow{⟶}^{Fuηu(M)}_{ηuFu(M)} & F_{u}^{2}(M) \end{matrix}  (7)

are isomorphisms. Since the horizontal subdiagrams in (7) are exact, this implies that s is an isomorphism.

10.3.3. Note. We shall also refer to 10.3.2 when the dual assertion is needed:

Let U ⟶^{u} X be a continuous morphism with an inverse image u^{∗} and a direct image u_{∗}. Suppose the category C_{U} has cokernels of reflexive pairs of arrows and the functor u_{∗} preserves these cokernels. Then the functor u_{∗} is conservative iff the canonical diagram

G_{u}^{2} \overrightarrow{⟶}^{Guϵu}_{ϵuGu} G_{u} ⟶^{ϵ_{u}} Id_{C_{U}}  (8)

is exact. Here G_{u} = u^{∗}u_{∗} and ϵ_{u} an adjunction morphism.

10.3.4. Proposition. Let U ⟶^{u} X and V ⟶^{v} Y be continuous morphisms.

(a) If u is weakly coflat (i.e. u_{∗} preserves cokernels of reflexive pairs of arrows), then the morphism

(u|v) : Hom(U, V) ⟶ Hom(X, Y)

has the same property.

(b) Suppose both u and v are weakly affine (i.e. they are weakly coflat and their direct image functors are conservative); and let the category C_{X} have cokernels of reflexive pairs of arrows. Then the morphism

(u|v)_{w} : Hom^{w}(U, V) ⟶ Hom^{w}(X, Y)

is weakly affine.

(c) Suppose that u and v are weakly flat and conservative, and the category C_{U} has limits of coreflexive pairs of arrows. Then the morphism

(u|v)^{w} : Hom_{w}(U, V) ⟶ Hom_{w}(X, Y)

is weakly flat and conservative.

Proof. (a) The functor

(u|v)_{∗} : C_{Hom(U,V)} ⟶ C_{Hom(X,Y)}, ϕ^{∗} ⟼ u_{∗}ϕ^{∗}v^{∗},

preserves all colimits the functor u_{∗} preserves, because the functor ϕ^{∗} ⟼ ϕ^{∗}v^{∗} preserves all (small) colimits and limits.

(b) Let G_{v} = v^{∗}v_{∗}, and let ϵ_{v} : G_{v} ⟶ Id_{C_{V}} be an adjunction morphism. If v_{∗} preserves cokernels of reflexive pairs of arrows and is conservative, then, by 10.3.3, the diagram

G_{v}^{2} \overrightarrow{⟶}^{Gv ϵv}_{ϵv Gv} G_{v} ⟶^{ϵ_{v}} Id_{C_{V}}  (9)

is exact. If ϕ^{∗} : C_{Y} ⟶ C_{X} is an object of C_{Hom^{w}(X,Y)}, i.e. ϕ^{∗} preserves cokernels of reflexive pairs of arrows, the diagram ϕ^{∗}(9) is exact. If, in addition, the morphism u_{∗} preserves cokernels of reflexive pairs of arrows and is conservative, hence the diagram

G_{u}^{2} \overrightarrow{⟶}^{Guϵu}_{ϵuGu} G_{u} ⟶^{ϵ_{u}} Id_{C_{U}} is exact, than the diagram

G_{u}^{2}ϕ^{∗}G_{v}^{2} = G_{(u|v)}^{2}(ϕ^{∗}) \overrightarrow{⟶}^{G_{(u|v)}ϵ_{(u|v)}}_{ϵ_{(u|v)}G_{(u|v)}} G_{(u|v)}(ϕ^{∗}) = G_{u}ϕ^{∗}G_{v} ⟶^{ϵ_{(u|v)}} ϕ^{∗}  (10)

is exact. If the category C_{X} has a certain type of colimits, the category C_{Hom(X,Y)} and its subcategory C_{Hom^{w}(X,Y)} have the same type of colimits. In particular, C_{Hom^{w}(X,Y)} has cokernels of reflexive pairs of arrows, because C_{X} has them.

Thus, the morphism (u|v)_{w} : Hom^{w}(U, V) ⟶ Hom^{w}(X, Y) satisfies the conditions of 9.3.3, hence its direct image functor is conservative.

(c) The assertion (c) is dual (therefore equivalent) to the assertion (b).

10.3.5. Corollary. Let U ⟶^{u} X and V ⟶^{v} Y be continuous morphisms.

(a) Suppose both u and v are weakly affine; and let the category C_{X} have cokernels of reflexive pairs of arrows. Then the morphism

(u|v)_{w} : Hom^{w}(U, V) ⟶ Hom^{w}(X, Y)

is monadic, i.e. it is isomorphic to the canonical morphism

Sp(F_{(u|v)_{w}}/Hom^{w}(X, Y)) ⟶ Hom^{w}(X, Y).

Here F_{(u|v)^{w}} is the monad on Hom^{w}(X, Y) associated with the (choice of an inverse and a direct image functors and adjunction arrows of the) morphism (u|v).

(b) Suppose that u and v are weakly flat and conservative, and the category C_{U} has limits of coreflexive pairs of arrows. Then the morphism

(u|v)^{w} : Hom_{w}(U, V) ⟶ Hom_{w}(X, Y)

is comonadic, i.e. it is isomorphic to the canonical morphism

Hom_{w}(U, V) ⟶ Sp^{o}(Hom_{w}(U, V)\G_{(u|v)^{w}}).

Here G_{(u|v)^{w}} is the comonad on Hom_{w}(U, V) associated with the morphism (u|v).

Proof. (a) By 10.3.4(b), the morphism (u|v)_{w} satisfies the conditions of the Beck’s theorem 5.4.2.

(b) Similarly, by 10.3.4(c), the morphism (u|v)^{w} satisfies the conditions of the (dual) Beck’s theorem 5.4.1.

The assertions follow from the respective versions of the Beck’s theorem.

Denote by Hom^{wfl}(X, Y) the object (in |Cat|^{o}) of weakly flat morphisms: objects of the category of quasi-coherent modules on Hom^{wfl}(X, Y) are functors ϕ^{∗} : C_{Y} ⟶ C_{X} which have a right adjoint and preserve kernels of coreflexive pairs of arrows.

10.3.6. Proposition. Let U ⟶^{u} X and V ⟶^{v} Y be affine weakly flat morphisms. Then a direct image functor of the morphism

(u|v)_{c} : Hom^{c}(U, V) ⟶ Hom^{c}(X, Y)

is conservative and preserves colimits of small diagrams.

The morphism (u|v)_{c} induces a weakly flat conservative morphism

(u|v)_{wfl} : Hom^{wfl}(U, V) ⟶ Hom^{wfl}(X, Y)

whose direct image functor preserves colimits of small diagrams.

Proof. Since u and v are affine, in particular u_{∗} and v_{∗} have right adjoints, it follows from 10.3.1(c) that (u|v)_{c} is a continuous morphism. By 10.3.4, a direct image functor of the morphism (u|v)_{c} is conservative.

10.3.7. Corollary. Let U ⟶^{u} X and V ⟶^{v} Y be affine weakly flat morphisms. Then

(a) The morphism

(u|v)_{c} : Hom^{c}(U, V) ⟶ Hom^{c}(X, Y)

is isomorphic to the canonical morphism

Sp(F_{(u|v)_{c}}/Hom^{c}(X, Y)) ⟶ Hom^{c}(X, Y),

where F_{(u|v)_{c}} = (F_{(u|v)_{c}}, μ_{(}u|v)) is a monad associated with (u|v)_{c}. The functor F_{(u|v)_{c}} preserves colimits.

(b) The morphism (u|v)_{c} is the composition of the standard morphism

Hom^{c}(U, V) ⟶ Sp^{o}(Hom^{c}(U, V)\G_{(u|v)_{c}})

and the canonical morphism

Sp^{o}(Hom^{c}(U, V)\G_{(u|v)_{c}}) ⟶ Hom^{c}(X, Y).  (11)

The morphism (11) is a localization.

(c) The morphism

(u|v)_{wfl} : Hom^{wfl}(U, V) ⟶ Hom^{wfl}(X, Y)

is isomorphic to the canonical morphism

Hom^{wfl}(U, V) ⟶ Sp^{o}(Hom^{wfl}(U, V)\G_{(u|v)_{wfl}}),

where G_{(u|v)_{wfl}} = (G_{(u|v)_{wfl}}, δ_{(u|v)}) is a comonad associated with (u|v)_{wfl}. The functor G_{(u|v)_{wfl}} preserves colimits.

Proof. The assertion is a consequence of 10.3.6 and the Beck’s theorem 5.4.1.

10.3.7.1. A description of the morphisms in 10.3.7. Since the morphisms U ⟶^{u} X and V ⟶^{v} Y are affine and weakly flat, X and Y can be identified with resp. Sp^{o}(U\G_{u}) and Sp^{o}(V \G_{v}). One can define a (fully faithful) direct image functor of the morphism (11) as follows.

Let M = (M, ζ) be a G_{(u|v)}-comodule; and let L = (L, ξ_{L}) be a G_{v}-comodule (i.e. an object of C_{Y}). Notice that the G_{(u|v)}-comodule structure on M is the composition of a right coaction, ζ^{v} : M ⟶ MG_{v}, and a left coaction, ζ_{u} : M ⟶ G_{u}M. In other words, M is naturally regarded as a ((U\G_{u}), (V \G_{v}))-bicomodule (M, ζ_{u}, ζ^{v}).

Thus, we have a coreflexive pair of morphisms

M(L) \overrightarrow{⟶}^{M(ξ_{L})}_{ζ^{v} (L)} MG_{v}(L).  (12)

The comodule structure ζ_{u} : M ⟶ G_{u}M induces G_{u}-comodule structure on M(L) and MG_{v}(L), so that (12) becomes a diagram of G_{u}-comudules. Since the functor G_{u} = u^{∗}u_{∗} preserves kernels of coreflexive pairs of arrows, the kernel M̃(L) of the (coreflexive) pair (12) has a (necessarily unique) G_{u}-comodule structure, ξ_{_{M}(L)} : M̃(L) ⟶ G_{u}M̃(L), such

that the canonical morphism M̃(L) ⟶ M(L) is a G_{u}-comodule morphism.

The correspondence (M, L) ⟼ (M̃(L), ξ_{M(L)}) is functorial in M and L. The functor which assigns to each (Hom^{c}(U, V)\G_{(u|v)})-comodule M the functor

(Y \G_{v}) − Comod ⟶ (X\G_{u}) − Comod, L ⟼ (M̃(L), ξ_{_{M}(L)})

is a direct image functor of the morphism (11).

10.3.7.2. The composition of morphisms. Let U ⟶^{u} X, V ⟶^{v} Y and W ⟶^{w} Z be affine weakly flat morphisms which allows to assume that X = Sp^{o}(U\G_{u}), Y = Sp^{o}(V \G_{v}), and Z = Sp^{o}(W \G_{w}). Let M = (M, ζ_{u}, ζ^{v}) be a ((U\G_{u}), (V \G_{v}))- bicomodule and N = (N, ζ_{v}, ζ^{w}) a ((V \G_{v}), (W \G_{w}))-bicomodule. Consider the diagram

MN \overrightarrow{⟶}^{ζ^{v}}_{ζv} MG_{v}N.

This diagram is coreflexive, hence by hypothesis it has a kernel, M ⊙ N. The coaction ζ^{w} : N ⟶ NG_{w} induces a right (W \G_{w})-comodule structure, M ⊙ ζ^{w} on M ⊙ N. Since the functor G_{u} = u^{∗}u_{∗} is weakly flat, by a standard argument, the left (U\G_{u})-comodule structure ζ_{u} : M ⟶ G_{u}M induces a left (U\G_{u})-comodule structure, ζ ⊙ N, on M ⊙ N. We denote by M ⊙ N the ((U\G_{u}), (W \G_{w}))-bicomodule (M ⊙ N, ζ_{u} ⊙ N, M ⊙ ζ^{w}).

The functor

ϕ_{M⊙N}^{∗} : (W \G_{w}) − Comod ⟶ (U\G_{u}) − Comod

corresponding to M ⊙ N is isomorphic to the composition ϕ_{M}^{∗} ∘ ϕ_{N}^{∗} of the functors corresponding to the resp. bicomodules N and M. This follows from general facts on commuting limits and from the fact that the limits of functors are computed object-wise (cf. [GZ], Glossary).

Complementary facts.

C1. Continuous morphisms to the categoric spectrum of a monoid. Let M be a monoid. Denote by Sp(M/E) the object of |Cat|^{o} corresponding to the category M − Sets of M-sets. We call Sp(M/E) the categoric spectrum of the monoid M.

C1.1. Proposition. Let X ⟶^{f} Sp(M/E) be a continuous morphism. Then

(a) The morphism f is determined by the object O = f^{∗}(M) uniquely up to isomorphism.

(b) There exists a coproduct of any set of copies of O.

(c) The object O has a structure of an M^{o}-module, i.e. there is a monoid morphism

M ⟶^{ϕ_{f}} C_{X}(O, O)^{o}.

Proof. These facts follow from the canonical isomorphisms

Hom_{X}(f^{∗}(M), −) ≃ Hom_{M}(M, f_{∗}(−)) ≃ f_{∗}(−).

Since f^{∗} has a right adjoint, it preserves colimits, in particular coproducts. Notice that the category of M-sets has small coproducts and cokernels of pairs of arrows, hence it has arbitrary small colimits. The rest of the argument follows the same lines as the argument of 4.1. Details are left to the reader.

For any object O of the category C_{X}, we denote by Γ_{X}O the monoid C_{X}(O, O)^{o}. We denote by |Cat|^{o}(M/E) the category of right (M/E)-modules. Its objects are

triples (X, O_{X}, ϕ), where X is an object of |Cat|^{o}, (O_{X}, ϕ) is a right M-module in the category C_{X}, i.e. ϕ is a monoid morphism M ⟶ C_{X}(O_{X}, O_{X})^{o} (cf. C1.1(c)). Morphisms from (X, O_{X}, ϕ) to (Y, O_{Y} , ψ) are given by morphisms X ⟶^{f} Y such that there exists an isomorphism λ : f^{∗}(O_{Y}) ⟶^{∼} O_{X} making the diagram

\begin{matrix}M^{o} & ⟶^{ϕ} & C_{X}(O_{X}, O_{X}) \\ ψ ↓ & & ↓ 𝔠_{λ} \\ C_{Y} (O_{Y} , O_{Y}) & ⟶^{f_{O_{Y} ,O_{Y}}^{∗}} & C_{X}(f^{∗}(O_{Y}), f^{∗}(O_{Y})) \end{matrix}  (2^{bis})

commute. Here 𝔠_{λ} denotes the conjugation by λ, h ⟼ λ^{−1} ∘ h ∘ λ.

Let |Cat|^{o}(M/E)_{𝔠} denote the full subcategory of the category |Cat|^{o}(M/E) whose objects are triples (X, O_{X}, ϕ) such that the following conditions hold:

(a) There exists a coproduct of any set of copies of O_{X}.

(b) Let Φ_{ϕ} denote the functor from the subcategory 𝔏_{M} of free (M/E)-sets to C_{X} which is uniquely defined by the action ϕ (thanks to (b) above). The image by Φ_{ϕ} of any pair of arrows X_{1} ⇉ X_{0} has a cokernel.

Let Cat^{op}(M/E)_{𝔠} denote the preimage of |Cat|^{o}(M/E)_{𝔠} in Cat^{op}(M/E). On the other hand, let (|Cat|^{o}/Sp(M/E)_{𝔠} denote the full subcategory of the category |Cat|^{o}/Sp(M/E) whose objects are continuous morphisms to Sp(M/E), and let (Cat^{op}/Sp(M/E))_{𝔠} denote its preimage in Cat^{op}/Sp(M/E).

C1.2. Proposition. The functor (3) induces an equivalence of the fibered categories

( (Cat^{op}/Sp(M/E))_{𝔠} ; ↓ ; (|Cat|^{o}/Sp(M/E))_{𝔠} ) ⟶ ( (Cat^{op}(M/E)_{𝔠} ; ↓ ; |Cat|^{o}(M/E)_{𝔠} ) .  (4^{bis})

Proof. The argument is similar to that of 4.4.2.

C1.3. E-Spaces. Denote by |Cat|_{E}^{o} the category whose objects are pairs (X, O), where X ∈ Ob|Cat|^{o} and O is an object of C_{X} such that there exist coproducts of small sets of copies of O and any pair of arrows O^{⊕I} \overrightarrow{⟶} O^{⊕J} between coproducts of copies of O has a cokernel in C_{X} (equivalently, every small diagram 𝔇 : D ⟶ C_{X} which maps all objects to O has a colimit in C_{X}). Morphisms from (X, O) to (X′, O′) are morphisms X ⟶^{f} X′ such that there exists an isomorphism f^{∗}(O′) ⟶^{∼} O. Composition is defined in an obvious way. Objects of the category |Cat|o_{E} will be called E-’spaces’.

It follows that E-’spaces’ are precisely those pairs (X, O) for which the functor

f_{_{O}∗} : C_{X} ⟶ Γ_{X}O − Sets, M ⟼ C_{X}(O, M),

is a direct image functor of a continuous morphism f_{O} : X ⟶ Sp(Γ_{X}O/E); i.e. f_{_{O}∗} has a left adjoint.

C1.4. Affine E-spaces. We call a E-space (X, O) affine if the canonical morphism f_{O} : X ⟶ Sp(Γ_{X}O) is an isomorphism; i.e. the functor f_{_{O}∗} (see (6)) is a category equivalence. We denote by Aff_{E} the full subcategory of the category |Cat|o_{E} objects of which are affine E-spaces.

Let 𝔄𝔣𝔣_{E} denote the subcategory of the category |Cat|o_{E} whose objects are pairs (Sp(M/E), M). Morphisms from (Sp(M/E), M) to (Sp(M′/E), M′) are morphisms Sp(M/E) ⟶ Sp(M′/E) corresponding to monoid morphisms M′ ⟶ M. There is an inclusion functor 𝔄𝔣𝔣_{E} ⟶^{γ_{∗}} |Cat|o_{E} which takes values in the subcategory of affine E-spaces.

C1.4.1. Proposition. The functor 𝔄𝔣𝔣_{E} ⟶^{γ_{∗}} |Cat|o_{E} is fully faithful and has a left adjoint. In particular, γ_{∗} induces an equivalence of 𝔄𝔣𝔣_{E} and the category of affine E-spaces.

Proof. The argument is similar to that of 4.5.4 and is left to the reader.

C2. Continuous morphisms, monads, and localizations. Let X ⟶^{q} Y be a localization with an inverse image functor C_{Y} ⟶^{q^{∗}} C_{X}. Let Σ_{q} denote the class of all morphisms s in C_{Y} such that q^{∗}(s) is invertible. A functor C_{Y} ⟶^{F} C_{Y} is compatible with

the localization q iff F (Σ_{q}) ⊆ Σ_{q}. In this case, there exists a unique functor C_{X} ⟶^{F̄} C_{X} such that q^{∗} ∘ F = F̄ ∘ q^{∗}.

C2.1. Proposition. Let X ⟶^{q} Y be a continuous localization.

(a) A functor C_{Y} ⟶^{F} C_{Y} is compatible with q iff the canonical morphism

q^{∗} ∘ F ⟶ q^{∗} ∘ F ∘ q_{∗}q^{∗}  (1)

is an isomorphism.

(b) Suppose C_{Y} ⟶^{F} C_{Y} is compatible with the localization q, and let F̄ be the functor C_{X} ⟶ C_{X} such that q^{∗} ∘ F = F̄ ∘ q^{∗}.

(i) If C_{Y} has colimits of certain type, then C_{X} has colimits of this type. If F preserves colimits of this type, then the functor F̄ has the same property.

