Noncommutative spaces.

Maxim Kontsevich, Alexander Rosenberg

Introduction

In commutative geometry, ”spaces” are understood either as

— locally ringed topological spaces (otherwise called geometric spaces), or as

— sheaves of sets on the category of affine schemes endowed with an appropriate topology. Here appropriate topology varies from the Zariski topology ([DeG, I.3.11]) to a flat (fpqc, or fppf) topology. In the case of the fpqc topology, spaces can be identified with functors X from the category CRings of commutative unital rings to Sets which preserve finite products and such that the diagram X(R) ⟶ X(T) \overrightarrow{⟶} X(T ⊗_{R} T) is exact for any faithfully flat ring morphism R ⟶ T. Respectively, commutative schemes are defined either as locally ringed spaces which are locally affine, or as sheaves of sets obtained by glueing affine schemes (i.e. representable functors) for Zariski topology.

Only comparatively few of the known examples of what might be regarded as noncommutative analogues of schemes, or formal schemes, can be realized as ringed spaces, like D-schemes of Beilinson-Bernstein [BB], [BD], ”virtual” noncommutative formal spaces introduced by Kapranov [Ka], and affine noncommutative schemes of P. Cohn [C]. Usually noncommutative analogues of schemes appear as

— categories (regarded as categories of quasi-coherent sheaves) over a base category, like the Proj of a graded noncommutative ring (see [M1], [V1], [V2], [A2], [AZ], [OW], and a number of other works) and the flag variety of a quantized enveloping algebra (see [LR2], [R3]),

— or functors from the category Alg_{k} of associative unital k-algebras to Sets, like projective spaces introduced in [KR1] and different Grassmannians and flag varieties constructed in [KR2].

We study the categorical approach to spaces in [KR3]. The purpose of this paper is to provide a geometric background for a number of examples of noncommutative spaces of the second type, i.e. spaces defined as functors Alg_{k} ⟶ Sets.

We define the category Aff_{k} of noncommutative affine k-schemes as the category of representable functors from Alg_{k} to Sets. Thus Yoneda imbedding R ⟼ Alg_{k}(R, −) induces an equivalence between Aff_{k} and the category Alg_{k}^{op} opposite to Alg_{k}. A flat cover on Aff_{k} is given by a finite set {A ⟶ A_{i} | i ∈ J} of flat algebra morphisms such that the corresponding morphism A ⟶ ∏_{i∈J} A_{i} is faithful. Flat covers play an important role in noncommutative setting due to the fact that the flat descent holds in the noncommutative case providing means for studying categories of quasi-coherent sheaves (this is used in [KR1]). But, in general, flat covers do not form a pretopology: the invariance under a base change fails. This leads to a weaker version of a (pre)topology based on the notion of a

Q-category (here ’Q’ stands for ’quotient’). By definition, a Q-category is a pair, Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A,

of functors such that the functor u^{∗} is fully faithful and left adjoint to u_{∗}, which implies that A is a quotient category of Ā and u_{∗} is a localization functor. Both Grothendieck sites and Grothendieck pretopologies can be viewed as Q-categories. For any category C, we have a Q-category C^{𝔸} = (CĀ ⇄ C^{A}) of functors on 𝔸 with values in C. In particular, we have the Q-category of presheaves on a Grothendieck site, or a pretopology.

The main basic notion in this formalism is the notion of a sheaf: given a Q-category

𝔸 = (Ā\overrightarrow{⟵}^{u∗}_{u^{∗}}A), an object x of the category A is called an 𝔸-sheaf if the canonical map

Ā(ȳ, u^{∗}(x)) ⟶ A(u_{∗}(ȳ), x), g ⟼ η_{x}^{−1} ∘ u_{∗}(g),  (1)

is an isomorphism for all ȳ ∈ ObĀ. Here η_{u} is an adjunction isomorphism Id_{A} ⟶^{∼} u_{∗}u^{∗}.

In the case of the Q-category of presheaves on a Grothendieck site, or a Grothendieck pretopology, we recover sheaves in the usual sense.

Flat covers on Aff_{k} give a rise to a Q-category, and noncommutative spaces represent sheaves of sets on this Q-category. Similarly to the commutative case, thus defined ’spaces’ are identified with functors Alg_{k} ⟶^{X} Sets which preserve finite products and such that the natural diagram

X(R) ⟶ X(T) \overrightarrow{⟶} X(T ⋆_{R} T)  (2)

is exact for any faithfully flat ring morphism R ⟶ T. Here T ⋆_{R}T is a traditional notation for the fiber coproduct T ∐_{R} T in the category of associative unital rings. One can show that all representable functors are spaces, i.e. the category Aff_{k} of noncommutative affine k-schemes is a full subcategory of the category of ’spaces’.