(ii) If C_{Y} has limits of certain type, then C_{X} has limits of this type. If F and q^{∗} preserve limits of this type (e.g. finite limits), then the functor F̄ has the same property.

(iii) If F has a right adjoint, then F̄ has a right adjoint.

Proof. (a) The assertion follows from C2.4.1.1 applied to f^{∗} = q^{∗} ∘ F.

(b) (i) Let D : D ⟶ C_{X} be a small diagram such that there exists colim(q_{∗}D). Then there exists the colimit of D and colim(D) = q^{∗}colim(q_{∗}D). Suppose the functor F preserves the colimit of q_{∗}D. Since F̄ ≃ q^{∗}F q_{∗}, we have:

F̄(colim(D)) ≃ q^{∗}F q_{∗}q^{∗}(colim(q_{∗}D)) ≃ q^{∗}F (colim(q_{∗}D)) ≃ q^{∗}(colim(F q_{∗}D)

(the second isomorphism here is due to the isomorphism q^{∗}F ⟶^{∼} q^{∗}F q_{∗}q^{∗}). Since q^{∗} has a right adjoint, it preserves colimits. Therefore

q^{∗}(colim(F q_{∗}D) ≃ colim(q^{∗}F q_{∗}D) ≃ colim(F̄D)

whence the assertion.

(ii) Let D : D ⟶ C_{X} be a small diagram such that there exists lim(q_{∗}D). Then, by [GZ, I.1.4], there exists the limit of D and lim(D) = q^{∗}(lim(q_{∗}D)). Let the functor F preserve the colimit of q_{∗}D. As in (i), we have:

F̄(lim(D)) ≃ q^{∗}F q_{∗}q^{∗}(lim(q_{∗}D)) ≃ q^{∗}F (lim(q_{∗}D)) ≃ q^{∗}(lim(F q_{∗}D)

If q^{∗} preserves limit of F q_{∗}D, we continue as follows:

q^{∗}(lim(F q_{∗}D) ≃ (lim(q^{∗}F q_{∗}D) ≃ lim(F̄D).

(iii) Let F^{!} be a right adjoint to F. Set F̄^{!} = q^{∗}F^{!}q_{∗}. By (a), F̄ ≃ q^{∗}F q_{∗}. Thus we have morphisms

F̄F̄^{!} ⟶^{∼} (q^{∗}F q_{∗}q^{∗})F^{!}q_{∗} ⟶^{∼} q^{∗}F F^{!}q_{∗} ⟶^{q^{∗}ϵ_{F} q_{∗}} q^{∗}q_{∗} ⟶^{ϵ_{q}} Id_{C_{X}} .  (3)

and

Id_{C_{X}} ⟶^{∼} q^{∗}q_{∗} ⟶^{q^{∗}η_{F} q_{∗}} q^{∗}F^{!}F q_{∗} ⟶^{q^{∗}F^{!}η_{q}F q_{∗}} q^{∗}F^{!}q_{∗}q^{∗}F q_{∗} ⟶^{∼} F̄^{!}F̄.  (4)

The compositions of the sequence of morphisms resp. (3) and (4) are adjunction arrows.

C2.2. Proposition. Let X ⟶^{q} Y be a localization and F = (F, μ) a monad on Y such that the endofunctor F is compatible with q. Then the monad F induces a monad, F̄ = (F̄, μ̄), on X defined uniquely up to isomorphism.

(i) If F is continuous (i.e. F has a right adjoint), then the monad F̄ is continuous.

(ii) If C_{Y} has colimits of certain type, then C_{X} has colimits of this type. If F preserves colimits of this type, then F̄ has the same property.

(iii) If C_{Y} has limits of certain type, then C_{X} has limits of this type. If F and q^{∗} preserve limits of this type, then F̄ has the same property.

Proof. Fix an inverse image, q^{∗}, of the localization q. Let F̄ be a unique endofunctor F̄ : C_{X} ⟶ C_{X} such that q^{∗} ∘ F = F̄ ∘ q^{∗}. Then q^{∗} ∘ F^{2} = F̄^{2} ∘ q^{∗}, and, by the universal property of localizations, there exists a unique morphism μ̄ : F̄^{2} ⟶ F̄ such that q^{∗}μ = μ̄q^{∗}. We leave to the reader verifying that μ̄ is a monad structure on F̄.

The assertions (i), (ii), (iii) follow from the corresponding assertions of C2.1.

C2.2.1. Remark. The same assertion holds for comonads. In fact, the first part is obtained by dualization. The parts (i) and (ii) are statements about endofunctors.

C3. Cones of non-unital monads and rings.

C3.1. Non-unital monads. Let X be a ’space’ such that C_{X} is an additive category, and let 𝔽_{+} = (F_{+}, μ) be a non-unital additive monad on X; i.e. F_{+} is an additive functor C_{X} ⟶ C_{X} and μ is a functor morphism F_{+}^{2} ⟶ F_{+} such that μ ∘ F_{+}μ = μ ∘ μF_{+}. Let 𝔽_{+} − mod_{1} denote the category of non-unital 𝔽_{+}-modules. Its objects are pairs (M, ξ), where M ∈ ObC_{X} and ξ a morphism F_{+}(M) ⟶ M such that ξ ∘ μ(M) = ξ ∘ F_{+}ξ. A morphism (M, ξ) ⟶ (M′, ξ′) is given by a morphism M ⟶^{f} M′ such that ξ′ ∘ F_{+}(f) = f ∘ ξ. Composition is defined naturally, so that map which assigns to each 𝔽_{+}-module (M, ξ) the object M and to every 𝔽_{+}-module morphism (M, ξ) ⟶^{f} (M′, ξ′) the morphism M ⟶^{f} M′ is a functor, 𝔽_{+} − mod_{1} ⟶^{𝔣_{∗}} C_{X}. This functor has a canonical left adjoint, 𝔣^{∗}, which maps every object N of C_{X} to the 𝔽_{+}-module (N ⊕ F (N), ξ_{N}), where the action F_{+}(N ⊕ F_{+}(N)) = F_{+}(N) ⊕ F_{+}^{2}(N) ⟶^{ξ_{N}} N ⊕ F_{+}(N) is the composition of the morphism

F_{+}(N) ⊕ F_{+}^{2}(N) ⟶^{(id_{F_{+}(N)},μ(N)} F_{+}(N) and the embedding F_{+}(N) ⟶ N ⊕ F_{+}(N).

Thus, 𝔣_{∗}𝔣^{∗} = Id_{C_{X}} ⊕ F. We denote Id_{C_{X}} ⊕ F_{+} by F and the monad corresponding to the pair of adjoint functors 𝔣_{∗}, 𝔣^{∗} by 𝔽 = (F, μ_{1}). It is easy to see that the category 𝔽_{+} − mod_{1} of non-unital 𝔽_{+}-modules is isomorphic to the category 𝔽 − mod of unital 𝔽-modules.

There is a natural embedding C_{X} ⟶ 𝔽_{+}−mod which assigns to each object M of C_{X} the 𝔽_{+}-module (M, 0). We denote the image of C_{X} in 𝔽_{+} − mod (i.e. the full subcategory generated by trivial modules) by T_{𝔽_{+}}.

C3.2. The cone of a non-unital monad. Suppose that the category C_{X} is abelian. We denote by C_{Cone(𝔽_{+}/X)} the quotient, 𝔽_{+} − mod_{1}/T_{𝔽_{+}}^{−}, of the category 𝔽_{+} − mod_{1} by the smallest Serre subcategory containing T_{𝔽_{+}}. This defines a ’space’ Cone(𝔽_{+}/X).

C3.2.1. Proposition. If 𝔽_{+} is a unital monad, then C_{Cone(𝔽_{+}/X)} is naturally equivalent to the category 𝔽_{+} − mod of unital 𝔽_{+}-modules, i.e. the ’space’ Cone(𝔽_{+}/X) is isomorphic to Sp(𝔽_{+}/X).

Proof. If the monad 𝔽_{+} = (F_{+}, μ) is unital with the unit element Id ⟶^{e} F_{+}, then there is a monad epimorphism F = Id⊕F_{+} ⟶^{γ} F_{+} defined by (e, id_{F_{+}}). The corresponding pull-back functor, γ_{∗}, is the inclusion functor 𝔽_{+} −mod into 𝔽_{+} −mod_{1}. Its left adjoint, γ^{∗}

assigns to each object (M, ξ) of 𝔽_{+}−mod_{1} the cokernel of the pair of morphisms M \overrightarrow{⟶}^{id_{M}}_{ξe(M)} M.

Since the functor γ_{∗} is fully faithful, its left adjoint γ^{∗} is an exact localization, and the kernel of γ^{∗} coincides with the subcategory T_{𝔽_{+}}. Therefore, T_{𝔽_{+}} is, in this case, a Serre subcategory, i.e. T_{𝔽_{+}} = T_{𝔽_{+}}^{−}, whence the assertion.

C3.2.2. Corollary. If 𝔽_{+} is a unital monad, then Sp(𝔽/X) ≃ Sp(𝔽_{+}/X) ∐ X.

Proof. If 𝔽_{+} is a unital monad, then (by the argument of C3.2.1) 𝔽 ≃ 𝔽_{+} ∏ Id_{X}, where Id_{X} denotes the identical monad (Id_{C_{X}} , id). This implies that the category 𝔽 − mod of 𝔽-modules is equivalent to the product 𝔽_{+} − mod ∏ C_{X}, hence the assertion.

C3.3. The cone of an associative ring. Let X = Sp(R), where R_{0} is a unital associative ring, and let R_{+} be an R_{0}-ring. The latter means that R_{0} is an associative ring, not unital in general, in the category of R_{0}-bimodules; i.e. the mutiplication in R_{+} is given by an R_{0}-bimodule morphism R_{+} ⊗_{R_{0}} R_{+} ⟶^{m} R_{+} satisfying the associativity condition. The R_{0}-ring R_{+} defines a non-unital monad 𝔽_{+} = (F_{+}, μ) on X, where F_{+} is the endofunctor R_{+} ⊗_{R_{0}} − on C_{X} = R_{0} −mod and F_{+}^{2} ⟶^{μ} F_{+} is induced by the multiplication R_{+} ⊗_{R_{0}} R_{+} ⟶^{m} R_{+}. The category 𝔽_{+} − mod_{1} of non-unital 𝔽_{+}-modules is the category of unital R_{0}-modules endowed with a non-unital R_{+}-module structure compatible with the action of R_{0} on the module and on R_{+}. We write R_{+} − mod_{1} instead of 𝔽_{+} − mod_{1} and T_{R_{+}} instead of T_{𝔽_{+}}. By definition T_{𝔽_{+}} is the full subcategory of R_{+} − mod_{1} spanned by modules with zero action.

The associated augmented monad 𝔽 (cf. C3.1) is isomorphic to the monad associated with the unital R_{0}-ring R = R_{0}⊕R_{+} which we call the augmented R_{0}-ring corresponding to R_{+}. The category R_{+} − mod_{1} is isomorphic to the category R − mod of unital R-modules.

We shall write Cone(R_{+}/R_{0}), or simply Cone(R_{+}), instead of Cone(𝔽_{+}/Sp(R_{0})). We shall identify R_{+} − mod_{1} with R − mod whenever it is convenient by some rea-

son. Thus, T_{R_{+}} is viewed as the full subcategory of R − mod whose objects are modules annihilated by the irrelevant ideal R_{+}; and we write C_{Cone(R_{+})} = R − mod/T_{R_{+}}^{−}, where T_{R_{+}}^{−} is the smallest Serre subcategory of the category R − mod containing T_{R_{+}} (getting

back the definition of a cone in 1.6). The localization functor R − mod ⟶^{u^{∗}} R − mod/T_{R_{+}}^{−} is an inverse image functor of a morphism of ’spaces’ Cone(R_{+}) ⟶^{u} Sp(R). The functor u^{∗} has a (necessarily fully faithful) right adjoint, i.e. the morphism u is continuous. The composition of the morphism u with the natural affine morphism Sp(R) ⟶ Sp(R_{0}) is a continuous morphism Cone(R_{+}) ⟶ Sp(R_{0}). Its direct image functor is (regarded as) the global sections functor.

C3.3.1. Proposition. If R_{+} is a unital ring, then Cone(R_{+}/R_{0}) ≃ Sp(R_{+}) and Sp(R) ≃ Sp(R_{+}) ∐ Sp(R_{0}).

Proof. The assertion follows from C3.2.1 and C3.2.2.

C3.3.2. Lemma. Let J be a two-sided ideal in the ring R contained in R_{+} (i.e. a two-sided ideal in R_{+} which is an R_{0}-bimodule). Let T_{R_{+}|J} denote the full subcategory of R − mod whose objects are R-modules annihilated by J; and let T_{R|J}^{−} be the Serre subcategory spanned by T_{R|J}. The quotient category R − mod/T_{R|J}^{−} is equivalent to C_{Cone(J)}.

Proof. The embedding J ↪ R induces a unital ring morphism J̃ ⟶^{ι} R, where J̃ is the ring R_{0} ⊕ J with natural multiplication. The pull-back functor R − mod ⟶^{ι_{∗}} J̃ − mod induces a functor from the subcategory T_{R|J} to the subcategory T_{J}. Since the functor ι_{∗} is exact (in a strong sense, that is it preserves small limits and colimits), it maps the Serre subcategory T_{R|J}^{−} to the Serre subcategory T_{J}^{−}. Thus we have a commutative diagram

\begin{matrix}R − mod & ⟶^{ι_{∗}} & J̃ − mod \\ ↑ & & ↑ \\ T_{R|J}^{−} & ⟶ & T_{J}^{−} \end{matrix}  (1)

of exact functors. Therefore the functor ι_{∗} induces a functor

R − mod/T_{R|J}^{−} ⟶ J̃ − mod/T_{J}^{−}.  (2)

The functor (2) is a category equivalence. In fact, let J̃−mod ⟶^{Ψ} R−mod be the functor which assigns to every J̃-module M the R-module J M. The cokernel of the embedding J M ↪ M belongs to T_{J}, hence the localization J̃ − mod ⟶ J̃ − mod/T_{J}^{−} maps this embedding to an isomorphism. We assign to each object M of J̃−mod the composition of the functor Ψ and the localization R−mod ⟶ R−mod/T_{R|J}^{−}. It follows that this functor

factors through the localization J̃ − mod ⟶ J̃ − mod/T_{J}^{−}, i.e. it defines (uniquely) a

functor J̃ − mod/T_{J}^{−} ⟶^{Φ} R − mod/T_{R|J}^{−} . The functor Φ is a quasi-inverse to the functor (2).

C3.3.3. Example: quasi-affine schemes. Quasi-affine schemes are defined (in [EGA II, 5.1.1]) as open quasi-compact subschemes ofaffine schemes. Open subschemes of SpecA are in one-to-onecorrespondence with the radical ideals in A. Quasi-compactness ofan open set defined by an ideal J means that J is the radicalof its finitely generated subideal (this holds in noncommutative case too, see [R, I.5.6]. One can show that the category ofquasi-coherent sheaves on the open subscheme of SpecA defined by the ideal J is equivalent to the quotient categoryA − mod/T_{A|J}^{−}. By C3.3.2, the latter category is

equivalent tothe category C_{Cone(J)} = J̃ − mod/T_{J}^{−} of modules on the cone Cone(J) of the (non-unital) R_{0}-ring J.

C3.3.4. Functoriality. Let R_{0} − Rings denote the category of (not necessarily unital) R_{0}-rings. A morphism of such rings, R_{+} ⟶^{ϕ} S_{+}, is an R_{0}-bimodule morphism compatible with multiplication. The morphism ϕ induces the pull-back functor

S_{+} − mod_{1} ⟶^{ϕ_{∗}} R_{+} − mod_{1}

which maps the subcategory T_{S_{+}} of trivial S_{+}-modules to the category T_{R_{+}} of trivial R_{+}- modules. Since ϕ_{∗} is exact and preserves small colimits, it maps the Serre subcategory T_{S_{+}}^{−} spanned by T_{S_{+}} to the Serre subcategory T_{R_{+}}^{−} spanned by T_{R_{+}}. Therefore, ϕ_{∗} induces a unique functor

S_{+} − mod_{1}/T_{S_{+}}^{−} ⟶^{ϕ̄_{∗}} R_{+} − mod_{1}/T_{R_{+}}^{−}  (3)

such that the diagram

\begin{matrix}S_{+} − mod_{1}/T_{S_{+}}^{−} & ⟶^{ϕ̄_{∗}} & R_{+} − mod_{1}/T_{R_{+}}^{−} \\ q∗_{S} ↑ & & ↑ q∗_{R} \\ S_{+} − mod_{1} & ⟶^{ϕ_{∗}} & R_{+} − mod_{1} \end{matrix}  (4)

commutes. In general, the functor ϕ̄_{∗} does not have a left adjoint, hence it cannot be interpreted as a direct image functor of a continuous morphism.

C3.3.4.1. The category R_{0} − Rings_{1}. We denote by R_{0} − Rings_{1} the subcategory of R_{0} − Rings formed by R_{0}-ring morphisms R_{+} ⟶^{ϕ} S_{+} whose inverse image functor,

R_{+} − mod_{1} ⟶^{ϕ^{∗}} S_{+} − mod_{1}, is compatible with the localizations at resp. T_{R_{+}}^{−} and T_{S_{+}}^{−}.

The compatibility means that there exists a functor C_{Cone(R_{+})} ⟶^{ϕ̄^{∗}} C_{Cone(S_{+})} such that the diagram

C_{Cone(R_{+})} = R_{+} − mod_{1}/T_{R_{+}}^{−} ⟶^{ϕ̄^{∗}} S_{+} − mod_{1}/T_{S_{+}}^{−} = C_{Cone(S_{+})}
q∗_{R} ↑  ↑ q∗_{S}
R_{+} − mod_{1} ⟶^{ϕ^{∗}} S_{+} − mod_{+}

(5)

commutes. Thanks to the universal property of localizations, the functor ϕ̄ is uniquely determined by the commutativity of (5).

Evidently, all ring isomorphisms belong to R_{0} − Rings_{1}. It follows from the universal property of localizations that the composition of morphisms of R_{0} − Rings_{1} belongs to R_{0} − Rings_{1}; i.e. R_{0} − Rings_{1} is, indeed, a subcategory of the category R_{0} − Rings. The map R_{+} ⟼ Cone(R_{+}) extends to a functor R_{0} − Rings_{1}^{op} ⟶ |Cat|^{o} which we denote by Cone.

C3.3.4.2. Remarks. (a) For any morphism R_{+} ⟶^{φ} S_{+} of R_{0} − Rings, the functor

φ̄^{∗} = q∗_{S}φ_{1}^{∗}q_{_{R}∗} : C_{Cone(R_{+})} ⟶ C_{Cone(S_{+})}  (6)

might be regarded as an inverse image functor of a morphism Cone(S_{+}) ⟶^{φ̄} Cone(R_{+}). Notice, however, that the map φ ⟼ φ̄ is not functorial, unless morphisms are picked from the subcategory R_{0} − Rings_{1}.

(b) For any morphism R_{+} ⟶^{φ} S_{+} of R_{0} − Rings_{1}, the corresponding morphism

Cone(S_{+}) ⟶^{φ̄} Cone(R_{+}) is continuous.

This follows from the fact that the functor C_{Cone(R_{+})} ⟶^{q_{_{R}∗}} R_{+} − mod_{1} has a right adjoint and from the formula (6).

C3.3.4.3. Proposition. Let S_{+} be an R_{0}-ring, e a central idempotent element in S_{+} (i.e. e^{2} = e). Then R_{+} = {r ∈ S | re = er = r} is an R_{0}-subring in S_{+}, and the inclusion R ↪ S is a morphism of R_{0} − Rings_{1}.

Proof is left to the reader.

C3.3.5. Remark. For any R_{0}-ring R_{+}, we have a canonical morphism (Zariski open immersion) Cone(R_{+}) ⟶ Sp(R), R = R_{0} ⊕ R_{+}, which depends functorially on R_{+} (a functor from R_{0} − Rings_{1}^{op}). This morphism can be regarded as a noncommutative analogue of the Stone compactification of a locally compact space. If the ring R_{+} is unital, then Sp(R) is the disjoint union of Sp(R_{+}) and Sp(R_{0}) (see C3.3).

C3.3.6. Hopf actions and cross-products. Let R_{0} be an associative unital kalgebra. We call an R_{0}-ring R_{+} an (R_{0}|k)-ring if the R_{0}-ring structure makes R_{+} a k-algebra, i.e. λr = rλ for all r ∈ R_{+} and λ ∈ k. Let H = (δ, H, μ) be a k-bialgebra. Here H ⟶^{δ} H ⊗_{k} H ⟶^{μ} H are resp. comultiplication and multiplication. Recall that a Hopf action of H on a k-algebra R_{+} is a unital H-module structure on R such that the multiplication R_{+} ⊗_{k} R_{+} ⟶ R_{+} is an H-module morphism. We assume that H acts trivially on R_{0}. Then the cross-product R_{+}#H is an (R_{0}|k)-ring.

The Hopf action of H on R_{+} induces an endofunctor, H̃, on the category R_{+} − mod_{1}. This endofunctor assigns to any (non-unital) R_{+}-module M = (M, R_{+} ⊗_{k} M →^{ξ} M) the R_{+}-module H ⊗_{k} M = (H ⊗_{k} M, ξ_{H}), where the action ξ_{H} is the composition of

R_{+}⊗_{k}H ⊗_{k}M ⟶^{∼} H ⊗_{k}R_{+}⊗_{k}M ⟶ H ⊗_{k}H ⊗_{k}R_{+}⊗_{k}M ⟶ H ⊗_{k}R_{+}⊗_{k}M ⟶ H ⊗_{k}M.

Here the second arrow is induced by the comultiplication δ, the third arrow by the action τ, and the fourth arrow by the R-module structure ξ. The multiplication H ⊗_{k} H ⟶^{μ} H

induces a monad structure, H̃^{2} \underrightarrow{_{μ}} H̃, on H̃. One can see that the category R_{+}#H−mod_{1} is isomorphic to the category H̃ − mod, where H̃ denotes the monad (H̃, μ̃). This follows from the observation that the functor H̃ is isomorphic to R#H ⊗_{R} −, where R = R_{0} ⊕ R_{+} is the augmented R_{0}-ring. This observation implies, on the other hand, that the functor H̃ is continuous (i.e. it has a right adjoint) and that there is a natural isomorphism between the category R_{+}#H −mod_{1} and the category (H̃/R)−mod of modules over the monad H̃. Here we write (H̃/R) instead of (H̃/Sp(R)) and identify R_{+} − mod with R − mod. Thus, we have a natural isomorphism Sp(H̃/R) ⟶^{∼} Sp(R#H) such that the diagram

\begin{matrix}Sp(H̃/R) ⟶^{∼} Sp(R#H) \\ ↘  ↙ \\ Sp(R) \end{matrix}  (7)

commutes.

The following assertion provides another family of morphisms of R_{0} − Rings_{1}.

C3.3.6.1. Proposition. Let H ⊗_{k} R_{+} ⟶^{τ} R_{+} be a Hopf action of an k-bialgebra H = (δ, H, μ) on a (R_{0}|k)-ring R_{+}. Suppose the functor H ⊗_{k} − is flat. Then the monad H̃ on Sp(R) induces a monad H̄ on Cone(R_{+}) such that there is a canonical commutative diagram

Sp(H̄/Cone(R_{+})) ⟶^{∼} Cone(R_{+}#H)
↘ ↙
Cone(R_{+})

(8)

of affine morphisms.

In particular, the canonical morphism R_{+} ⟶ R_{+}#H belongs to R_{0} − Rings_{1}.

Proof. It follows that the functor H̃ maps the subcategory T_{R_{+}} to itself. Since the functor H ⊗_{k} − : k − mod ⟶ k − mod is flat (i.e. it is exact and preserves colimits of

small diagrams), the functor R_{+} − mod_{1} \underrightarrow{_{H}} R_{+} − mod_{1} is flat too. Therefore, the Serre subcategory T_{R_{+}}^{−} is stable under H̃, and the functor H̃ induces a continuous functor

R_{+} − mod_{1}/T_{R_{+}}^{−} = C_{Cone(R_{+})} ⟶^{H̄} C_{Cone(R_{+})}. By C2.2, the multiplication H̃^{2} \underrightarrow{_{μ}} H̃

induces a multiplication H̄^{2} ⟶^{μ̄} H̄. The isomorphism of categories

R_{+}#H − mod_{1} ⟶^{∼} (H̃/R) − mod

mentioned above induces an isomorphism C_{Cone(R#H)} ⟶^{∼} (H̄/Cone(R))−mod, regarded as an inverse image functor of an isomorphism Sp(H̄/Cone(R)) ⟶^{∼} Cone(R#H) such that the diagram (8) commutes. The monad (H̃/R) is continuous (i.e. the functor H̃ has a right adjoint), because H̃ is isomorphic to the (obviously) continuous functor R#H ⊗_{R} −. By C2.2(i), this implies that the monad H̄ on Cone(R) is continuous. By 6.2, the latter means precisely that the natural morphism Sp(H̄/Cone(R)) ⟶ Cone(R) is affine. Therefore, by the commutativity of (8), the morphism Cone(R#H) ⟶ Cone(R) is affine. This shows, in particular, the canonical morphism R_{+} ⟶ R_{+}#H belongs to R_{0} − Rings_{1}.

C4. Noncommutative projective spectra.

C4.1. Proj_{G}. Fix a monoid G. Let 𝔽_{+} = (F_{+}, μ) be a G-graded (non-unital in general) monad on X. Let gr_{G}𝔽_{+} − mod_{1} denote the category of G-graded non-unital 𝔽_{+}-modules and preserving gradings morphisms. Let

gr_{G}𝔽_{+} − mod_{1} ⟶^{π^{∗}} 𝔽_{+} − mod_{1}  (1)

be the functor forgetting the grading. We denote by gr_{G}T_{𝔽_{+}} the preimage of the subcategory T_{𝔽_{+}} in gr_{G}𝔽_{+}−mod_{1}. Let C_{Proj_{G}}(𝔽_{+}) be the quotient category gr_{G}𝔽_{+}−mod_{1}/gr_{G}T_{𝔽_{+}}^{−}. This defines a ’space’ Proj_{G}(𝔽_{+}) = Proj_{G}(𝔽_{+}/X).

C4.2. Actions. Let G be a monoid. An action of G on a ’space’ X is a monoidal functor G ⟶^{𝔏} Ẽnd(C_{X}). Here G is viewed as a discrete monoidal category and Ẽnd(C_{X}) denote the (strict) monoidal category of endofunctors C_{X} ⟶ C_{X}; i.e. Ẽnd(C_{X}) = (End(C_{X}), ∘).

C4.2.1. Examples. (a) Let 𝔽_{+} be a (non-unital in general) G-graded monad on a ’space’ X; and let C_{Y} = gr_{G}𝔽_{+} − mod_{1}. For any G-graded 𝔽_{+}-module N = ⊕_{ν∈G}N_{ν} and any γ ∈ G, we denote by N[γ] the G-graded 𝔽_{+}-module defined by N[γ]_{σ} = N_{σγ}. This defines a strict action of G on the ’space’ Y. Here strict means that the monoidal functor G ⟶^{𝔏} Ẽnd(C_{X}) is strict, that is N[γ_{1}γ_{2}] = (N[γ_{2}])[γ_{1}] for all N.

(b) The action of G on the ’space’ Y in (a) (i.e. on the category gr_{G}𝔽_{+} − mod_{1}), induces an action of G on Proj_{G}(𝔽_{+}).

C4.3. Proposition. Let 𝕋 be a G-stable, topologizing subcategory of gr_{G}𝔽_{+} − mod_{1}, and let 𝕋̃ denote the image of 𝕋 in 𝔽_{+} − mod_{1}. Then π^{∗^{−1}}(𝕋̃^{−}) ⊆ 𝕋^{−}.

If the ’space’ X has the property (sup) and the functor F_{+} preserves supremums of subobjects, then π^{∗^{−1}}(𝕋̃^{−}) = 𝕋^{−}.

Proof. (a) Since the functor (1) is exact, the preimage, π^{∗^{−1}}(𝕋̃^{−}), of the Serre subcategory 𝕋̃^{−} is a Serre subcategory of the category gr_{G}𝔽_{+} − mod_{1}. The inclusion π^{∗^{−1}}(𝕋̃^{−}) ⊆ 𝕋^{−} is equivalent to that every nonzero object of π^{∗^{−1}}(𝕋̃^{−}) has a nonzero subobject which belongs to 𝕋; or, what is the same, for any nonzero object, N, of π^{∗^{−1}}(𝕋̃^{−}), there exists a nonzero morphism L ⟶^{g} N, with L ∈ Ob𝕋. We can and will assume that L is generated by one of its homogeneous components. Then 𝔽_{+} − mod_{1}(L, N) is a Ggraded ℤ-module, and some of homogeneous components of the morphism g are nonzero. Replacing the module L by the module L[γ] for an appropriate γ ∈ G, we can assume that the homogeneous component of g of zero degree is nonzero. Thus, there exists a nonzero morphism L[γ] ⟶ N of graded 𝔽_{+}-modules. Since the subcategory 𝕋 is stable under the action of G, the object L[γ] belongs to 𝕋.

(b) Since X has the property (sup) and F preserves supremums, both categories, gr_{G}𝔽_{+} − mod_{1} and 𝔽_{+} − mod_{1} possess this property too. Therefore, every object, M, of 𝕋^{−} has a filtration, {M_{i} | i ≥ 0} such that M_{i} = sup(M_{j}|j < i), if i is a limit ordinal, and M_{i+1}/M_{i} belongs to 𝕋. But, this implies that M is an object of 𝕋̃^{−}; i.e. we have the inverse inclusion, 𝕋^{−} ⊆ π^{∗^{−1}}(𝕋̃^{−}).

C4.3.1. Corollary. (a) π^{∗^{−1}}(T_{𝔽_{+}}^{−}) ⊆ gr_{G}T_{𝔽_{+}}^{−}.

(b) If the ’space’ X has the property (sup) and the functor F_{+} preserves supremums of subobjects, then gr_{G}T_{𝔽_{+}}^{−} = π^{∗^{−1}}(T_{𝔽_{+}}^{−}).

Proof. Set 𝕋 = gr_{G}T_{𝔽_{+}}. Then 𝕋̃^{−} coincides with T_{𝔽_{+}}^{−}, hence the assertion.

C4.3.2. Corollary. Suppose that X has the property (sup) and F_{+} preserves supremums of subobjects. Then the forgetful functor (1) induces a faithful exact functor

C_{Proj_{G}(𝔽_{+})} ⟶^{𝔭^{∗}} C_{Cone(𝔽_{+})}  (2)

Proof. By C4.3.1(b), gr_{G}T_{𝔽_{+}}^{−} = π^{∗^{−1}}(T_{𝔽_{+}}^{−}), where gr_{G}𝔽_{+}−mod_{1} ⟶^{π^{∗}} 𝔽_{+}−mod_{1} is the

forgetful functor. The functor π^{∗^{−1}} induces a faithful functor between quotient categories

gr_{G}𝔽_{+} − mod_{1}/gr_{G}T_{𝔽_{+}}^{−} ⟶ 𝔽_{+} − mod_{1}/T_{𝔽_{+}}^{−}.

This functor is exact because the inclusion functor gr_{G}𝔽_{+} − mod_{1} ⟶ 𝔽_{+} − mod_{1} is exact. Hence the assertion.

The functor (2) is regarded as an inverse image functor of a morphism (’projection’) Cone(𝔽_{+}) ⟶^{𝔭} Proj_{G}(𝔽_{+}).

C4.3.3. The Proj of an associative ring. Let R_{0} be an associative unital ring and G a monoid. Let R_{+} be a G-graded R_{0}-ring, which is, by definition, a G-graded ring in the category of R_{0}-bimodules. Then we have the category gr_{G}R_{+} − mod_{1} of G-graded R_{+}-modules and its subcategory gr_{G}T_{R_{+}} = T_{R_{+}} ∩ gr_{G}R_{+} − mod_{1}. We obtain the ’space’ Proj_{G}(R_{+}) = Proj_{G}(R) defined by

C_{Proj_{G}(R_{+})} = gr_{G}R_{+} − mod_{1}/gr_{G}T_{R_{+}}^{−}.

Since the conditions of C4.3.1(b) hold, gr_{G}T_{R_{+}}^{−} = gr_{G}R_{+} −mod_{1} ∩T_{R_{+}}^{−}, and, therefore, we have a canonical projection

Cone(R_{+}) ⟶^{𝔭} Proj_{G}(R_{+}).

Taking X = Sp(R_{0}) (i.e. C_{X} = R_{0} − mod), we can identify R_{+} with the monad 𝔽_{+} = (F_{+}, μ), where F_{+} = R_{+} ⊗_{R_{0}} − and F_{+}^{2} ⟶^{μ} F_{+} is determined by the multiplication R_{+} ⊗_{R_{0}} R_{+} ⟶^{m} R_{+}. If R_{+} is G-graded, then the monad 𝔽_{+} is G-graded. We have natural isomorphisms Cone(R_{+}) ≃ Cone(𝔽_{+}) and Proj_{G}(R_{+}) ≃ Proj_{G}(𝔽_{+}).

We have recovered the construction 1.7 illustrated by examples 1.8, 1.9, and 1.10. One can approach to these examples from a different side, via Hopf actions.

C5. Hopf actions. Let G be a monoid and R_{0} an associative unital k-algebra. For a G-graded (R_{0}|k)-ring R_{+}, we denote by gr_{G}R_{+} − mod_{1} the category of non-unital G-graded R_{+}-modules. Let H = (δ, H, μ) be a G-graded k-bialgebra with comultiplication δ and multiplication μ; and let H ⊗_{k} R_{+} ⟶^{τ} R_{+} is a Hopf action compatible with grading. Recall that a Hopf action of H on a k-algebra R_{+} is a unital H-module structure on R such that the multiplication R_{+} ⊗_{k} R_{+} ⟶ R_{+} is an H-module morphism. We assume that H acts trivially on R_{0}. Then the cross-product R_{+}#H is a G-graded (R_{0}|k)-ring.

The Hopf action of H on R_{+} induces an endofunctor, H_{G}, on the category gr_{G}R_{+} − mod_{1} which assigns to any (non-unital) G-graded R_{+}-module M = (M, R_{+} ⊗_{k} M →^{ξ} M) the G-graded R_{+}-module H ⊗_{k} M = (H ⊗_{k} M, ξ_{H}), where the action ξ_{H} is same as in the non-graded case (cf. C3.3.6). The multiplication H ⊗_{k} H ⟶^{μ} H gives rise to a monad H_{G} = (H_{G}, μ_{G}) (like in C3.3.6); and the category gr_{G}R_{+}#H − mod_{1} is isomorphic to the category H_{G} −mod. By an argument similar to that of C3.3.6, the monad H_{G} is continuous (i.e. the functor H_{G} has a right adjoint) which is equivalent to that the forgetful functor H_{G} − mod ⟶ gr_{G}R_{+} − mod_{1} is a direct image functor of an affine morphism.

C5.1. Proposition. Let H ⊗_{k} R_{+} ⟶^{τ} R_{+} be a Hopf action of an k-bialgebra H = (δ, H, μ) on a G-graded (R_{0}|k)-ring R_{+}. Suppose the functor H ⊗_{k} − is flat. Then the monad H̃ on Sp(R) induces a monad H̄ on Proj_{G}(R_{+}) such that there is a canonical commutative diagram

Sp(H̄/Proj_{G}(R_{+})) ⟶^{∼} Proj_{G}(R_{+}#H)
↘ ↙
Proj_{G}(R_{+})

(8)

of affine morphisms.

Proof. The argument is similar to that of C3.3.6.1. Details are left to the reader.

C5.2. Example. Let G be a connected reductive algebraic group over an algebraicly closed field k of zero characteristic. Fix a Borel subgroup B, a maximal unipotent subgroup U, and a maximal torus H chosen in a compatible way: H and U are subgroups of B, and B = HU). Let R be the algebra of regular functions on the homogeneous space G/U (called after I. M. Gelfand the ’base affine space’). The algebra R is the direct sum of all simple finite dimensional modules, each appears once; i.e. R = ⊕_{λ≥0}R_{λ}, where λ runs through nonnegative integral weights. Then R_{0} = k, and R_{+} = ⊕_{λ>0}R_{λ} is a G-graded k-algebra. Here G is the group of intergral weights of the group G.

The category C_{Cone(R_{+})} is equivalent to the category of quasi-coherent sheaves on the base affine space G/U. The category C_{Proj_{G}(R_{+})} is equivalent to the category of quasi-coherent sheaves on the flag variety G/B. We refer for details to [LR4].

C5.2.1. Note. If the group G is simply connected, this construction can be given in terms of the Lie algebra 𝔤 of G and its Cartan subalgebra 𝔥, as it is done in 1.9.

C5.2.2. D-modules. By construction, there is a Hopf action on R of the universal enveloping (Hopf) algebra U(𝔤). Consider instead of R the crossed product R_{+}#U(𝔤).

The universal enveloping algebra, H, of the Cartan subalgebra, 𝔥, acts on the algebra R according the decomposition R = ⊕_{λ≥0}R_{λ}: each R_{λ} is a one-dimensional representation of H with the weight λ tensored by the vector space R_{λ}. This is a Hopf action commuting with the action of U(𝔤), hence it determines to a Hopf action of Ũ(𝔤) = U(𝔤) ⊗_{k} H on R_{+}.

The category C_{Cone(R_{+}#_{U}(𝔤))} is equivalent to the category of D-modules on the base affine space G/U.

The category C_{Proj_{G}(R_{+}#_{U}(𝔤))} is equivalent to the category of D-modules on the flag variety G/B.

We can express these facts saying that the category of D-modules on the base affine space G/U is the category of quasi-coherent sheaves on the noncommutative quasi-affine ’space’ Cone(R_{+}#Ũ(𝔤)) and the category of D-modules on the flag variety G/B is the category of quasi-coherent sheaves on the noncommutative ’space’ Proj_{G}(R_{+}#Ũ(𝔤)). Both are semiseparated (actually, separated) noncommutative schemes.

C5.3. Example: quantum affine base space and quantum flag variety. Let U_{q}(𝔤) be the quantized enveloping algebra of a semisimple Lie algebra, 𝔤, and let H be its maximal torus (this time canonical). We define R and R_{+} as in C5.2; i.e. R = ⊕_{λ≥0}R_{λ} and R_{+} = ⊕_{λ>0}R_{λ}, where R_{λ} is the simple U_{q}(𝔤)-module with the highest weight λ. The multiplication is given by choosing projections R_{λ} ⊗ R_{μ} ⟶ R_{λ+μ} for different λ and μ in an appropriate way (see [LR4] for details).

We define the quantum base affine space of 𝔤 as the ’space’ Cone(R_{+}) and the quantum flag variety of 𝔤 as the ’space’ Proj_{G}(R_{+}).

C5.3.1. D-modules on the quantum base affine space and the quantum flag variety. Let U_{q}(𝔤) be the quantized enveloping algebra of a semisimple Lie algebra, 𝔤, and let H be its maximal a torus. Let R = ⊕_{λ≥0}R_{λ} be the algebra of functions on the quantum base affine space, and R_{+} = ⊕_{λ≥0}R_{λ} the quantum base affine space of 𝔤 which is by difinition the spectrum Cone(R_{+}) of the algebra R_{+} (see C5.2).