As in commutative case, fpqc covers are not always the best choice, and there are noncommutative versions of other types of covers on the category of noncommutative affine schemes which are used to define other categories of ’spaces’. For instance, smooth covers seem to be a more sensible choice for a big part of examples we consider (in [KR] and [KR2]). These covers form a Q-category, but, in general, not a pretopology.

There is another interpretation of Q-categories illustrated by the following example: A is the category CRings of commutative unital rings, Ā the category of (commutative) ring epimorphisms with a nilpotent kernel, the functor A → Ā maps any ring to the identical endomorphism of this ring, and Ā → A maps a ring epimorphism S → R to its target, R. In this case, sheaves (resp. monopresheaves) on 𝔸 turn out to be formally étale (resp. formally unramified) functors CRings ⟶ Sets; and epipresheaves (defined in 3.1.4) are formally smooth functors. This example suggests that Q-categories might be regarded also as ”categories of thickennings”. An appropriate (not quite obvious) choice of a Q-category produces a noncommutative version of formally étale, formally unramified, and formally smooth functors.

The paper is organized as follows.

First three sections contain preliminaries on Q-categories. In Sections 1 and 2, we define a Q-category and morphisms of Q-categories and give a number of examples. In Section 3, we introduce the notions of a sheaf, a monopresheaf and an epipresheaf in a Q-category and illustrate these notions using examples of Section 2.

In Section 4, we interpret epipresheaves, monopresheaves and sheaves of sets on a Q-category as resp. formally smooth, formally unramified, and formally étale functors. In this case, the Q-category is thought as the Q-category of thickennings.

Most of Section 5 is dedicated to definition and basic properties, and some examples of formally smooth, formally unramified and formally étale morphisms of presheaves of sets on a Q-category. We introduce locally finitely presentable morphisms and define smooth (resp. unramified, resp. étale) morphisms as locally finitely presentable morphisms which are formally smooth (resp. formally unramified, resp. formally étale). We define open immersions as smooth monomorphisms and obtain general properties of open immersion as a consequence of those of smooth morphisms.

In Section 6, we introduce formally 𝔸-infinitesimal morphisms (which might be regarded as formal thickenings of spaces) as the dual notion to that of formally 𝔸-smooth morphisms. We make one more step giving a symmetric form to the duality between formally smooth and formally infinitesimal morphisms. Curiously, this leads to the interpretation of separated (resp. universally closed, resp. proper) morphisms of schemes as formally unramified (resp. formally smooth, resp. formally étale) morphisms for an appropriate choice of the class of formally infinitesimal morphisms of schemes.

In Section 7, we discuss closed immersions and separated morphisms.

The second part of the work is dedicated to noncommutative locally affine spaces and schemes. In Section 8, we introduce locally affine spaces and schemes. With any Q-category, we associate another Q-category of a ’topological nature’ called a quasi-cosite. We define ”𝔸-spaces” as sheaves of sets on the quasi-cosite associated with 𝔸. For any quasi-topology on the category of 𝔸-spaces, we define locally affine spaces and schemes in the most direct way. Starting with another Q-category, 𝔸_{1} = (Ā_{1} ⇄ A) (having the same underlying category A) regarded as a Q-category of thickennings, we define three natural quasi-topologies in which covers are sets of resp. 𝔸_{1}-smooth morphisms, 𝔸_{1}-étale morphisms, and 𝔸_{1}-open immersions.

In Section 9, we consider basic applications of this formalism. We recover commututative schemes and algebraic spaces taking as 𝔸 the Q-category of commutative rings with fpqc cocovers. This means that A is the category of commutative rings and objects of Ā are finite conservative families of flat ring morphisms. And Ā_{1} is the category of ring surjective morphisms with a nilpotent kernel. An appropriate noncommutative generalization of this setting produces (via the formalism of Section 8) the notions of noncommutative schemes and algebraic spaces.

Section 10 is devoted to the noncommutative Grassmannian which is one of important examples of a noncommutative locally affine space.

The paper has two appendices. The first appendix contains some complementary facts about Q-categories. In Appendix 2, we discuss finiteness conditions.