The maximal torus H acts on R_{+}, and this action commutes with the action of U_{q}(𝔤). Thus R_{+} has a structure of a Ũ_{q}(𝔤)-module, where Ũ_{q}(𝔤) = U_{q}(𝔤) ⊗_{k} H. By C2.2, Ũ_{q}(𝔤) induces a continuous monad, U_{q}^{∼}(𝔤), on Cone(R_{+}). And we have the commutative diagram

Sp(U_{q}^{∼}(𝔤)/Cone(R_{+})) ⟶^{∼} Cone(R_{+}#Ũ_{q}(𝔤))
↘ ↙
Cone(R_{+})

The action of Ũ_{q}(𝔤) on R_{+} respects G-grading, hence it induces a continuous monad, Ū_{q}(𝔤), on Proj_{G}(R_{+}); and we have a commutative diagram

Sp(Ū_{q}(𝔤)/Proj_{G}(R_{+})) ⟶^{∼} Proj_{G}(R_{+}#Ũ_{q}(𝔤))
↘ ↙
Proj_{G}(R_{+})

It is shown in [LR4] that the monad Ū_{q}(𝔤) on Proj_{G}(R_{+}) is compatible with the affine localizations

Sp((S_{w}^{−1}R)_{0}) ⟶ Proj_{G}(R_{+}), w ∈ W,  (1)

of the quantum flag variety described in 1.10.1. Similarly, the monad U_{q}^{∼}(𝔤) on Cone(R_{+}) is compatible with the affine localizations

Sp(S_{w}^{−1}R) ⟶ Cone(R_{+}), w ∈ W,  (2)

of the quantum base affine ’space’ (see 1.10.1).

Applying C2.2, we obtain that the affine cover (1) of the quantized flag variety, Proj_{G}(R_{+}), induces an affine cover of Proj_{G}(R_{+}#Ũ_{q}(𝔤)).

Similarly, the affine cover (1) of the quantum base affine space Cone(R_{+}) induces an affine cover of the ’space’ Cone(R_{+}#Ũ_{q}(𝔤)).

Moreover, it follows from C2.2 that in both cases the morphisms of affine cover are affine localizations. Therefore, Proj_{G}(R_{+}#Ũ_{q}(𝔤)) and Cone(R_{+}#Ũ_{q}(𝔤)) are examples of semiseparated noncommutative D-schemes.

C6. The category of coalgebras and the category of flat, affine covers.

C6.1. The category of coalgebras. Let R be an associative, unital ring, H = (H, δ) a coalgebra in the category of R-bimodules, ϕ : R ⟶ S a ring morphism. The comultiplication δ induces a comultiplication, δ_{ϕ} : H_{ϕ} ⟶ H_{ϕ} ⊗_{S} H_{ϕ}, on the S-bimodule H_{ϕ} = S ⊗_{R} H ⊗_{R} S. In fact, H_{ϕ} ⊗_{S} H_{ϕ} ≃ S ⊗_{R} H ⊗_{R} S ⊗_{R} H ⊗_{R} S, and the comultiplication δ_{ϕ} is determined by the composition of δ : H ⟶ H ⊗_{R} H and H ⊗_{R} H ⟶ H ⊗_{R} S ⊗_{R} H given by a ⊗ b ⟼ a ⊗ 1 ⊗ b.

We denote by ℭ𝔬𝔞𝔩𝔤 the category of coalgebras: its objects are pairs (R\H), where R is an associative ring and H = (H, δ) is a coalgebra in the category of R-bimodules. Morphisms from (R\H) to (R′\H′) are pairs (ϕ, λ_{ϕ}), where ϕ is a ring morphism R ⟶ R′, λ_{ϕ} is a morphism of coalgebras H_{ϕ} ⟶ H′ in the category of coalgebras in R-bimodules. The composition is defined in an obvious way.

We denote by ℭ𝔬𝔞𝔩𝔤_{𝔣𝔩} the full subcategory of ℭ𝔬𝔞𝔩𝔤 whose objects are (R\H), H = (H, δ), such that H is flat as a right R-module.

There is a forgetful functor

Φ^{∗} : ℭ𝔬𝔞𝔩𝔤 ⟶ Rings, (R\H) ⟼ R, (ϕ, λ_{ϕ}) ⟼ ϕ  (1)

which is right adjoint to the functor

Φ_{∗} : Rings ⟶ ℭ𝔬𝔞𝔩𝔤, R ⟼ (R\R), ψ ⟼ (ψ, ψ).  (2)

The adjunction morphisms are ϵ_{Φ} = id : Φ^{∗}Φ_{∗} ⟶ Id_{Rings} and η_{Φ} : Id_{ℭ𝔬𝔞𝔩𝔤} ⟶ Φ_{∗}Φ^{∗}; the latter morphism assigns to each object (R\H) of ℭ𝔬𝔞𝔩𝔤 the morphism (id_{R}, 𝔢_{H}), where 𝔢_{H} denotes the counit H ⟶ R of the coalgebra H (which is a coalgebra morphism). Since the first adjunction arrow is an isomorphism, the functor Φ_{∗} is fully faithful. In other words, the pair of functors ℭ𝔬𝔞𝔩𝔤 ⇄ Rings is a Q^{o}-category.

C6.2. The category 𝔄ℭov of affine covers. On the other hand, consider the category 𝔄ℭov of affine covers whose objects are flat, conservative, affine morphisms Sp(R) ⟶ X. Morphisms from Sp(R) ⟶^{ϕ} X to Sp(S) ⟶^{ψ} Y are commutative diagrams

\begin{matrix}Sp(R) & ⟶^{f} & Sp(S) \\ ϕ ↓ & & ↓ ψ \\ X & ⟶^{h} & Y \end{matrix}  (3)

such that the morphism Sp(R) ⟶^{f} Sp(S) is affine.

For each object (R\H) of the category ℭ𝔬𝔞𝔩𝔤, the forgetful functor (R\H)−Comod ⟶ R − mod is an inverse image functor of an affine morphism Sp(R) ⟶^{f} Sp(R\H). The morphism f is flat iff H is flat as a right R-module (see 7.3.1). Any morphism (ϕ, λ_{ϕ}) : (R\H) ⟶ (S\G) induces a functor

|ϕ, λ_{ϕ}|^{∗} : (R\H) − Comod ⟶ (S\G) − Comod,  (4)

(M, M →^{ξ} H ⊗_{R} M) ⟼ (ϕ^{∗}(M), ϕ^{∗}(M) → G ⊗_{S} ϕ^{∗}(M)),

where ϕ^{∗}(M) = S ⊗_{R} M, and the morphism ϕ^{∗}(M) → G ⊗_{S} ϕ^{∗}(M)) is the composition of

ϕ^{∗}(ξ) : ϕ^{∗}(M) ⟶ ϕ^{∗}(H ⊗_{R} M) = S ⊗_{R} H ⊗_{R} M,
S ⊗_{R} H ⊗_{R} M ⟶ S ⊗_{R} H ⊗_{R} S ⊗_{R} M,

and

λ_{ϕ} ⊗_{S} M : S ⊗_{R} H ⊗_{R} S ⊗_{R} M ⟶ G ⊗_{S} ϕ^{∗}(M) = G ⊗_{R} M.

The functor |ϕ, λ_{ϕ})|^{∗} is (regarded as) an inverse image functor of a morphism |ϕ, λ_{ϕ}| : Sp(S\G) ⟶ Sp(R\H). The map (ϕ, λ_{ϕ}) ⟼ |ϕ, λ_{ϕ}|^{∗} extends to a pseudo-functor ℭ𝔬𝔞𝔩𝔤 ⟶ Cat which induces a functor ℭ𝔬𝔞𝔩𝔤^{op} ⟶ |Cat|^{o}. It follows that the diagram

\begin{matrix}R − mod & ⟶^{ϕ^{∗}} & S − mod \\ ↑ & & ↑ \\ (R\H) − Comod & ⟶^{|ϕ,λ_{ϕ}|^{∗}} & (S\G) − Comod \end{matrix}  (5)

commutes. In particular, the corresponding diagram in |Cat|^{o},

Sp(S) ⟶^{|ϕ|} Sp(R)
↓  ↓
Sp(S\G) ⟶^{|ϕ,λ_{ϕ}|} Sp(R\H)

(6)

commutes. Thus we have a contravariant functor

ℭ𝔬𝔞𝔩𝔤^{op} ⟶ 𝔄ℭov  (7)

from the category of coalgebras to the category of affine covers.

C6.3. Proposition. The image of the functor (7) is equivalent to the category of affine covers.

Proof. Let

\begin{matrix}Sp(S) & ⟶^{f} & Sp(R) \\ ϕ ↓ & & ↓ ψ \\ X & ⟶^{h} & Y \end{matrix}  (8)

be a morphism of covers, i.e. the morphisms ϕ, ψ, f are affine and, in addition, ϕ and ψ are flat and conservative.

Choosing inverse image functors of each of the morphisms in the diagram, we obtain a quasi-commutative diagram

\begin{matrix}R − mod & ⟶^{f^{∗}} & S − mod \\ ϕ^{∗} ↑ & & ↑ ψ^{∗} \\ C_{Y} & ⟶^{h^{∗}} & C_{X} \end{matrix}  (9)

(a) Since f is affine, its direct image functor, f_{∗} is the composition of and equivalence of the categories (Morita equivalence) S − mod ⟶ S′ − mod and the pull-back functor S′ − mod ⟶ R − mod by a ring morphism f̂ : R ⟶ S′ (cf. 6.6.1). Thus the inverse image functor f^{∗} of f is the composition of f̂^{∗} = S′ ⊗_{R} − : R − mod ⟶ S′ − mod and an equivalence S′ − mod ⟶ S − mod. Replacing S by S′, we shall assume that the functor

f^{∗} in (9) is S ⊗_{R} − for a ring morphism R \underrightarrow{_{f}} S.

(b) The inverse image functor h^{∗} defines a canonical functor morphism

(ψf)^{∗}(ψf)_{∗} ⟶^{λ_{h}} ϕ^{∗}ϕ_{∗}.  (10)

Denote for convenience the composition ψf by g. The quasi-commutativity of (9) means that there is an isomorphism 𝔠 = 𝔠_{ϕ,h} : ϕ^{∗}h^{∗} ⟶^{∼} g^{∗}. This isomorphism induces a morphism

h^{∗} ⟶^{fc′} ϕ_{∗}g^{∗}. We define λ_{h} as the composition of the following morphisms:

g^{∗}g_{∗} ⟶^{𝔠^{−1}g_{∗}} ϕ^{∗}h^{∗}g_{∗} ⟶^{ϕ^{∗}𝔠′g_{∗}} ϕ^{∗}ϕ_{∗}g^{∗}g_{∗} ⟶^{ϕ^{∗}ϕ_{∗}} ϕ^{∗}ϕ_{∗}.  (11)

(c) The morphism g^{∗}g_{∗} ⟶^{λ_{h}} ϕ^{∗}ϕ_{∗} is a comonad morphism.

The proof of this fact follows from the argument of Proposition 1.0.7.1 in [R4], where the siminar fact is proven for the dual situation. One needs only to switch to dual categories and functors.

(d) Since the functors ϕ^{∗} and ψ^{∗} are flat and conservative, the diagram (9) can be included into the diagram

\begin{matrix}R − mod & ⟶^{f^{∗}} & S − mod \\ ϕ^{∗} ↑ & & ↑ ψ^{∗} \\ C_{Y} & ⟶^{h^{∗}} & C_{X} \\ ↑ & & ↑ \\ (R\G_{ϕ}) − Comod & ⟶^{𝔥^{∗}} & (S\G_{ψ}) − Comod \end{matrix}  (12)

Here G_{ϕ} is the comonad (ϕ^{∗}ϕ_{∗}, δ_{ϕ}) and G_{ψ} = (ψ^{∗}ψ_{∗}, δ_{ψ}), lower vertical arrows are canonical category equivalences given by Beck’s theorem. The functor 𝔥^{∗} is induced by the monad morphism λ_{h} defined by (11) as follows. To any G_{ψ}-comodule L = (L, L →^{ξ} ψ^{∗}ψ_{∗}(L)), the functor 𝔥^{∗} assigns the G_{ϕ}-comodule (f^{∗}(L), ξ′), where the G_{ϕ}-coalgebra structure, ξ′ : f^{∗}(L) ⟶ ϕ^{∗}ϕ_{∗}f^{∗}(L), is the composition of

f^{∗}(L) ⟶^{f^{∗}(ξ)} f^{∗}ψ^{∗}ψ_{∗}(L) ⟶^{f^{∗}ψ^{∗}ψ_{∗}η_{f}} f^{∗}ψ^{∗}ψ_{∗}f_{∗}f^{∗}(L) ⟶^{∼} g^{∗}g_{∗}f^{∗}(L) ⟶^{λ_{h}f^{∗}(L)} ϕ^{∗}ϕ_{∗}f^{∗}(L).

The diagram

\begin{matrix}R − mod & ⟶^{f^{∗}} & S − mod \\ ϕ_{1}^{∗} ↑ & & ↑ ψ_{1}^{∗} \\ (R\G_{ϕ}) − Comod & ⟶^{𝔥^{∗}} & (S\G_{ψ}) − Comod \end{matrix}  (13)

obtained from (12) by composing the vertical arrows commutes. Here ψ_{1}^{∗} and ϕ_{1}^{∗} are functors forgeting the respective comodule structures.

(e) It follows from (the argument of) 7.3.1 that the comonad G_{ϕ} is isomorphic to the comonad H_{ϕ}⊗_{R}− for a coalgebra H_{ϕ} = (H_{ϕ}, δ_{ϕ}) in the category of R-bimodules determined uniquely up to isomorphism (see 7.3.1(2)). Therefore the category (R\G_{ϕ}) − Comod is naturally isomorphic to the category (R\H_{ϕ}) − Comod of (R\H_{ϕ})-comodules. Similarly, (S\G_{ψ}) − Comod ≃ (S\H_{ψ}) − Comod for a coalgebra H_{ψ} in the category of S-bimodules.

The morphism λ_{h} (defined by (11)) together with the ring morphism R ⟶^{f̂} S (cf. (a)) define a coalgebra morphism (f̂, λ′_{h}) : H_{ϕ} ⟶ H_{ψ} such that the diagram

\begin{matrix}(R\G_{ϕ}) − Comod & ⟶^{𝔥^{∗}} & (S\G_{ψ}) − Comod \\ ↑ & & ↑ \\ (R\H_{ϕ}) − Comod & ⟶^{(f̂,λ′_{h})^{∗}} & (S\H_{ψ}) − Comod \end{matrix}  (14)

commutes. Combining (13) and (14), we obtain a commutative diagram

\begin{matrix}R − mod & ⟶^{f^{∗}} & S − mod \\ ↑ & & ↑ \\ (R\H_{ϕ}) − Comod & ⟶^{(f̂,λ′_{h})^{∗}} & (S\H_{ψ}) − Comod \end{matrix}  (15)

in which the vertical arrows are functors forgetting the comodule structrure. This implies the assertion.

C6.4. Corollary. Let

\begin{matrix}Sp(R) & ⟶^{f} & Sp(S) \\ ϕ ↓ & & ↓ ψ \\ X & ⟶^{h} & Y \end{matrix}  (3)

be a morphism of affine flat covers. Then the morphism X ⟶^{h} Y has a direct image functor.

Proof. Thanks to C6.3, it suffices to show that for any coalgebra morphism

(ϕ, λ_{ϕ}) : (R\H) ⟶ (S\G)

the functor

|ϕ, λ_{ϕ}|^{∗} : (R\H) − Comod ⟶ (S\G) − Comod

has a right adjoint. This follows from 5.3.2.2.

C6.5. Proposition. Let Sp(R) ⟶^{ϕ} X and Sp(S) ⟶^{ψ} Y be affine flat covers. Then X is isomorphic to Y iff there exist affine flat covers Sp(R) ⟵^{f} Sp(T) ⟶^{g} Sp(S) such that the associated coalgebras H_{ϕf} and H_{ψg} in the category of T -bimodules are isomorphic.

Proof. (a) For any two affine flat covers, Sp(R) ⟶^{ϕ} X and Sp(T) ⟶^{f} Sp(R), the categories H_{ϕ} − Comod and H_{ϕf} − Comod are equivalent, or, what is the same, Sp(R\H_{ϕ}) ≃ Sp(T, H_{ϕf}).

In fact, the composition of affine flat covers is an affine flat cover. Therefore, by Beck’s theorem, the category C_{X} is equivalent both to H_{ϕ} − Comod and H_{ϕf} − Comod.

(b) Let

X ⟵^{ϕ} Sp(R) ⟵^{f} Sp(T) ⟶^{g} Sp(S) ⟶^{ψ} Y

be affine flat covers such that the coalgebras H_{ϕf} and H_{ψg} are isomorphic. By (a), the category C_{X} is equivalent to (T \H_{ϕf})−Comod, the category C_{Y} is equivalent to (T \H_{ψg})− Comod, and the isomorphism of the coalgebras induces an isomorphism of the categories of comodules. Hence C_{X} is equivalent to C_{Y}.

(c) Let Sp(R) ⟶^{ϕ} X and Sp(S) ⟶^{ψ} Y be affine covers and α an isomorphism Y ⟶^{∼} X. We claim that there exist affine covers Sp(R) ⟵^{f} Sp(T) ⟶^{g} Sp(S) such that the diagram

\begin{matrix}Sp(T) \\ ^{f} ↙  ↘^{g} \\ Sp(R)  Sp(S) \\ ϕ ↓  ↓ ψ \\ X ⟵^{α} Y \end{matrix}  (17)

commutes. Replacing ψ by αψ, we assume that α is the identical isomorphism X ⟶ X.

The category C_{X} is equivalent to the category (R\H_{ϕ}) − Comod for a coalgebra H_{ϕ} = (H_{ϕ}, δ_{ϕ}) in the category of R-bimodules such that H_{ϕ} is flat as a right R-module. This implies that C_{X} is a Grothendieck category. In particular, it has arbitrary (small) colimits.

Let F_{ϕ} = (F_{ϕ}, μ_{ϕ}) = (ϕ_{∗}ϕ^{∗}, μ_{ϕ}) and F_{ψ} = (F_{ψ}, μ_{ψ}) = (ψ_{∗}ψ^{∗}, μ_{ψ}) be monads associated with resp. ϕ and ψ. By 2.6.2.3 in [R4], there exists a free product, F_{ϕ} ⋆ F_{ψ}, of the monads F_{ϕ} and F_{ψ}. The canonical monad morphisms F_{ϕ} ⟶ F_{ϕ} ⋆ F_{ψ} ⟵ F_{ψ} induce a commutative diagram

\begin{matrix}(F_{ϕ} ⋆ F_{ψ}/X) − mod & ⟶ & (F_{ϕ}/X) − mod \\ ↓ & & ↓ \\ (F_{ψ}/X) − mod & ⟶ & C_{X} \end{matrix}  (18)

of the corresponding direct image functors. Since the monads F_{ϕ} and F_{ψ} are continuous, their coproduct, F_{ϕ} ⋆ F_{ψ}, is continuous too (by 2.6.2.3 in [R4]). This implies that all morphisms in the diagram (18) are affine. By Beck’s theorem, the category (F_{ϕ}/X) − mod is equivalent to the category R − mod and (F_{ψ}/X) − mod is equivalent to the category S − mod. Applying the Beck’s theorem to the (direct image of the) affine morphism (F_{ϕ} ⋆ F_{ψ}/X) − mod ⟶ R − mod, we obtain the commutative diagram

\begin{matrix}T − mod & ⟶ & (F_{ϕ} ⋆ F_{ψ}/X) − mod \\ f_{∗} ↓ & & ↓ \\ R − mod & ⟶ & (F_{ϕ}/X) − mod \end{matrix}  (19)

of (direct image) functors in which the horizontal arrows are category equivalences and the ring, the morphism f_{∗} is the pull-back of a ring morphism f : R ⟶ T which is defined uniquely up to isomorphism. Combining (18) and (19) and using an equivalence of (F_{ψ}/X) − mod to S − mod, we obtain a quasi-commutative diagram

\begin{matrix}T − mod & ⟶^{g_{∗}} & S − mod \\ f_{∗} ↓ & & ↓ ψ_{∗} \\ R − mod & ⟶^{ϕ_{∗}} & C_{X} \end{matrix}  (20)

Since the inverse image functors (F_{ϕ}/X) − mod ⟵ C_{X} ⟶ (F_{ψ}/X) − mod are flat and conservative, the inverse image functors

(F_{ϕ}/X) − mod ⟶ (F_{ϕ} ⋆ F_{ψ}/X) − mod ⟵ (F_{ψ}/X) − mod

have the same property (by 2.6.4.1, or 4.3.2 in [R4]). Therefore f_{∗} and g_{∗} in (19) are direct image functors of affine flat covers.

C6.5.1. The dual description. Let Sp(R) ⟶^{f} X be an affine morphism. Denote by f^{∧} the dual morphism X ⟶ Sp(R) with an inverse image functor f_{∗} (cf. 3.4). Since f is affine, the morphism f^{∧} is continuous: f_{∗}^{∧} = f^{!}. By 4.1, the morphism f^{∧} (hence the morphism f) is uniquely defined by the right R-module (O_{∗}, ϕ̂), where O_{∗} = f^{∧∗}(R) = f_{∗}(R) and ϕ̂ is the canonical ring morphism R ⟶ C_{X}(O_{∗}, O_{∗})^{o} = C_{X}(f_{∗}(R), f_{∗}(R))^{o}. The functor f_{∗}^{∧} = f^{!} is the composition of the functor

C_{X}(O_{∗}, −) : C_{X} ⟶ C_{X}(O_{∗}, O_{∗})^{o} − mod

and the pull-back

ϕ̂_{∗} : C_{X}(O_{∗}, O_{∗})^{o} − mod ⟶ R − mod

by the ring morphism ϕ̂. Let X = Sp(R\H), where H = (H, δ) is a coalgebra in the category of R-bimodules, and let f be the standard morphism Sp(R) ⟶ Sp(R\H) having as an inverse image functor the forgetful functor (R\H)−Comod ⟶ R−mod (cf. 7.3.1(1)) and as a direct image functor the functor L ⟼ H ⊗_{R} L (cf. 7.3.1(2)). Then the object O_{∗} coincides with the H-comodule H = (H, δ), and the functor f̃_{∗} = f^{!} is isomorphic to

(R\H) − Comod ⟶ R − mod, L ⟼ Hom_{R}(H, L)  (1)

C6.6. Affine covers over the spectrum of a ring.

C6.6.1. Affine covers over Sp(S). Fix an associative ring S and denote by 𝔄ℭ𝔬𝔳/Sp(S) the category of affine covers over Sp(S). Its objects are diagrams

Sp(R) ⟶^{ϕ} X ⟶^{f} Sp(S)

such that ϕ is affine, flat, and conservative, and the composition f ∘ ϕ is affine. Morphisms from Sp(R) →^{ϕ} X →^{f} Sp(S) to Sp(T) →^{ψ} Y →^{g} Sp(S) are commutative diagrams

\begin{matrix}Sp(R) ⟶^{γ} Sp(T) \\ ϕ ↓  ↓ ψ \\ X ⟶^{h} Y \\ _{f} ↘  ↙_{g} \\ Sp(S) \end{matrix}  (1)

The composition is defined in an obvious way. Notice that, since the compositions f ∘ ϕ and g ∘ ψ in (1) are affine, the morphism γ is affine too (see 6.4).

C6.6.2. Coalgebras over a ring. For convenience, we denote by S\ℭ𝔬𝔞𝔩𝔤 the category (S\S)\ℭ𝔬𝔞𝔩𝔤 of coalgebras over the coalgebra (S\S). It follows from definitions that the category S\ℭ𝔬𝔞𝔩𝔤 admits the following description. Objects of S\ℭ𝔬𝔞𝔩𝔤 are triples ((R\H), ϕ, τ), where (R\H) is an object of the category ℭ𝔬𝔞𝔩𝔤 (i.e. H = (H, δ) is a coalgebra in the category of R-bimodules), ϕ is a ring morphism S ⟶ R, and τ is a coalgebra morphism (R\R ⊗_{S} R) ⟶ (R\H). In particular, ϵ ∘ τ : R ⊗_{S} R ⟶ R is the morphism induced by the multiplication, μ_{R}, on R (cf. C6.1). Here ϵ denotes the counit, H ⟶ R, of the coalgebra H.

For any ring S, the map which assigns to every object ((R\H), S →^{ϕ} R, τ,) of the category S\ℭ𝔬𝔞𝔩𝔤 the diagram

Sp(R) = Sp^{o}(R\R) ⟶^{Sp^{o}(ϵ)} Sp^{o}(R\H) ⟶^{Sp^{o}(ϕ,τ)} Sp^{o}(S\S) = Sp(S)  (2)

extends naturally to a functor

(S\ℭ𝔬𝔞𝔩𝔤)^{op} ⟶ 𝔄ℭ𝔬𝔳/Sp(S)  (3)

C6.6.3. Proposition. The image of the functor (3) is equivalent to the category 𝔄ℭ𝔬𝔳/Sp(S).

Proof. Let Sp(R) ⟶^{ϕ} X ⟶^{f} Sp(S) be an affine cover over Sp(S). Replacing it by an isomorphic cover, we can assume that X = Sp^{o}(R\H) for a coalgebra (R\H), the inverse image ϕ^{∗} of ϕ is the forgetful functor (R\H) − Comod ⟶ R − mod, and the composition f ∘ ϕ corresponds to a ring morphism S ⟶ R. The latter means that (f ∘ ϕ)^{∗}(S) ≃ ϕ^{∗}(f^{∗}(S)) ≃ R, where the isomorphisms are (R, S)-bimodule isomorphisms. We can assume that ϕ^{∗}(f^{∗}(S)) = R. Since ϕ^{∗} is the functor forgetting the coaction, the (R\H)-comodule f^{∗}(S) is of the form (R, γ) for some coaction γ : R ⟶ H ⊗_{R}R ≃ H. The fact that the R-module morphism γ is an H-comodule structure and an (R, S)-bimodule morphism implies that its composition with the natural isomorphism H ⊗_{R} R ⟶^{∼} H induces a coalgebra morphism (R\R⊗_{S} R) ⟶ (R\H). Thus we have assigned to an affine cover over Sp(S) a coalgebra over S which the functor (3) maps to an affine cover that is isomorphic to the affine cover we have started with.

C6.6.4. Remark. Proposition C6.6.3 can be also deduced directly from C6.3 as follows. Since ℤ is the initial object of the category Rings of unital rings, the fully faithful functor

Φ_{∗} : Rings ⟶ ℭ𝔬𝔞𝔩𝔤, R ⟼ (R\R),

induces a fully faithful functor Φ_{∗}^{+} : Rings ⟶ ℤ\ℭ𝔬𝔞𝔩𝔤 which is right adjoint to the forgetful functor

Φ^{+∗} : ℤ\ℭ𝔬𝔞𝔩𝔤 ⟶ Rings, (R\H, R ⊗_{ℤ} R →^{τ} H) ⟼ R.

(cf. C6.1). Moreover, for any ring S the natural functor

(S\S)\(ℤ\ℭ𝔬𝔞𝔩𝔤) ⟶ (S\S)\ℭ𝔬𝔞𝔩𝔤

is an isomorphism of categories.

C7. Affine flat covers and descent of flat morphisms. If U = Sp(R) and V = Sp(S), then the category C_{Hom^{c}(U,V)} is equivalent to the category of (R, S)-bimodules, or, equivalently, Hom^{c}(U, V) is isomorphic to Sp(R ⊗ S^{o}) (cf. 9.2.1).

The object Hom^{wfl}(U, V) corresponds to the full subcategory of the category of (R, S)-bimodules whose objects are bimodules which are flat as right S-modules. We shall write Hom^{fl}(U, V) instead of Hom^{wfl}(U, V).

Notice that since the categories C_{X} and C_{Y} are abelian, weakly flat morphisms are just flat. By this reason, we shall write Hom^{fl}(X, Y) instead of Hom^{wfl}(X, Y).

Let U ⟶^{u} X and V ⟶^{v} Y be affine flat covers (i.e. affine, flat, conservative morphisms); hence X ≃ Sp^{o}(R\H_{u}) and Y ≃ Sp^{o}(S\H_{v}), where H = (H_{u}, δ_{u}) is a coalgebra in the category of R-bimodules and H_{v} = (H_{v}, δ_{v}) is a coalgebra in the category of Sbimodules. The coalgebras H_{u}, H_{v} determine a coalgebra H_{(u|v)} = (H_{(u|v)}, δ_{(u|v)}) in the category of R⊗S^{o}-bimodules which is naturally identified with H_{u}⊗H_{v}^{o}. The latter follows from the fact that the tensoring by H_{(u|v)} maps the bimodule R ⊗ S^{o} to H_{u} ⊗ H_{v} (i.e. H_{(u|v)} = H_{u} ⊗ H_{v}), and the comultiplication δ_{(u|v)} on H_{u} ⊗ H_{v} is the one induced by the comultiplications (δ_{u}, δ_{v}).

C7.1. Proposition. Let Sp(R) ⟶^{u} X and Sp(S) ⟶^{v} Y be affine flat covers. Then (a) The pair (u, v) determines an affine morphism

Sp(R ⊗ S^{o}) ⟶ Hom^{c}(X, Y).  (1)

which is the decomposition of

Sp(R ⊗ S^{o}) ⟶ Sp^{o}(R ⊗ S^{o}\H_{u} ⊗ H_{v}^{o})  (2)

and

Sp^{o}(R ⊗ S^{o}\H_{u} ⊗ H_{v}^{o}) ⟶ Hom^{c}(X, Y)  (3)

(b) The morphism (3) is an exact localization (i.e. its inverse image functor is an exact localization). Equivalently, an inverse image functor of (3) is exact and its direct image functor is fully faithful.

(c) The morphism (1) (or (3)) induces an equivalence of the category C_{Hom^{fl}(X,Y)} of flat morphisms X ⟶ Y and the full subcategory of the category (R⊗S^{o}\H_{u}⊗H_{v}^{o})−Comod whose objects are comodules (M, ξ) such that the R ⊗ S^{o}-module M is flat as a right Smodule.

Proof. The assertion follows from 9.7 and the Beck’s theorem 5.4.1.

C7.2. Morphisms corresponding to bicomodules. Under the conditions of C7.1, we can identify X with Sp^{o}(R\H_{u}) and Y with Sp^{o}(S\H_{v}).

One can define a (fully faithful) direct image functor of the morphism (11) as follows. Let M be a H_{(u|v)}-comodule which is convenient to regard as an (H_{u}, H_{v})-bicomodule,

M = (M, H_{u} ⊗_{S} M ←^{ζ_{u}} M →^{ζ_{v}} M ⊗_{S} H_{v}). It follows from 10.7.1 that a direct image functor of the morphism (3) assigns to M a functor

C_{Y} = (S\H_{v}) − Comod ⟶ (R\H_{u}) − Comod = C_{X}

which maps each H_{v}-comodule, L = (L, ξ_{L}) to a kernel of the pair of morphisms

M ⊗_{S} L \overrightarrow{⟶}^{M⊗_{S} ξ_{L}}_{ζv ⊗L} M ⊗_{S} H_{v} ⊗_{S} L.  (4)

with a comodule structure induced by the coaction ζ_{u} : M ⟶ H_{u} ⊗_{S} M.

C7.2.1. Composition of morphisms. Let u : Sp(R) ⟶ X, v : Sp(S) ⟶ Y and w : Sp(T) ⟶ Z be affine flat morphisms which allows to assume that X = Sp^{o}(R\H_{u}), Y = Sp^{o}(S\H_{v}), and Z = Sp^{o}(T \H_{w}). Let M = (M, ζ_{u}, ζ^{v}) be a ((R\H_{u}), (S\H_{v}))- bicomodule and N = (N, ζ_{v}, ζ^{w}) a ((S\H_{v}), (T \H_{w}))-bicomodule.

Denote by M ⊠_{(S\H_{v})} N, or simply by M ⊠ N, the kernel of the diagram

M ⊗_{S} N \overrightarrow{⟶}^{ζ^{v}}_{ζv} M ⊗_{S} H_{v} ⊗_{S} N.

The right coaction ζ^{w} : N ⟶ N ⊗_{S} H_{w} induces a right (T \H_{w})-comodule structure, and the left (R\H_{u})-comodule structure ζ_{u} : M ⟶ H_{u} ⊗_{S} M induces a left (R\H_{u})- comodule structure on M ⊠N. Thus we obtain a ((R\H_{u}), (T \H_{w}))-bicomodule M⊠N = (M ⊠ N, ζ_{u} ⊠ N, M ⊠ ζ^{w}).

There is a natural morphism from the composition ϕ_{M}^{∗} ∘ ϕ_{N}^{∗} of the functors corresponding to the resp. bicomodules N and M to the functor

ϕ_{M⊠N}^{∗} : (T \H_{w}) − Comod ⟶ (R\H_{u}) − Comod

corresponding to the (R\H_{u}), (T \H_{w}))-bicomodule M ⊠ N. This morphism is an isomorphism if M is flat as a right S-module.

C7.3. The bicategory of affine flat covers. We define the bicategory of affine flat covers as follows. Its objects are affine flat covers Sp(R) →^{u} X. The category of morphisms from Sp(R) →^{u} X to Sp(S) →^{v} Y is the category of ((R\H_{u}), (S\H_{v}))-bicomodules. The composition is given by the tensor product (⊠, see C7.2.1) of bicomodules.

C7.3.1. Proposition. The category of morphisms from a flat cover Sp(R) →^{u} X to a flat cover Sp(S) →^{v} Y is determined by the objects X and Y uniquely up to equivalence.

Proof. Let Sp(R) →^{u} X and Sp(R)′ →^{u′} X be two affine flat covers of X. By C6.5,

there exists affine flat covers Sp(R) ←^{p} Sp(T) →^{p′} Sp(R)′ such that u ∘ p = u′ ∘ p′. Thus, it suffices to show that the category of morphisms from Sp(R) →^{u} X to Sp(S) →^{v} Y is equivalent to the category of morphisms from Sp(T) →^{up} X to Sp(S) →^{v} Y.

The functor which assigns to every ((R\H_{u}), (S\H_{v}))-bicomodule M = (M, ζ_{u}, ζ_{v}) the ((T \H_{up}), (S\H_{v}))-bicomodule T ⊗_{S} M = (M, ζ_{up}, ζ_{v}) is a category equivalence. This follows from 7.4.5, or directly from the Beck’s theorem (see the part (a) of the argument of 7.4.5).

C7.4. Descent of morphisms over Sp(T). Fix an object Z of the category |Cat|^{o}. For any two objects X = (X, X →^{f} Z), Y = (Y, Y →^{f} Z) of the category |Cat|^{o}/Z, we denote by Hom_{Z}(X, Y) (resp. Hom_{Z}^{c}(X, Y), resp. Hom_{Z}^{fl}(X, Y)) the object of |Cat|^{o} corresponding to the category of functors ϕ^{∗} : C_{Y} ⟶ C_{X} (resp. having a right adjoint, resp. exact and having a right adjoint) such that the diagram

\begin{matrix}C_{Y} ⟶^{ϕ^{∗}} C_{X} \\ _{g^{∗}} ↖ ↗_{f^{∗}} \\ C_{Z} \end{matrix}  (5)

quasi-commutes.

Suppose that Z = Sp(T) for an associative unital ring T; and let u : Sp(R) ⟶ X and v : Sp(S) ⟶ Y be affine flat covers such that the compositions Sp(R) ⟶^{fu} Sp(T) ⟶^{gv} Sp(S) are affine. Thanks to 7.6.3 (see also 7.6.2), we can assume that the compositions f ∘u, g∘v correspond to ring morphisms resp. T ⟶ R and T ⟶ S, and X = Sp^{o}(R\H_{u}) and Y = Sp^{o}(S\H_{v}) for coalgebras (R\H_{u}) and (S\H_{v}) in resp. the category of R- and S-bimodules. The morphisms X ⟶ Sp(T) and Y ⟶ Sp(T) are described by resp.

(T →^{ϕ_{f}} R, (R\R ⊗_{T} R) ⟶^{τ_{u}} (R\H_{u})) and (T →^{ϕ_{f}} S, (S\S ⊗_{T} S) ⟶^{τ_{u}} (S\H_{v})),

(cf. C6.6.2). The quasi-commutativity of the diagram (5) implies in particular that ϕ^{∗}(O_{Y}) ≃ O_{X}. Here O_{X} = f^{∗}(O_{T}) = f^{∗}(T), and O_{Y} = g^{∗}(O_{T}) = g^{∗}(T). The morphisms f and g are continuous (this follows from 7.6.3). Therefore, by 3.4.1, the object O_{X} (resp. O_{Y}) determines the morphism f (resp. g) uniquely up to isomorphism. This, in turn, implies that the diagram (5) quasi-commutes if and only if ϕ^{∗}(O_{Y}) ≃ O_{X}. The

object O_{X} = f^{∗}(O_{T}) = f^{∗}(T) is (isomorphic to) the (R\H_{u})-comodule (R, R →^{τ′_{u}} H_{u}),

and O_{Y} = g^{∗}(O_{T}) = g^{∗}(T) is (isomorphic to) the (S\H_{v})-comodule (S, S →^{τ′_{v}} H_{v}). The coactions τ′_{u} and τ′_{v} induce the coalgebra morphisms (R\R ⊗_{T} R) ⟶^{τ_{u}} (R\H_{u}) and (S\S ⊗_{T} S) ⟶^{τ_{v}} (S\H_{v}) which appear above (cf. C6.6.2).

Suppose the functor ϕ^{∗} : C_{Y} ⟶ C_{X} is given by an ((R\H_{u}), (S\H_{v}))-bicomodule M = (M, H_{u} ⊗_{R} M ←^{ζ_{u}} M →^{ζ_{v}} M ⊗_{S} H_{v}). Then by C7.2 applied to the S\H_{v})-comodule O_{Y} = (S, τ_{v}), the (R\H_{u})-comodule ϕ^{∗}(O_{Y}) is a kernel of the pair of morphisms

M \overrightarrow{⟶}^{M⊗_{S} τv}_{ζv} M ⊗_{S} H_{v}.  (6)

with a comodule structure induced by the coaction ζ_{u} : M ⟶ H_{u} ⊗_{S} M.

Therefore, the diagram (5) is quasi-commutative iff the kernel of (6) is isomorphic to u^{∗}(O_{X}) = R and the (R\H_{u})-comodule structure induced on R by M ⟶^{ζ_{u}} H_{u} ⊗_{R} M is isomorphic to τ_{u} : R ⟶ H_{u}.

The argument above proves the following

C7.4.1. Proposition. An ((R\H_{u}), (S\H_{v}))-bicomodule M = (M, ζ_{u}, ζ_{v}) determines a continuous morphism from X = (X, X →^{f} Sp(T)) to Y = (Y, Y →^{g} Sp(T)) iff there exists an (R\H_{u})-comodule morphism λ : (R, τ_{u}) ⟶ (M, ζ_{u}) such that the diagram

R ⟶^{λ} M \overrightarrow{⟶}^{M⊗_{S} τv}_{ζv} M ⊗_{S} H_{v}  (7)

is exact.