A considerable part of this manuscript was written while the second author was visiting Max-Planck Institute für Mathematik in Bonn during the summer of 2001. He would like to thank the Institute for excellent working conditions.

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

I. Q-categories and sheaves.

Q-categories were introduced (initially in [R5]) as a milder version of Grothendieck sites. We define noncommutative spaces as sheaves on Q-categories. Another interpretation of Q-categories leads to (a generalization of) the notions of formally smooth, formally unramified, and formally étale morphisms which play a significant role in this work and its sequals. We establish, among other things, the following dictionary:

sheaves of sets (on Q-categories) ⟷ formally étale spaces
monopresheaves of sets ⟷ formally unramified spaces
epipresheaves of sets ⟷ formally smooth spaces

Thus, formally étale (resp. formally unramified, resp. formally smooth) morphisms might be regarded as relative versions of sheaves (resp. monopresheaves, resp. epipresheaves) on a Q-category.

1. Q-categories. A Q-category is a pair of functors Ā ⟶^{u_{∗}} A ⟶^{u^{∗}} Ā such that the functor u^{∗} is fully faithful and left adjoint to u_{∗}. We shall regard functors u_{∗}, u^{∗} as resp.

direct and inverse image functors of a morphism Ā ⟶^{u} A and write this data as Ā ⇄^{u} A.

A morphism from a Q-category Ā ⇄^{u} A to a Q-category Ā′ ⇄^{u′} A′ is a triple (Φ, Φ̄, ϕ),

where A ⟶^{Φ} A′ and Ā ⟶^{Φ̄} Ā′ are functors and ϕ is a functor isomorphism Φu_{∗} ⟶ u′_{∗}Φ̄. The composition of two morphisms, is defined by

(Φ′, Φ̄′, ϕ′) ∘ (Φ, Φ̄, ϕ) = (Φ′Φ, Φ̄′Φ̄, Φ̄′ϕ ∘ ϕ′Φ)

For a universum 𝔘, we denote by QCat_{𝔘}, or simply by QCat, the category whose

objects are Q-categories Ā ⇄^{u} A such that A and Ā belong to Cat_{𝔘}.

1.1. Remark. Since the functor u^{∗} is fully faithful, the direct image functor u_{∗} is

a localization. A morphism (Φ, Φ̄, ϕ) from Ā ⇄^{u} A to Ā′ ⇄^{u′} A′ is defined uniquely up to

isomorphism by the functor Ā ⟶^{Φ̄} Ā′ and the compatibility of Φ̄ with the corresponding localizations (expressed by ϕ).

In fact, the isomorphism Φu_{∗} ⟶^{ϕ} u′_{∗}Φ̄ induces an isomorphism Φ ⟶ u′_{∗}Φ̄u^{∗}. This follows from the fact that, since the functor u^{∗} is fully faithful, an adjunction morphism Id_{A} ⟶^{η_{u}} u_{∗}u^{∗} is an isomorphism. The isomorphism ϕ is equivalent (after replacing Φ by u′_{∗}Φ̄u^{∗}) to the canonical morphism

u′_{∗}Φ̄u^{∗}u_{∗} ⟶ u′_{∗}Φ̄  (1)

induced by an adjunction morphism u^{∗}u_{∗} ⟶^{ϵ_{u}} IdĀ. The compatibility of Φ̄ with localizations means exactly that (1) is an isomorphism.

1.2. The 2-category of Q-categories. The category QCat is the category of 1-morphisms of a 2-category, QCat^{2}: given a pair of 1-morphisms (Φ, Φ̄, ϕ), (Ψ, Ψ̄, ψ) from Ā ⇄^{u} A to Ā′ ⇄^{u′} A′, a 2-arrow from (Φ, Φ̄, ϕ) ⟶ (Ψ, Ψ̄, ψ) is a pair of morphisms, Φ ⟶^{α} Ψ, Φ̄ ⟶^{ᾱ} Ψ̄ such that the diagram

Φu_{∗}  ⟶^{ϕ} u′_{∗}Φ̄
αu_{∗} ↓  ↓ u′_{∗}ᾱ
Ψu_{∗}  ⟶^{ϕ} u′_{∗}Ψ̄

commutes.