C7.4.2. Corollary. The category of flat morphisms from X = (X, X →^{f} Sp(T)) to Y = (Y, Y →^{g} Sp(T)) is equivalent to the category whose objects are pairs (M, λ), where M = (M, ζ_{u}, ζ_{v}) is an ((R\H_{u}), (S\H_{v}))-bicomodule such that M is flat as a right S-module, and λ an (R\H_{u})-comodule morphism (R, τ_{u}) ⟶ (M, ζ_{u}) such that the diagram

R ⟶^{λ} M \overrightarrow{⟶}^{M⊗_{S} τv}_{ζv} M ⊗_{S} H_{v}  (7)

is exact. Morphisms (M, λ) ⟶ (M′, λ′) are ((R\H_{u}), (S\H_{v}))-bicomodule morphisms ψ : M ⟶ M′ which make the diagram

\begin{matrix}(M, ζ_{u}) ⟶^{ψ} (M′, ζ′_{u}) \\ _{λ} ↖  ↗_{λ′} \\ (R, τ_{u}) \end{matrix}  (8)

commute.

Proof. The fact follows from C7.1(c) and C7.4.1. Details are left to the reader.

C8. Flat descent and relations.

C8.1. Proposition. (a) Let 𝔛 \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔜 ⟶^{f} X be an exact diagram in |Cat|^{o}. Then

(i) The morphism f (i.e. its inverse image functor f^{∗}) is conservative.

(ii) If the functors C_{𝔜} \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} C_{𝔛} preserve limits (resp. colimits) of a certain type, then

f^{∗} has the same property.

(b) Let 𝔛 ⟵^{p} ℨ ⟶^{q} 𝔜 be a diagram in |Cat|^{o}. Suppose the functors p^{∗} and q^{∗} preserve limits (resp. colimits) of a certain type, then the inverse image functors of the canonical coprojections

𝔛 ⟶^{π_{p}} 𝔛 ∐_{p,q} 𝔜 ⟵^{π_{q}} 𝔜

have the same property.

Proof. (a) By 2.2, the category C_{X} is the kernel of the pair C_{𝔜} \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} C_{𝔛} of inverse

image functors of p_{1} and p_{2}. This means that the category C_{X} can be described as follows: its objects are pairs (M, ϕ), where M ∈ ObC_{𝔜} and ϕ is an isomorphism p_{1}^{∗}(M) →^{∼} p_{2}^{∗}(M). A morphism from (M_{1}, ϕ_{1}) to (M_{2}, ϕ_{2}) is given by a morphism M_{1} ⟶^{g} M_{2} such that the diagram

\begin{matrix}p_{1}^{∗}(M_{1}) & ⟶^{p_{1}^{∗}(g)} & p_{1}^{∗}(M_{1}) \\ ϕ_{1} ↓ & & ↓ ϕ_{2} \\ p_{2}^{∗}(M_{1}) & ⟶^{p_{2}^{∗}(g)} & p_{2}^{∗}(M_{1}) \end{matrix}  (1)

commutes. A natural inverse image functor, C_{X} ⟶^{f^{∗}} C_{𝔜}, of the morphism f assigns to every object (M, ϕ) of C_{X} the object M.

(i) It follows from the construction that f^{∗} is conservative.

(ii) If p_{1}^{∗}, p_{2}^{∗} preserve a certain type of limits (resp. colimits), it follows from the

description of the category C_{X} that the functor (M, ϕ) ⟼^{f^{∗}} M preserves the same type of limits (resp. colimits).

(b) The category C_{𝔛 ∐_{p,q} 𝔜} is described as follows. Its objects are triples (M, ϕ, L),

where M ∈ ObC_{𝔛}, L ∈ ObC_{𝔜}, and ϕ is an isomorphism p^{∗}(M) ⟶^{∼} q^{∗}(L). Morphisms (M, ϕ, L) ⟶ (M′, ϕ′, L′) are pairs of morphisms (M →^{u} M′, L →^{v} L′) such that the diagram

p^{∗}(M) ⟶^{ϕ} q^{∗}(L)
p^{∗}(u) ↓  ↓ q^{∗}(v)
p^{∗}(M′) ⟶^{ϕ′} q^{∗}(L′)

commutes. Composition is defined in an obvious way. A natural inverse image functor of the projection π_{p} (resp. π_{q}) maps an object (M, ϕ, L) to M (resp. to L) and a morphism

(M, ϕ, L) ⟶^{(u,v)} (M′, ϕ′, L′) to M →^{u} M′ (resp. to L →^{v} L′).

It follows from this description (as in (ii) above) that if the functors p^{∗} and q^{∗} preserve (co)limits of certain type, then the functors π_{p}^{∗} and π_{q}^{∗} have the same property.

C8.1.1. Corollary. Let 𝔉 = (F →^{π} E) be a fibered category.

(a) Let ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔘 ⟶^{f} 𝔛 be an exact diagram of presheaves of sets on the category E.

Then (i) The morphism f is conservative.

(ii) If the functors Qcoh(𝔉/ℜ) \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} Qcoh(𝔉/𝔘) preserve a certain type of limits

(resp. colimits), then f^{∗} has the same property.

(b) Let X ⟵^{p} Z ⟶^{q} Y be a diagram of presheaves of sets on the category E. If the functors p^{∗} and q^{∗} preserve limits (resp. colimits) of a certain type, then the inverse image functors of the canonical coprojections

X ⟶^{π_{p}} X ∐_{p,q} Y ⟵^{π_{q}} Y

have the same property.

Proof. By [KR3, 11.1.5.2(b)], the category Qcoh(𝔉/𝔛) of quasi-coherent modules on 𝔛 is the kernel of the pair Qcoh(𝔉/G) \overrightarrow{⟶} Qcoh(𝔉/ℜ) of inverse image functors of p_{1} and p_{2}, and the category Qcoh(𝔉/X ∐_{p,q} Y) is the pull-back of the inverse image functors

Qcoh(𝔉/X) ⟶^{p^{∗}} Qcoh(𝔉/Z) ⟵^{q^{∗}} Qcoh(𝔉/Y). The assertion follows now from C8.1.

C8.1.1.1. Proposition. Let 𝔉 = (F →^{π} E) be a fibered category, and let 𝔗 be a topology on E which is coarser than the topology of effective descent.

(a) Let ℜ \overrightarrow{⟶}^{p_{1}}_{p_{2}} G ⟶^{f} X be an exact diagram of sheaves of sets on (E, 𝔗). Then

(i) The morphism f is conservative.

(ii) If the functors Qcoh(𝔉/ℜ) \overrightarrow{⟶}^{p^{∗}_{1}}_{p^{∗}_{2}} Qcoh(𝔉/G) preserve a certain type of limits

(resp. colimits), then f^{∗} has the same property.

(b) Let

ℨ ⟶^{p}  𝔛
q ↓  ↓ π_{p}
𝔜 ⟶^{π_{q}} 𝔛 ∐_{p,q} 𝔜

be a cartesian coproduct of sheaves of sets on (E, 𝔗). If the functors p^{∗}, q^{∗} preserve limits (resp. colimits) of a certain type, then the functors π_{p}^{∗}, π_{q}^{∗} have the same property.

Proof. Let X′ be the cokernel of the pair ℜ ⇉ G in the category E^{∧} of presheaves on E. Since the topology𝔗 is coarser than the topology of effective descent and X is asheaf associated with the presheaf X′, the unique presheaf morphism X′ ⟶ X induces an equivalence of categories Qcoh(𝔉/X) ⟶^{∼} Qcoh(𝔉/X′) (see [KR3, 2.6]. The assertion follows now from C8.1.1.

C8.2. Proposition. Let 𝔛 \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔜 ⟶^{f} X be an exact diagram in |Cat|^{o} such that

(i) inverse image functors of p_{1} and p_{2} preserve kernels of coreflexive pairs of arrows; (ii) the morphism f is continuous; (iii) The category C_{𝔜} has kernels of coreflexive pairs of arrows.

Then the morphism f is comonadic, i.e. 𝔜 ⟶^{f} X is isomorphic to the canonical morphism 𝔜 ⟶ Sp^{o}(𝔜\H_{f}), where H_{f} is a comonad associated with (a choice of inverse and direct image functors and adjunction morphisms of) the morphism f.

Proof. By C8.1, the morphism f is weakly flat, i.e. its inverse image functor preserves kernels of coreflexive pairs of arrows. The assertion now follows from the Beck’s theorem (see 5.4.1).

C8.2.1. Remark. Let

𝔛 \overrightarrow{⟶}^{p_{1}}_{p_{2}} 𝔜 ⟶^{f} X

be an exact diagram in |Cat|^{o}. Suppose the morphisms p_{1}, p_{2} in the diagram are continuous. Then their inverse image functors resp. p_{1}^{∗} and p_{2}^{∗} preserve colimits of any small diagram. By C8.1(ii), this implies that f^{∗} preserves colimits of any small diagram. Recall that if the category C_{X} has colimits of small diagrams, f is continuous iff the following conditions hold (cf. [BD]):

(a) f^{∗} preserves colimits of small diagrams;

(b) for any W ∈ ObC_{𝔜}, there exists a morphism f^{∗}(V) ⟶ W for some V ∈ ObC_{X}. Notice that the condition (b) is fulfilled if the category C_{X} has an initial object (for

instance, C_{X} is (pre)additive). In fact, every functor having a right adjoint maps an initial object to initial object. In particular, each of the inverse image functors, p_{1}^{∗} and p_{2}^{∗}, maps an initial object, •, of C_{X} to an initial object of C_{𝔜}. Since all initial objects are isomorphic, there is an isomorphism ϕ : p_{1}^{∗}(•) ⟶^{∼} p_{1}^{∗}(•). In other words, (•, ϕ) is an (initial) object of the category C_{X} which f^{∗} maps to •, an initial object of C_{𝔜}, hence the condition (b).

C8.3. Example: quasi-compact quasi-separated schemes. Let X be a quasicompact, quasi-separated scheme or algebraic space. Then X has a finite affine cover {U_{i} ⟶ X | i ∈ J} and for any i, j ∈ J, the intersection U_{i} ⋂ U_{j} has a finite affine cover {U_{ij}^{k} ⟶ U_{i} ⋂ U_{j} | k ∈ J_{ij}}. Set U = ∐_{i∈J} U_{i} and R = ∐_{i,j∈J} ∐_{k∈J_{ij}} U_{ij}^{k}. Then we have an exact diagram of schemes (resp. algebraic spaces)

R \overrightarrow{⟶} U ⟶ X  (1)

Since R and U are affine schemes, the projections R \overrightarrow{⟶}^{p_{1}}_{p_{2}} U are affine morphisms. By

construction, inverse image functors of p_{1} and p_{2} are conservative and exact, as well as the functor π^{∗}. As any morphism to a quasi-separated quasi-compact scheme, π has a direct image functor (this, by the way, follows from the argument of C8.2.1) which makes π (or ruther the corresponding morphism of categories of quasi-coherent sheaves) satisfy the conditions of Beck’s theorem; hence Qcoh_{X} is equivalent to the category of G_{π}-comodules, where G_{π} = (π^{∗}π_{∗}, δ_{π}) is the comonad associated with the continuous morphism π. This, however, does not imply that the morphism U ⟶^{π} X is affine.

The morphism π being not affine is equivalent to any of the following conditions:

(a) the direct image functor π_{∗} is not right exact;

(b) the comonad G_{π} is not continuous;

(c) the functor G_{π} = π^{∗}π_{∗} is not exact;

(d) the functor F_{π} = π_{∗}π^{∗} is not exact. This follows from the equivalence ot the following conditions on a scheme morphism π:

(i) π is affine (in the conventional sense);

(ii) its direct image functor is exact (Serre’s criterion);

(iii) π induces an affine morphism of categories of quasi-coherent sheaves. Notice, however, that the flatness of the morphism π (i.e. the exactness of π^{∗}) is equivalent to that the functor G_{π} is left exact.

Thus, for any quasi-compact quasi-separated scheme or algebraic space X, the object |Qcoh_{X}| of |Cat|^{o} corresponding to the category of quasi-coherent sheaves on X is isomorphic to the cospectrum of a left exact comonad on Sp(R) for some (commutative) ring R which is not, in general, given by a coalgebra in the category of R-bimodules.

C9. Monads, comonads, and relations. For any object X of the category |Cat|^{o}, we denote by 𝔐𝔬𝔫_{X} the category of monads on X and by ℭ𝔐𝔬𝔫_{X} the category of comonads on X. We have two functors

Sp_{X} : 𝔐𝔬𝔫_{X}^{op} ⟶ |Cat|^{o}/X, (F/X) ⟼ (Sp(F/X) ⟶ X),  (1)

and

Sp_{X}^{o} : ℭ𝔐𝔬𝔫_{X}^{op} ⟶ X\|Cat|^{o}, (X\G) ⟼ (X ⟶ Sp^{o}(X\G)).  (2)

C9.1. Base change. Let X ⟶^{f} Y be a continuous morphism with an inverse image functor f^{∗}, a direct image functor f_{∗} and adjunction arrows Id_{C_{X}} ⟶^{η_{f}} f_{∗}f^{∗} and f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{Y}}. Let G = (G, δ) be a comonad on Y with counit ε. The comultiplication δ and the adjunction arrow η_{f} induce a comultiplication δ^{f} on G^{f} = f^{∗}Gf_{∗} defined as the composition

G^{f} = f^{∗}Gf_{∗} ⟶^{f^{∗}δf_{∗}} f^{∗}G^{2}f_{∗} ⟶^{f^{∗}Gη_{f} Gf_{∗}} (f^{∗}Gf_{∗})^{2} = (G^{f})^{2}.  (1)

The counit ε and the adjunction arrow ϵ_{f} determine the counit, ε^{f}, of (G^{f}, δ^{f}) given by the composition

G^{f} = f^{∗}Gf_{∗} ⟶^{f^{∗}εf_{∗}} f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{X}} .  (2)

C9.1.1. Lemma. Let X ⟶^{f} Y be a continuous morphism with inverse and direct image functors resp. f^{∗} and f_{∗} and adjunction arrows Id_{C_{X}} ⟶^{η_{f}} f_{∗}f^{∗} and f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{Y}}. Let G = (G, δ) be a comonad on Y with counit ε.

(a) For any comonad G = (G, δ) on Y having the counit ε, the pair G^{f} = (G^{f}, δ^{f}) is a comonad on X with the counit ε^{f}.

(b) The correspondence G ⟼ G^{f} extends naturally to a functor ℭ𝔐𝔬𝔫_{Y} ⟶ ℭ𝔐𝔬𝔫_{X}.

(c) For every comonad G = (G, δ) on Y, the inverse image functor f^{∗} induces a functor

f_{G}^{∗} : (Y \G) − Comod ⟶ (X\G^{f}) − Comod  (3)

which can be regarded as an inverse image functor of a morphism

f_{G} : Sp^{o}(X\G^{f}) ⟶ Sp^{o}(Y \G).  (4)

(d) If the category C_{X} has kernels of coreflexive pairs of arrows, then the functor (3) has a right adjoint, i.e. the morphism (4) is continuous.

(e) If the comonad G = (G, δ) and the morphism f are weakly flat (i.e. the functors G and f^{∗} preserve kernels of coreflexive pairs of arrows), then G^{f} is weakly flat too.

Proof. (a) The comonad G can be regarded as the one obtained from an adjoint pair of functors, i.e. G = g^{∗}g_{∗}, δ = g^{∗}η_{g}g_{∗}, where η_{g} is an adjunction arrow, and the counit ε is a complementary adjunction morphism ϵ_{g} (see 5.3). Then the G^{f} becomes the comonad corresponding to the pair of adjoint functors f^{∗}g^{∗}, g_{∗}f_{∗} and the adjunction morphisms g_{∗}η_{f}g^{∗} ∘ η_{g} and ϵ_{f} ∘ f^{∗}ϵ_{g}f_{∗}.

(b) The functoriality of the map G ⟼ G^{f} is evident.

(c) The functor (3) assigns to every (Y \G)-comodule (L, ξ) the (X\G^{f})-comodule (f^{∗}(L), ξ^{f}), where the coaction ξ^{f} : f^{∗}(L) ⟶ G^{f}(f^{∗}(L) = f^{∗}Gf_{∗}f^{∗}(L) is the composition Gη_{f}(L) ∘ f^{∗}(ξ).

(d) The comonad G corresponds to an adjoint pair of functors, g^{∗} ⊣ g_{∗}, regarded as resp. inverse and direct image functors of a continuous morphism Y ⟶^{g} Z = Sp^{o}(Y \G). By (a), G^{f} is the comonad corresponding to the adjoint pair f^{∗}g^{∗}, g_{∗}f_{∗} of resp. inverse and direct image functors of the continuous morphism gX ⟶^{f} Z. By 5.4, the inverse image functor f^{∗}g^{∗} of gf decomposes canonically into

C_{Z} = (Y \G) − Comod ⟶^{f_{G}^{∗}} (X\G_{gf}) − Comod ⟶ C_{Y} .

Since C_{X} has kernels of coreflexive pairs of arrows, it follows from 5.4.1(a) (Beck’s theorem) that the functor f_{G}^{∗} in this decomposition has a right adjoint.

(e) If the functors f^{∗} and G preserve limits of certain type, the functor G^{f} = f^{∗}Gf_{∗} does the same, since f_{∗} preserves limits of any small diagram.

C9.2. Morphisms (f\λ). Let X, Y be objects of |Cat|^{o}; and let G and H be comonads resp. on X and Y. A morphism (Y \H) ⟶ (X\G) is a pair (f\λ), where f is a continuous morphism X ⟶ Y, λ is a comonad morphism (X\H^{f}) ⟶ (X\G). It follows from this definition that the morphism (f\λ) is decomposed into a morphism of underlying objects and a morphism of comonads:

\begin{matrix}(Y \H) ⟶^{(f\λ)} (X\G) \\ _{(f\H^{f})} ↘  ↗_{(X\λ)} \\ (X\H^{f}) \end{matrix}  (5)

Here (X\λ) = (id_{X}\λ) and (f\H^{f}) = (f\id_{H^{f}}). Any morphism (f\λ) : (X\G) ⟶ (Y \H) induces a functor

|f\λ|^{∗} : (Y \H) − Comod ⟶ (X\G) − Comod  (6)

which is the composition of the functor

f_{H}^{∗} : (Y \H) − Comod ⟶ (X\H^{f}) − Comod  (7)

(cf. C9.1) and the inverse image

λ^{∗} : (X\H^{f}) − Comod ⟶ (X\G) − Comod  (8)

of the morphism λ (see (5) above and 5.3.2). The functor |f\λ|^{∗} is regarded as an inverse image functor of a morphism

Sp^{o}(f\λ) : Sp^{o}(X\G) ⟶ Sp^{o}(Y \H).  (9)

We shall use also a short-hand notation |f\λ| instead of Sp^{o}(f\λ).

C9.2.1. Proposition. Let G = (G, δ_{G}) and H = (H, δ_{H}) be comonads on resp. Y and X, and let (f\λ) : (X\G) ⟶ (Y \H) be a comonad morphism. If the category C_{X} has kernels of coreflexive pairs of arrows and the functors f^{∗} and H preserve these kernels, then the morphism Sp^{o}(f\λ) is continuous, i.e. the functor |f\λ|^{∗} has a right adjoint.

Proof. By definition, the functor |f\λ|^{∗} is the composition of the functors (7) and (8). By C9.1(d), the functor (7) has a right adjoint if C_{X} has kernels of coreflexive pairs of arrows. By 5.3.2.2, the functor (8) has a right adjoint if the category C_{X} has kernels of coreflexive pairs of arrows and the functor H^{f} preserves these kernels. The latter condition holds if H and f have this property.