1.3. Induced Q-categories. Fix a Q-category 𝔸 = (Ā ⇄^{u} A). Let Φ : B ⟶ A be a functor. Let B̄ denote the fiber product of B ⟶^{Φ} A ⟵^{u_{∗}} Ā. Recall that objects of B̄ are triples (x, α, ȳ), where x ∈ ObB, ȳ ∈ ObĀ, and α is an isomorphism Φ(x) ⟶^{∼} u_{∗}(ȳ). Morphisms are defined in an obvious way. There are natural projection functors

B̄ ⟶^{Φ̄} Ā, (x, α, ȳ) ⟼ ȳ, and B̄ ⟶^{u_{_{Φ}∗}} B, (x, α, ȳ) ⟼ x.

We define a functor B ⟶^{u_{Φ}^{∗}} B̄ by x ⟼ (x, η_{u}Φ(x), u^{∗}Φ(x)). It follows that u_{_{Φ}∗} ∘ u∗_{Φ} = Id_{B} and u∗_{Φ} ∘ u_{_{Φ}∗}(x, α, ȳ) = (x, η_{u}Φ(x), u^{∗}Φ(x)). We have a canonical morphism

ϵ_{u_{Φ}}(x, α, ȳ) = (id_{x}, ϵ_{u}(ȳ) ∘ u^{∗}(α) : u∗_{Φ} ∘ u_{_{Φ}∗}(x, α, ȳ) ⟶ (x, α, ȳ)

functorial in (x, α, ȳ). Hence ϵ_{u_{Φ}} = {ϵ_{u_{Φ}}(x, α, ȳ)| (x, α, ȳ) ∈ ObB̄} is a funcrtor morphism u∗_{Φ} ∘ u_{_{Φ}∗} ⟶ IdB̄. One can see that ϵ_{u_{Φ}} and the identical functor Id_{B} ⟶ Id_{B} = u_{_{Φ}∗}u∗_{Φ} are adjunction morphisms for the pair of functors u_{_{Φ}∗}, u∗_{Φ}. In particular, the functor

B ⟶^{u_{_{Φ}∗}} B̄ is fully faithful.

Notice that u_{∗} ∘ Φ̄(x, α, ȳ) = u_{∗}(ȳ), and Φ ∘ u_{_{Φ}∗}(x, α, ȳ) = Φ(x). Thus

ϕ(x, α, ȳ) = α : Φ ∘ u_{_{Φ}∗}(x, α, ȳ) = Φ(x) ⟶ Φ(x) = u_{∗} ∘ Φ̄(x, α, ȳ), (x, α, ȳ) ∈ ObB̄,

defines a functor isomorphism Φ ∘ u_{_{Φ}∗} ⟶^{ϕ} u_{∗} ∘ Φ̄.

Altogether, we have obtained a Q-category 𝔸_{Φ} = 𝔹 = (B̄ ⇄^{u_{Φ}} B) induced by the functor Φ and a canonical Q-category morphism (Φ̄, ϕ, Φ) : 𝔹 ⟶ 𝔸.

1.3.1. Two special cases. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. For any object x of the category A, we have the categories A/x and x\A and the canonical functors A/x ⟶ A ⟵ x\A. We denote the corresponding induced Q-categories resp. by 𝔸/x and x\𝔸 .

1.4. Q^{o}-categories. A Q^{o}-category is a pair of functors Ā ⟶^{u_{∗}} A ⟶^{u^{∗}} Ā such that the functor u^{∗} is fully faithful and a right adjoint to u_{∗}. In other words, the data

Ā ⟶^{u_{∗}} A ⟶^{u^{∗}} Ā is a Q^{o}-category iff the dual data, Ā^{op} ⟶^{u_{∗}^{op}} A^{op} ⟶^{u^{∗op}} Ā^{op} is a Q-category.

All facts and constructions on Q-categories (resp. Q^{o}-categories) have their dual versions which will be used when needed.

2. Examples of Q-categories and Q^{o}-categories. Given a category A, there are two families of basic examples of Q-categories (resp. Q^{o}-categories) having A as the underlying category:

– the Q-category of cosieves on A (cf. 2.1) and its Q-subcategories,

– dually, the Q^{o}-category of sieves on A and its Q^{o}-subcategories (among them Grothendieck sites with the base A);

– the Q-category (resp. the Q^{o}-category) of morphisms in A (see 2.5) and its Qsubcategories (resp. Q^{o}-subcategories).

Our main class of examples are Q-categories of functors on a Q-category (or Qcategories of presheaves on a Q^{o}-category) (see 2.6).

2.1. The Q-category of cosieves. Fix a category A. Let SA denote the category of cosieves on A defined as follows. Objects of SA are pairs (x, R), where x ∈ ObA and R is a cosieve in x\A. Morphisms from (x, R) to (x′, R′) are given by arrows x ⟶^{f} x′ such that R′_{f} ⊆ R. Here R′_{f} is a cosieve in x\A whose objects are all pairs (v, ξ ∘ f) such that

(v, ξ) ∈ ObR′. There is a functor A ⟶^{u^{∗}} SA which assigns to each object x of A the pair (x, x\A). The functor u^{∗} is fully faithful and has a canonical right adjoint,

SA ⟶^{u_{∗}} A, (x, R) ⟼ x.

This defines a Q-category of cosieves, SA ⇄ A.

Cosieves in x\A are in a natural one-to-one correspondence with subfunctors of the functor A(x, −). Thus the Q-category of cosieves is isomorphic to a Q-category, Ā ⇄ A, defined as follows. Objects of Ā are pairs (x, R), where x ∈ ObA, R is a subfunctor of A(x, −). Morphisms from (x, R) to (y, S) are morphisms x ⟶^{f} y such that the functor

morphism A(y, −) ⟶^{A(f,−)} A(x, −) induces a morphism S ⟶ R of the subfunctors. The functor u_{∗} maps a pair (x, R) to x. The functor u^{∗} assigns to any object x of A the pair (x, A(x, −)).

2.2. Quasi-(co)sites and (co)sites. Let A be a category, and let 𝔗 be a map which assigns to every object x of A a set 𝔗(x) of subfunctors of A(x, −) which contains A(x, −) itself. We shall identify the pair (A, 𝔗) with the full Q-subcategory Ā_{𝔗} ⇄ A of the Q-category SA ⇄ A of cosieves objects of which are all pairs (x, R), where x ∈ ObA and R ∈ 𝔗(x).

We call the pair (A, 𝔗) a quasi-cosite if two conditions hold:

(a) for any pair R, R′ ∈ 𝔗(x), R ∩ R′ ∈ 𝔗(x),

(b) if R ∈ 𝔗(x) and R′ is a subfunctor of A(x, −) containing R, then R′ ∈ 𝔗(x).

Quasi-sites correspond to quasi-cosites on the dual category A^{op}.

Grothendieck sites are quasi-sites. Recall that a site is a pair (A, 𝔗), where 𝔗 is a topology, i.e. a map which assigns to each x ∈ ObA a set 𝔗(x) of subfunctors of A(−, x) (called refinements of x) satisfying the conditions:

(i) for any R ∈ 𝔗(x) and any arrow f : y ⟶ x, the subfunctor R_{f} = R×_{A(−,x)}A(−, y) of A(−y) is a refinement of y (i.e. it belongs to 𝔗(y).

(ii) If R ∈ 𝔗(x) and R̃ is a subfunctor of A(−, x) such that R̃_{f} ∈ 𝔗(y) for any f ∈ R(y), y ∈ ObA,, then R̃ ∈ 𝔗(x).

Q-categories dual to Grothendieck sites are called cosites. The Q-category of cosieves, (SA ⇄ A) and its Q-subcategory 𝔸_{dis} = (Ā_{dis} ⇄ A), where Ā_{dis} is formed by all pairs (x, x\A), x ∈ ObA, are two extreme examples of cosites.

Cosites might be regarded as a topology in terms of “closed sets”. If (Ā ⇄^{u} A) is a cosite, then A might be viewed as the category of closed sets of a would-be space.

2.3. A quasi-cosite associated with a Q-category. Fix a Q-category (Ā ⇄^{u} A). To any ȳ ∈ ObĀ, we assign the category ȳ\u^{∗} of pairs (f, x), where f is a morphism

ȳ ⟶ u^{∗}(x). The functor u_{∗} induces a morphism, Φ^{∼} = (Φ, id_{u_{∗}}, Id_{A}), from 𝔸 = (Ā ⇄^{u} A) to the cosite SA ⇄ A. Here Φ is a functor Ā ⟶ SA which assigns to any object ȳ of Ā the pair (u_{∗}(ȳ), Rȳ), where Rȳ denotes the cosieve in u_{∗}(ȳ)\A formed by all (v, u_{∗}(ȳ) →^{ξ} v)

such that ξ = η_{u}^{−1}(v) ∘ ξ̄ for some ȳ ⟶^{ξ̄} u^{∗}(v). The quasi-cosite, 𝔗𝔸 = (𝔗_{𝔸} ⇄ A), associated with 𝔸 is the smallest quasi-cosite containing the image of the functor Φ. The triple (Id_{A}, Φ, id) is a canonical morphism from 𝔸 to the Q-category (SA ⇄ A) of cosieves on A.