C9.3. Proposition. Let

(Y \H) \overrightarrow{⟶}^{(p\λ)}_{(q\γ)} (X\G)

be comonad morphisms such that (i) The pull-back, (X\H^{p} ×_{λ,γ} H^{q}), of comonad morphisms

(X\H^{p}) ⟶^{(X\λ)} (X\G) ⟵^{(X\γ)} (X\H^{q})  (3)

exists.

(ii) The category C_{Y} has kernels of coreflexive pairs of arrows, andthe morphisms

X \overrightarrow{⟶}^{p}_{q} Y are weakly flat and conservative.

Then the pushforward of

Sp^{o}(Y \H) ⟵^{Sp^{o}(q\γ)} Sp^{o}(X\G) ⟶^{Sp^{o}(p\λ)} Sp^{o}(Y \H)

is naturally isomorphic to Sp^{o}(X\H^{p} ×_{λ,γ} H^{q}).

Proof. The pair of morphisms (p\λ), (q\γ) is described by the diagram

\begin{matrix}(X\H^{p}) & ⟶^{(Xλ)} & (X\G) & ⟵^{(X\γ)} & (X\H^{q}) \\ (p\H^{p}) ↑ & & & & ↑ (q\H^{q}) \\ (Y \H) & & & & (Y \H) \end{matrix}  (4)

If the condition (a) holds, i.e. a pull-back of the pair of comonad morphisms λ, γ exists, we complete (4) to a commutative diagram

\begin{aligned}& (X\G) & ⟵^{(X\γ)} & (X\H^{q}) & ⟵^{(q\H^{q})} & (Y \H) \\ & (X\λ) ↑ & & ↑ (X\π_{q}) \\ (Y \H) ⟶^{(p\H^{p})} & (X\H^{p}) & ⟵^{(X\π_{p})} & (X\H^{p} ×_{λ,γ} H^{q}) \end{aligned}  (5)

Applying the functor Sp^{o} to (5), we obtain the diagram

Sp^{o}(X\G) ⟶^{Sp^{o}(X\γ)} Sp^{o}(X\H^{q}) ⟵^{Sp^{o}(q\H^{q})} Sp^{o}(Y \H)
Sp^{o}X\λ) ↓  ↓ Sp^{o}(X\π_{q})
Sp^{o}(Y \H) ⟶^{Sp^{o}(p\H^{p})} Sp^{o}(X\H^{p}) ⟶^{Sp^{o}(X\π_{p})} Sp^{o}(X\H^{p} ×_{λ,γ} H^{q})

(6)

By condition (b) and Beck’s theorem, both Sp^{o}(p\H^{p}) and Sp^{o}(q\H^{q}) are isomorphisms, hence the assertion.

C9.3.1. Corollary. Under the conditions of C9.3, the cokernel of

Sp^{o}(Y \H) \overrightarrow{⟶}^{Sp^{o}(p\λ)}_{Sp^{o}(q\γ)} Sp^{o}(X\G)

is naturally isomorphic to the cokernel of

Sp^{o}(X\H^{p}) \overrightarrow{⟶} Sp^{o}(X\H^{p} ×_{λ,γ} H^{q}),

where one arrow is Sp^{o}(X\π_{p}) and another arrow is the composition of

Sp^{o}(X\π_{q}) : Sp^{o}(X\H^{q}) ⟶ Sp^{o}(X\H^{p} ×_{λ,γ} H^{q})

and the isomorphisms

Sp^{o}(X\H^{p}) ⟶^{Sp^{o}(p\π_{p})^{−1}} Sp^{o}(Y \H) and Sp^{o}(Y \H) ⟶^{Sp^{o}(q\π_{q})} Sp^{o}(X\H^{q})

(see the diagram (6)).

Proof. The assertion follows from the diagram (6) and the general nonsense fact about the connection of the push-forward of a pair of arrows x ⟵^{α} y ⟶^{β} x and the cokernel of this pair.

C9.4. The case of affine covers and relations. Let

Sp(R) \overrightarrow{⟶}^{p}_{q} Sp𝔄 ⟶^{f} X  (1)

be an exact diagram in |Cat|^{o} such that all morphisms are faithfully flat. To the pair of morphisms p, q, there corresponds the pair of morphisms

(R\H^{p}) ⟶^{(R\μ_{p})} (R\R) ⟵^{(R\μ_{q})} (R\H^{q})  (2)

(cf. C9.3(a)), where H^{p} is (R ⊗_{(𝔄,p)} R, δ_{p}) and H_{q} = (R ⊗_{(𝔄,q)} R, δ_{q}), and morphisms μ_{p} and μ_{q} are induced by the multiplication R⊗R ⟶ R. The pull-back of (2) is the coalgebra (R\(H_{p,q}, δ_{p,q}). Here H_{p,q} = (R ⊗ R)/(I_{p} ⋂ I_{q}), where I_{p} (resp. I_{q}) is the kernel of the epimorphism R ⊗ R ⟶ R ⊗_{(𝔄,p)} R (resp. R ⊗ R ⟶ R ⊗_{(𝔄,q)} R); and the comultiplication δ_{p,q} is uniquely determined by the condition that the natural epimorphism R ⊗R ⟶ H_{p,q} is a coalgebra morphism (R\(R ⊗ R, δ_{R})) ⟶ (R\(H_{p,q}, δ_{p,q})).

Set H_{p,q} = (H_{p,q}, δ_{p,q}). It follows from the construction that the square

\begin{matrix}(R\H_{p,q}) & ⟶ & (R\H^{p}) \\ ↓ & & ↓ \\ (R\H^{q}) & ⟶ & (R\R) \end{matrix}  (3)

commutes. This implies that there is a natural morphism Sp𝔄 ⟶ Sp^{o}(R\H_{p,q})

which equalizes the pair of arrows Sp(R) \overrightarrow{⟶}^{p}_{q} Sp𝔄. This morphism is the composi-

tion of the morphism Sp^{o}(R\H^{p}) ⟶ Sp^{o}(R\H_{p,q}) induced by the coalgebra morphism (R\H_{p,q}) ⟶ (R\H^{p}) and the isomorphism Sp𝔄 ⟶ Sp^{o}(R\H^{p}).

It is not true in general that the functor Sp^{o} transforms (3) into a cocartesian square,

or, equivalently, that Sp^{o}(R\H_{p,q}) is a cokernel of the pair of arrows Sp(R) \overrightarrow{⟶}^{p}_{q} Sp𝔄.

C9.4.1. Proposition. The square (3) is cocartesian, or, equivalently, the diagram

Sp(R) \overrightarrow{⟶}^{p}_{q} Sp𝔄 ⟶ Sp^{o}(R\H_{p,q})  (4)

is exact, iff H_{p,q} is flat as a right R-module.

Proof. The fact follows from C9.3 (and holds in a more general situation). Details are left to the reader.

C9.4.2. Remark. The realization of the cokernel of the pair Sp(R) \overrightarrow{⟶}^{p}_{q} Sp𝔄 as

Sp^{o}(R\H_{p,q}) (under the conditions of C9.4.1) might be not the best choice. For instance, suppose that the object X in the exact sequence (1) is also affine, i.e. X = Sp(S) for some ring S. In this case, we can assume (after replacing some of the rings in (1) by Morita equivalent rings) that the diagram (1) is obtained by applying the functor Sp

to the exact diagram of rings S ⟶^{ϕ} 𝔄 \overrightarrow{⟶}^{𝔭}_{𝔮} R. It follows (from Beck’s theorem) that

Sp(S) ≃ Sp^{o}(𝔄\𝔄 ⊗_{S} 𝔄).

C9.5. Quasi-coherent modules on a Grassmannian. Let R ⟶^{ϕ} S be a k-algebra morphism. Let M be a left R-module, and let L, N be projective left R-modules of finite type. We have canonical isomorphisms

Hom_{S}(S ⊗_{R} M, S ⊗_{R} L) ≃ Hom_{R}(M, S ⊗_{R} L) ≃ Hom_{R^{e}}(M ⊗_{k} L_{R}^{∨}, S)  (1)

(cf. the argument of [KR3, C9.1.3]). Here R^{e} = R ⊗_{k} R^{op}. In particular, the composition,

Hom_{S}(S ⊗_{R} M, S ⊗_{R} L) × Hom_{S}(S ⊗_{R} L, S ⊗_{R} N) ⟶^{c_{M,L,N}^{S}} Hom_{S}(S ⊗_{R} M, S ⊗_{R} N),

determines a unique map

Hom_{R^{e}}(M ⊗_{k} L_{R}^{∨}, S) × Hom_{R^{e}}(L ⊗_{k} N_{R}^{∨}, S) ⟶^{λ_{M,L,N}} Hom_{R^{e}}(M ⊗_{k} N_{R}^{∨}, S)

The map λ_{M,L,N} assigns to any pair of R^{e}-module morphisms, M ⊗_{k} L_{R}^{∨} ⟶^{u} S and L ⊗_{k} N_{R}^{∨} ⟶^{v} S, the composition

M ⊗_{k} N_{R}^{∨} ⟶ M ⊗_{k} L_{R}^{∨} ⊗_{R} L ⊗_{k} N_{R}^{∨} ⟶^{u⊗_{R}v} S ⊗_{R} S ⟶ S

Here the first arrow is induced by the canonical map k ⟶ L_{R}^{∨} ⊗_{R} L (– the composition of the morphism k ⟶ Hom_{R}(L, L) sending the unit of k to the identity morphism, id_{L}, and the isomorphism Hom_{R}(L, L) ⟶ L_{R}^{∨} ⊗_{R} L) and the last arrow is induced by the multiplication on S.

The relation u ∘ v = id_{S⊗_{R}L} (defining the functor G_{M,L}) is expressed by the commutative diagram

\begin{matrix}L ⊗_{k} M_{R}^{∨} ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{v⊗_{R}u} & S ⊗_{R} S \\ λ_{L,M,L} ↑ & & ↓ \\ L ⊗ L_{R}^{∨} & ⟶^{ev_{L}} R ⟶^{ϕ} & S \end{matrix}  (2)

Here ev_{L} denotes the evaluation morphism, a ⊗ λ ⟼ ⟨λ, a⟩.

Taking as S the R-ring T(L ⊗_{k} M_{R}^{∨} ⊕ M ⊗_{k} L_{R}^{∨}) representing the functor

R\Alg_{k} ⟶ Sets, U ⟼ Hom_{U}(U ⊗_{R} L, U ⊗_{R} M) × Hom_{U}(U ⊗_{R} M, U ⊗_{R} L),

and as u, v canonical R-bimodule morphisms resp. L ⊗_{k} M_{R}^{∨} ⟶ U and M ⊗_{k} L_{R}^{∨} ⟶ U, one might discribe an R-ring, G = G_{M,L}, representing the functor G_{M,L} : R\Alg_{k} ⟶ Sets (cf. C7.1) as the colimit of the (non-commutative) diagram (2).

The relations u_{j} ∘ (v_{i} ∘ u_{i}) = u_{i}, i, j = 1, 2, defining (together with relations u_{i} ∘ v_{i} = id_{S⊗_{R}L}, i = 1, 2) the functor ℜ_{M,L} are expressed by the commutative diagrams

\begin{matrix}M ⊗_{k} L_{R}^{∨} ⊗_{R} L ⊗_{k} M_{R}^{∨} ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{u_{i}⊗_{R}v_{i}⊗_{R}u_{j}} & S ⊗_{R} S ⊗_{R} S \\ ↑ & & ↓ \\ M ⊗_{k} M_{R}^{∨} ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{(v_{i}∘u_{i})⊗_{R}u_{j}} & S ⊗_{R} S \\ ↑ & & ↓ id \\ R ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{ϕ⊗_{R}u_{j}} & S ⊗_{R} S \end{matrix}  (3)

Fix an R-ring G = G_{M,L} corepresenting the functor G_{M,L} : R\Alg_{k} ⟶ Sets. And let G ⊗_{R} L ⟶^{v} G ⊗_{R} M ⟶^{u} G ⊗_{R} L be a canonical splitting — the image of id_{G} under the isomorphism Alg(G, G) ⟶^{∼} G_{M,L}(G). Set p = v ∘ u : G ⊗_{R} M ⟶ G ⊗_{R} M. Denote by H a colimit of the diagram

\begin{matrix}M ⊗_{k} M_{R}^{∨} ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{p⊗_{R}u} & G ⊗_{R} G \\ ↑ & & ↓ id \\ M ⊗_{k} L_{R}^{∨} ≃ R ⊗_{R} M ⊗_{k} L_{R}^{∨} & ⟶^{ϕ⊗_{R}u} & G ⊗_{R} G \end{matrix}  (4)

Note that H is a quotient of the bimodule G ⊗_{R} G. The comultiplication, δ, on H is uniquely defined by the condition: the canonical G-bimodules epimorphism G ⊗_{R} G ⟶ H is a coalgebra morphism.

The category of quasi-coherent sheaves on the locally affine space Gr+_{M,L} is equivalent to the category (H, δ) − comod of (H, δ)-comodules.

C9.5.1. Example: the zero dimensional projective space. Let R = k. The zero dimensional projective space is by definition ℙ_{k^{1}} = Gr+_{k^{1},k^{1}} . The algebra G representing the functor G_{k^{1},k^{1}} is the quotient of the free k-algebra k⟨x, y⟩ in two variables by the two-sided ideal generated by xy − 1: G = k⟨x, y⟩/(xy − 1). Denote by e the element 1 − yx of G. It follows that e is an idempotent, e^{2} = e. The coalgebra corresponding to the projection π : Spec(A) ⟶ ℙ_{k^{1}} is (H, δ), where the bimodule H enters into the exact sequence

0 ⟶ G ⊗ eG ⟶^{i} G ⊗ G ⟶ H ⟶ 0  (1)

Here ⊗ = ⊗_{k}, and i is defined by a ⊗ eb ⟼ ax ⊗ eb. The comultiplication δ is uniquely determined by the commutativity of the diagram

H ⟶^{δ} H ⊗_{G} H
↑  ↑
G ⊗ G ⟶^{δ^{∼}} G ⊗ G ⊗ G

where δ^{∼} sends a ⊗ b to a ⊗ 1 ⊗ b. The counit ϵ : H ⟶ G is induced by the multiplication G ⊗ G ⟶ G in the algebra G.

C10. Miscellaneous.

C10.1. Connections. Let X ⟶^{f} Y be a continuous morphism. Fix inverse and direct image functors of f and adjunction arrows, Id_{C_{Y}} ⟶^{η_{f}} f_{∗}f^{∗} and f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{X}}. Let G_{f} be the comonad associated with this data: G_{f} = (G_{f}, δ_{f}), where G_{f} = f^{∗}f_{∗} and δ_{f} = f^{∗}η_{f}f_{∗}.

A (X\G_{f})-connection is a pair (M, ρ), where M is an object of C_{X} and ρ a morphism M ⟶ G_{f}(M) such that ϵ_{f}(M) ∘ ρ = id_{M}. A morphism from an f-connection (M, ρ) to an f-connection (M′, ρ′) is a morphism g : M ⟶ M′ such that the diagram

M ⟶^{g} M′
ρ ↓  ↓ ρ′
G_{f}(M) ⟶^{G_{f} (g)} G_{f}(M′)

commutes. Composition is defined in a natural way. We denote the category of G_{f}- connections by ℭ𝔬𝔫𝔫(X\G_{f}).

A G_{f}-connection (M, ρ) is called integrable if it is an (X\G_{f})-comodule. Thus, the category (X\G) − Comod is a full subcategory of the category ℭ𝔬𝔫𝔫(X\G_{f}) formed by connections and morphisms between them.

C10.1.1. Connections and monadic morphisms. Suppose that the morphism X ⟶^{f} Y is monadic, i.e. the arrow f̄in the diagram 5.4(3) is an isomorphism. Then we can assume (for our immediate purposes) that X = Sp(F_{f}/Y) and take the standard inverse and direct image functors of the canonical morphism f : Sp(F_{f}/Y) ⟶ Y. The functor G_{f} : (F_{f}/Y) − mod ⟶ (F_{f}/Y) − mod assigns to each (F_{f}/Y)-module M = (M, ξ) the (F_{f}/Y)-module (F_{f}(M), μ_{f}(M)). The comultiplication δ_{f} : G_{f} ⟶ G_{f}^{2} is F_{f}η_{f}, where η_{f} is the unit of the monad F_{f}; the counit ϵ_{f} : G_{f} ⟶ Id_{(F_{f} /Y)−mod} is defined by ϵ_{f}(M, ξ) = ξ : F_{f}(M) ⟶ M.

A (X\G_{f})-connection is a pair (M, ρ), where M = (M, ξ) is an (F_{f}/Y)-module and ρ an (F_{f}/Y)-module morphism (M, ξ) ⟶ (F_{f}(M), μ_{f}(M)) such that ξ ∘ ρ = id_{M}. In other words, ρ is a morphism M ⟶ F_{f}(M) satisfying the equations

μ_{f}(M) ∘ F_{f}(ρ) = ρ ∘ ξ, ξ ∘ ρ = id_{M}.  (1)

The connection (M, ρ) is integrable iff F_{f}η_{f}(M) ∘ ρ = F_{f}(ρ) ∘ ρ. It follows from (1) that (M, ξ) is the kernel of the pair of arrows

F_{f}η_{f}(M), F_{f}(ρ) : F_{f}(M) \overrightarrow{⟶} F_{f}^{2}(M)  (2)

and ρ : (M, ξ) ⟶ F_{f}(M) = (F_{f}(M), μ_{f}(M)) is a universal arrow. The pair

η_{f}(M), ρ : M \overrightarrow{⟶} F_{f}(M)  (3)

is coreflexive, since ξ∘F_{f}η_{f}(M) = id_{M} = ξ∘ρ. Suppose C_{Y} has kernels of coreflexive pairs, and let L be a kernel of the pair (3). If the functor F_{f} preserves the kernels of coreflexive pairs, then F_{f}(L) = (F_{f}(L), μ_{f}(L)) is a kernel of the pair (2) which implies that (M, ξ) is isomorphic to F_{f}(L).

C10.2. Weakly quasi-affine morphisms. We call a continuous morphism X ⟶^{f} Y weakly quasi-affine if the canonical diagram

(G_{f})^{2} \overrightarrow{⟶}^{ϵ_{f} G_{f}}_{G_{f} ϵ_{f}} G_{f} ⟶^{ϵ_{f}} Id_{C_{X}}  (1)

is exact. Here G_{f} = f^{∗}f_{∗} and ϵ_{f} is an adjunction arrow.

C10.2.1. Proposition. Let Y ∈ Ob|Cat|^{o} be such that the category C_{Y} has cokernels of reflexive pairs of arrows. The following conditions on a continuous morphism X ⟶^{f} Y are equivalent:

(a) The canonical morphism X̄ ⟶^{f} Sp(F_{f}/Y) (cf. 5.4.2) is a localization.

(b) The morphism f is weakly quasi-affine.

Proof. Since the category C_{Y} has cokernels of reflexive pairs of arrows, the functor f̄_{∗} : C_{Y} ⟶ (F_{f}/Y) − mod, L ⟼ (f_{∗}(L), f_{∗}ϵ_{f}(L)), has a left adjoint, f̄^{∗}, which assigns to each (F_{f}/Y)-module (M, ξ) a cokernel of the pair of arrows

f^{∗}f_{∗}f^{∗}(M) \overrightarrow{⟶}^{ϵ_{f} f^{∗}(M)}_{f^{∗}(ξ)} f^{∗}(M).  (2)

(b)⇔(a). The exactness of the diagram (1) means precisely that the adjunction morphism ϵf̄: f̄^{∗}f̄_{∗} ⟶ Id_{C_{X}} is an isomorphism. The latter is equivalent to that f̄_{∗} is a fully faithful functor, i.e. f̄^{∗} is a localization.

C10.2.2. Remark. If a morphism X ⟶^{f} Y is weakly quasi-affine, than an adjunction arrow ϵ_{f} : f^{∗}f_{∗} ⟶ Id_{C_{X}} is a strict epimorphism. Notice that ϵ_{f} is an epimorphism iff the functor f_{∗} is faithful. The exactness of the diagram (1) implies that f_{∗} is conservative. On the other hand, if the functor f_{∗} reflects cokernels of reflexive pairs of arrows, then the diagram (1) is exact, i.e. f is weakly quasi-affine. This follows from the observation that

the pair of morphisms (f^{∗}f_{∗})^{2} \overrightarrow{⟶}^{ϵ_{f} G_{f}}_{G_{f} ϵ_{f}} f^{∗}f_{∗} is reflexive, and the diagram f_{∗}(1),

f_{∗}(f^{∗}f_{∗})^{2} \overrightarrow{⟶}^{f∗ϵ_{f} G_{f}}_{f∗G_{f} ϵ_{f}} f_{∗}f^{∗}f_{∗} ⟶^{f_{∗}ϵ_{f}} f_{∗},

is exact for any pair of adjoint functors f_{∗}, f^{∗} and an adjunction morphism ϵ_{f}.

By Beck’s theorem, a morphism X ⟶^{f} Y satisfying the equivalent conditions of C10.2.1 is monadic if its direct image functor f_{∗} preserves cokernels of reflexive pairs of arrows.

C10.2.3. Weakly quasi-affine morphisms to the spectrum of a ring. Let R be an associative unital ring and f a continuous morphism X ⟶ Sp(R) with an inverse image functor f^{∗} and a direct image f_{∗}; and let O = f^{∗}(R). Consider the decomposition

\begin{matrix}C_{X} ⟶^{f̂_{∗}} Γ_{X}O − mod \\ f_{∗} ↘  ↙ ϕ_{∗} \\ R − mod \end{matrix}  (3)

Here Γ_{X}O denotes the ring C_{X}(O, O)^{o}, ϕ_{∗} is the pull-back functor by the ring morphism ϕ : R ⟶ Γ_{X}O defining a right R-module structure on O, and f̂_{∗} = C_{X}(O, −) (see 4.5).

C10.2.3.1. Proposition. Let C_{X} be a Grothendieck category and R an associative unital ring.

(a) A continuous morphism X ⟶^{f} Sp(R) is weakly quasi-affine iff its direct image functor is faithful.

(b) Every weakly quasi-affine morphism X ⟶ Sp(R) is a composition of an affine morphism and a flat localization.

Proof. By C10.2.2, direct image functor of a weakly quasi-affine morphism is faithful. Conversely, if f_{∗} is a faithful functor and C_{X} is a Grothendieck category, then, by

4.8.2, the canonical morphism X ⟶ Sp(Γ_{X}O) is a flat localization. Thus the first arrow in the canonical decomposition X ⟶ Sp(Γ_{X}O) ⟶ Sp(R) of the morphism f is a flat localization and the second one an affine morphism.

C10.2.3.2. Two decompositions of a continuous morphism to the spectrum of a ring. Let X ⟶^{f} Sp(R) be continuous morphism to the spectrum of a unital associative ring. We have two canonical decompositions of the morphism f incorporated in the commutative diagram

\begin{matrix}X & ⟶^{f_{O}} & Sp(Γ_{X}O) \\ f̄ ↓ & & ↓ ϕ_{f} \\ Sp(F_{f}/R) & ⟶^{f̂} & Sp(R) \end{matrix}  (4)

where F_{f} is the monad (f_{∗}f^{∗}, μ_{f}) associated with f. Let Γ_{X}O^{∼} denote the monad on Sp(R) (i.e. on R − mod) determined by the ring morphism ϕ_{f}. Then we have a natural isomorphism Sp(Γ_{X}O^{∼}/R) ⟶^{∼} Sp(Γ_{X}O) which makes the diagram

Sp(Γ_{X}O^{∼}/R) ⟶^{∼} Sp(Γ_{X}O)

↘ ↙_{ϕ_{f}}

Sp(R)

(5)

commute. The monad Γ_{X}O^{∼} is exactly the continuous monad associated with the monad F_{f} (cf. C2.1). Thus, there exists a canonical monad morphism Γ_{X}O^{∼} ⟶ F_{f} to which there corresponds a morphism of their spectra,

Sp(F_{f}/R) ⟶ Sp(Γ_{X}O^{∼}/R).  (6)

Adjoining the morphism (6) (or ruther the composition of (6) with the isomorphism Sp(Γ_{X}O^{∼}/R) ⟶^{∼} Sp(Γ_{X}O)), we obtain a commutative diagram

\begin{matrix}X ⟶^{f_{O}} Sp(Γ_{X}O) \\ f̄ ↓ ^{λ_{f}} ↗  ↓ ϕ_{f} \\ Sp(F_{f}/R) ⟶^{f̂} Sp(R) \end{matrix}  (7)

If f is weakly quasi-affine, then, by C10.2.1, the morphism f̄ is a localization. If, in addition, C_{X} is a Grothendieck category, then the morphism f_{O} is a flat localization.

The canonical morphism λ_{f} : Sp(F_{f}/R) ⟶ Sp(Γ_{X}O) is an isomorphism iff the morphism f is affine. In this case, f_{O} : X ⟶ Sp(Γ_{X}O) is an isomorphism.

C10.2.4. Relatively ample morphisms. Let X ⟶^{f} Y be a continuous morphism. We call a continuous morphism ϕ : U ⟶ X f-ample if the canonical diagram

(G_{fϕ}^{2} \overrightarrow{⟶}^{ϵ_{fϕ}G_{fϕ}}_{G_{fϕ}ϵ_{fϕ}} G_{fϕ} ⟶^{ϵ_{fϕ}} Id_{C_{X}} )ϕ^{∗}  (1)

(where G_{fϕ} = (fϕ)^{∗}(fϕ)_{∗} and ϵ_{fϕ} is an adjunction arrow) is exact.

It follows that id_{X} is f-ample iff f is weakly quasi-affine.

Denote by U_{fϕ} the object of |Cat|^{o} such that C_{U_{fϕ}} is the full subcategory of C_{U} whose objects are all M ∈ ObC_{U} for which the diagram

G_{fϕ}^{2}(M) \overrightarrow{⟶}^{ϵ_{fϕ}G_{fϕ}}_{G_{fϕ}ϵ_{fϕ}} G_{fϕ}(M) ⟶^{ϵ_{fϕ}} M  (2)

is exact. The condition ’ϕ is f-ample’ implies that ϕ induces a continuous morphism φ : U_{fϕ} ⟶ X such that fφ is weakly quasi-affine.

C10.2.4.1. Note. The exactness of the diagram (1) implies that the functor f_{∗}ϕ_{∗}ϕ^{∗} is faithful and conservative.

C10.2.4.2. Example. Let X ⟶^{f} Y be a continuous morphism, and let θ : X ⟶ X be a continuous endomorphism. Suppose C_{X} has countable coproducts. Then F_{θ} = ⊕_{n≥0}θ^{∗n} has a natural structure of a monad on X. Let U = Sp(F_{θ}/X), and let ϕ = ϕ_{θ} be a canonical morphism U ⟶ X. Thus ϕ_{∗}ϕ^{∗} = F_{θ}. If f_{∗} preserves countable coproducts, f_{∗}ϕ_{∗}ϕ^{∗} ≃ ⊕_{n≥0}f_{∗}θ^{∗n}. In this case, if the morphism ϕ is f-ample, than the family {f_{∗}θ^{∗n} | n ≥ 0} is conservative.

C10.2.4.2.1. Example. Let X = (X, O_{X}) be a scheme and X ⟶^{f} Y a quasicompact scheme morphism. Let L be an invertible sheaf on X. Set θ = L ⊗_{O_{X}} −. The sheaf L is f-ample in the usual sense iff the morphism ϕ = ϕ_{θ} constructed in C10.2.4.2 is f-ample.

C10.3. Continuous, affine, and flat morphisms in a fibered category. Let 𝔓 be a subcategory of |Cat|^{o}, and let p : F ⟶ B be a fibered category. We say that a morphism f of B belongs to 𝔓 if its image by by the canonical functor Ψ_{p} : B ⟶ |Cat|^{o} belongs to 𝔓. In particular, we call a morphism f of B continuous (resp. affine, resp. flat, resp. faithful) if its image by Ψ_{p} : B ⟶ |Cat|^{o} is continuous (resp. affine, resp. flat, resp. faithful).

C10.3.1. Continuous, affine, and flat morphisms in the fibered category of modules on affine schemes. Consider the fibered category 𝔐(A, O) over the category A = Aff_{k} of affine k-schemes (cf. [KR3, 11.4.1]). All morphisms of Aff_{k} are affine. In fact, any morphism f : SpecR ⟶ SpecS is given by an algebra morphism S ⟶ R. By 3.8, the corresponding inverse image functor f^{∗} is R⊗_{S} : S − mod ⟶ R − mod, the direct image functor f_{∗} is the ’pull-back’ functor R − mod ⟶ S − mod, and a right adjoint to f_{∗} is given by f^{!} : L ⟼ Hom_{S}(R, L).

A morphism f : SpecR ⟶ SpecT is flat (resp. faithful) iff the corresponding algebra morphism T ⟶ R turns R into a flat right T -module, that is the functor R ⊗_{T} − is exact (resp. faithful).

C10.3.2. Proposition. Let X, Y be presheaves of sets on Aff_{k} and let X ⟶^{f} Y be a morphism of presheaves. Suppose Y is affine (i.e. representable). Then X is affine iff the morphism |Qcoh_{X}| ⟶ |Qcoh_{Y} | in |Cat|^{o} induced by f is affine.

Proof. Let Y be isomorphic to SpecR for some k-algebra R. By [KR3, 11.1.5.1], the category Qcoh_{Y} is equivalent to the category R − mod.

(a) If X is affine, i.e. X ≃ SpecS for a k-algebra S, then Qcoh_{X} is equivalent to the category S − mod and we have a quasi-commutative diagram

\begin{aligned}S − mod & ⟶^{∼} & Qcoh_{X} \\ ϕ^{∗} ↑ & & ↑ f^{∗} \\ R − mod & ⟶^{∼} & Qcoh_{X} \end{aligned}  (1)

where ϕ^{∗} is the inverse image functor corresponding to a k-algebra morphism R ⟶^{ϕ} S; i.e. ϕ^{∗} = S ⊗_{R} −. Since ϕ^{∗} is an inverse image of an affine morphism (by the argument above or 3.8), f is affine.

(b) The converse assertion follows from 6.6.1.

Let 𝔄ss_{k} be the category whose objects are associative k-algebras. Morphisms from a k-algebra R to a k-algebra S are equivalence classes of algebra morphisms R ⟶ S by the following equivalence relation: two algebra morphisms, f, g : R ⟶ S are equivalent if they are conjugated, i.e. g(−)t = tf(−) for an invertible element t of S.

Note that the restriction of the natural functor Ass_{k} ⟶ 𝔄ss_{k} to the subcategory of commutative algebras is a strict fully faithful functor (’strict’ means that it is injective on objects).

C10.3.3. Proposition. The canonical functor Aff_{k} ⟶ |Cat|^{o} induces a faithful functor from the category 𝔄ss_{k}^{op} to the subcategory |Cat|_{aff}^{o} of |Cat|^{o} formed by affine morphisms.

Proof. The assertion is a corollary of [KR3, 5.10.1] and C10.3.2.

Denote by 𝔅𝔄ss the category whose objects are associative unital rings and morphisms from R to S isomorphism classes of (S, R)-bimodules. The composition is induced by tensoring bimodules. Notice that there is a natural embedding 𝔄ss ↪ 𝔅𝔄ss. Denote by 𝔐𝔄ss class of morphisms of the form f ∘ m, where m is the isomorphism class of an invertible bimodule (Morita equivalence) and f belongs to Hom(𝔄ss).

C10.3.4. Proposition. (a) 𝔐𝔄ss is a subcategory of 𝔅𝔄ss.

(b) The natural functor 𝔅𝔄ss^{op} ⟶ |Cat|_{𝔠}^{o} is fully faithful.

(c) The restriction of the functor (b) to the subcategory 𝔐𝔄ss^{op} induces an equivalence 𝔐𝔄ss^{op} ⟶ (|Cat|^{o}/Spℤ)_{aff}.

Proof. The assertion follows from C10.3.3 and 6.4.1.

C10.4. Additive monads and continuous monads. For any X ∈ Ob|Cat|^{o} such that the category C_{X} is (pre)additive, denote by 𝔐𝔬𝔫_{X}^{a} the category of additive monads on X, i.e. monads (F, μ) on C_{X} with an additive functor F. We denote by 𝔐𝔬𝔫_{X}^{c} the category of continuous monads on X.

C10.4.1. Proposition. Let R be an associative unital ring and X = Sp(R). The inclusion functor 𝔧^{∗} : 𝔐𝔬𝔫_{X}^{c} ↪ 𝔐𝔬𝔫_{X}^{a} has a left adjoint.

Proof. Let F = (F, μ) be an additive monad on C_{Sp(R)}, and let f : Sp(F/R) ⟶ Sp(R) denote the canonical morphism with inverse image functor V ⟼ (F (V), μ(V)). In particular, the object O = f^{∗}(R) is (F (R), μ(R)), and C_{Sp(F/R)}(O, O) ≃ f_{∗}f^{∗}(R) = F (R). The latter means that the R-bimodule F (R) has a structure of a ring (in the category of R-bimodules) such that the adjunction arrow R ⟶ F (R) is a ring morphism. Let f_{_{O}∗} denote the functor C_{Sp(F/R)}(O, −) and f ∗_{O} its left adjoint. The functor f_{_{O}∗}f ∗_{O} is isomorphic to the functor F_{c} = F (R) ⊗_{R} − : R − mod ⟶ R − mod. Thus we obtain a monad 𝔉_{c} on R − mod. The map 𝔉 ⟼ 𝔉_{c} is functorial. We denote this functor by 𝔧_{∗}.

Since the functor F_{c} preserves colimits of small diagrams, the natural isomorphism F_{c}(R) ⟶^{∼} F (R) extends canonically to the functor morphism φ_{F} : F_{c} ⟶ F such that φ_{F} (M) is an isomorphism for any projective R-module M of finite type.

In fact, the isomorphism F_{c}(R) ⟶^{∼} F (R) yields an isomorphism φ_{F} (L) : F_{c}(L) ⟶ F (L) for any free R-module L of finite rank functorial in L. Since every free module of infinite rank is a colimit of its free submodules of finite rank, φ_{F} extends to all free modules. Since every R-module is a cokernel of a pair of R-module morphisms L_{1} \overrightarrow{⟶} L_{0} with L_{0}, L_{1} free R-modules, and F_{c} preserves cokernels, φ_{F} extends to all R-modules in a unique way (thanks to universal property of colimits).

One can see that φ_{F} is a monad morphism. The function φ : F ⟼ φ_{F} defines a functor morphism 𝔧^{∗}𝔧_{∗} ⟶ Id_{𝔐𝔬𝔫_{X}^{a}} which is an adjunction morphism. The other adjunction morphism, Id_{𝔐𝔬𝔫_{X}^{c}} ⟶ 𝔧_{∗}𝔧^{∗} is identical.

Let C_{X} be an additive category. Let End_{a}(C_{X}) denote the category of additive functors C_{X} ⟶ C_{X} and End_{c}(C_{X}) its full subcategory formed by continuous endofunctors.

C10.4.2. Proposition. Let C_{X} be an additive category with small colimits. Then (a) The inclusion functor End_{c}(C_{X}) ⟶ End_{a}(C_{X}) has a left adjoint.

(b) The inclusion functor 𝔧^{∗} : 𝔐𝔬𝔫_{X}^{c} ↪ 𝔐𝔬𝔫_{X}^{a} has a left adjoint.

Proof. (a) Let F be a functor C_{X} ⟶ C_{X}. Consider the category End_{c}(C_{X})/F whose objects are pairs (G, ξ), where G is a continuous functor C_{X} ⟶ C_{X} and ξ is a functor morphism, morphisms (G, ξ) ⟶ (G′, ξ′) are functor morpisms ψ : G ⟶ G′ such that ξ = ξ′ ∘ ψ. Let D_{F} denote the diagram

End_{c}(C_{X})/F ⟶ End(C_{X}), (G, ξ) ⟼ G.

We denote a colimit of the diagram D_{F} by F_{c}^{∗}. The functor F_{c}^{∗}, being a colimit of a diagram of continuous functors, preserves colimits of small diagrams. Since for any x, y ∈ ObC_{X}, there exist morphisms F_{c}^{∗}(x) ⟶ y, i.e. the category F_{c}^{∗}/y is non-empty, the assignment to every y ∈ ObC_{X} a colimit of the functor F_{c}^{∗}/y ⟶ C_{X}, (x, F_{c}^{∗}(x) →^{ξ} y) ⟼ x, determines a right adjoint, F_{c∗}, to F_{c}^{∗}.

It follows from the construction that there is a canonical morphism ϵ_{𝔧}(F) : F_{c}^{∗} ⟶ F which depends functorially on F. If F is continuous, then ϵ_{𝔧}(F) can be chosen to be an identical morphism. This shows that 𝔧^{∗} : F ⟼ F_{c} is a left adjoint to the inclusion functor 𝔧_{∗} : End_{c}(C_{X}) ↪ End_{a}(C_{X}) and ϵ_{𝔧} is an adjunction morphism. The other adjunction morphism is identical.

(b) If F = (F, μ) is a monad on C_{X}, then there is a unique monad structure, μ_{c} : (F_{c}^{∗})^{2} ⟶ F_{c}^{∗} such that the adjunction morphism F_{c}^{∗} ⟶ F is a monad morphism. It follows from (a) that the map F = (F, μ) ⟼ F_{c} = (F_{c}^{∗}, μ_{c}) extends to a functor which is left adjoint to the inclusion functor 𝔧^{∗} : 𝔐𝔬𝔫_{X}^{c} ↪ 𝔐𝔬𝔫_{X}^{a}. Details are left to the reader.

C10.4.3. Proposition. Suppose the category C_{X} has limits and colimits of small diagrams. Then the inclusion functor 𝔧^{∗} : 𝔐𝔬𝔫_{X}^{c} ↪ 𝔐𝔬𝔫_{X} has a left adjoint.

Proof. Let F = (F, μ) be a monad on X. Since Id_{C_{X}} is a continuous functor and there is a morphism η : Id_{C_{X}} ⟶ F (the unit of F), the category End_{c}(C_{X})/F (see the part (a) of the argument of C10.4.2) is non-empty. Let D_{F} denote the standard functor End_{c}(C_{X})/F ⟶ End(C_{X}). For any two objects, x, y of the category C_{X}, we have isomorphisms:

C_{X}(F_{c}(x), y) ≃ lim C_{X}(D_{F} (x), y) ≃ lim C_{X}(x, D_{F}^{∨}(y)).

Here D_{F}^{∨} denotes the diagram End_{c}(C_{X})/F ⟶ End(C_{X}) which assigns to any object (G^{∗}, ξ) of End_{c}(C_{X})/F a right adjoint, G_{∗}, to G^{∗} and to any morphism (G^{∗}, ξ) ⟶ (H^{∗}, ν) the corresponding morphism H_{∗} ⟶ G_{∗} of right adjoint functors. If the category C_{X} has limits, there exists a limit of D_{F}^{∨}(y). Choosing this limit for each y, we define a functor which is a right adjoint to F_{c}.

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