2.3.1. Note. If 𝔸 is a quasi-cosite, then 𝔗𝔸 is naturally isomorphic to 𝔸. Dually, with every Q^{o}-category 𝔸, one can associate a quasi-site which is naturally isomorphic to 𝔸 if 𝔸 is a quasi-site.

2.3.2. Proposition. Suppose 𝔸 has the property:

(*) for any ȳ ∈ ObĀ and any morphism x ⟶^{f} u_{∗}(ȳ), there exists a morphism x̄ ⟶^{f̄} ȳ and an isomorphism u_{∗}(x̄) ⟶^{α} x such that u_{∗}(f̄) = f ∘ α.

Then the quasi-cosite associated with 𝔸 is a cosite.

Proof is left to the reader.

2.4. Quasi-(co)sites and (co)covers. Let τ be a function which assigns to each object, x, of the category A a family, τ_{x}, of sets of arrows to x which contains {x →^{id_{x}} x}. This data defines a category, A_{τ}, whose objects are all pairs (x, U), where x ∈ ObA, U ∈ τ_{x}; we shall call them covers. Morphisms from (x, U) to (y, V) are morphisms x ⟶^{f} y such that for any arrow x_{u} →^{u} x in U there exists an arrow y_{v} →^{v} y in V and a morphism x_{u} ⟶^{g_{uv}} y_{v} such that f ∘ u = v ∘ g_{uv}. The functor A_{τ} ⟶ A which assigns to every pair (x, U) the object x and to every morphism (x, U) ⟶^{f} (y, V) the morphism x ⟶^{f} y is a right adjoint to the fully faithful functor A ⟶ A_{τ} which maps every object x of A to (x, {id_{x}}). This defines a Q^{o}-category 𝔸_{τ} = (A_{τ} ⇄ A).

Consider the quasi-site 𝔗𝔸_{τ} associated with 𝔸_{τ}. The functor A_{τ} ⟶^{Φ} 𝔗A_{τ} assigns to every cover (x, U) the pair (x, R_{U}), where R_{U} is the sieve associated with the set of arrows U: it consists of all arrows to x which factor through some of the arrows of U.

If for any morphism y ⟶^{f} x and any U ∈ τ_{x}, there exists V ∈ τ_{y} such that f is a morphism (y, V) ⟶ (x, U), then the quasi-site associated with 𝔸_{τ} is a site. In particular, if τ is a Grothendieck pretopology, we obtain this way the site associated with τ.

2.4.1. Covers. Let 𝔸 = (Ā ⇄^{u} A) be a quasi-site. A set of arrows U = {x_{i} → x| i ∈ J} in A is called a cover (or an 𝔸-cover) of x, if the pair (x, R_{U}), where R_{U} is the sieve associated to U, is an object of Ā.

It follows from the definition of a quasi-site that

(i) every set of arrows to x which contains a cover is a cover;

(ii) if U and U′ are covers of x, then U ×_{x} U′ = {x_{u} ×_{x} x_{v} ⟶ x | u ∈ U, v ∈ U′} is a cover of x, provided the pull-backs x_{u} ×_{x} x_{v} exist for all u ∈ U, v ∈ U′.

2.5. The Q-category and the Q^{o}-category of morphisms of a category. Fix a category A. Consider the category A^{2} objects of which are morphisms of the category

A, and morphisms from x →^{f} y to x′ →^{f′} y′ are commutative squares

\begin{matrix}x & ⟶^{g} & x′ \\ f ↓ & & ↓ f′ \\ y & ⟶^{h} & y′ \end{matrix}  (1)

Denote by u^{∗} the functor A ⟶ A^{2} which assigns to any object x of A the object x ⟶^{id_{x}} x and to any morphism f the corresponding commutative square. The functor u^{∗} is fully faithful and has a right adjoint, u_{∗}, which maps any object x →^{f} y of A^{2} to x and any morphism (1) to x →^{g} x′. In fact, u_{∗}u^{∗} = Id_{A}, and there is a natural morphism u^{∗}u_{∗} ⟶^{ϵ_{u}} Id_{A^{2}} which assigns to any object x →^{f} y of the category A^{2} the morphism

x ⟶^{id_{x}} x
id_{x} ↓  ↓ f
x ⟶^{f} y

from u^{∗}u_{∗}(x →^{f} y) to (x →^{f} y). One can see that Id_{A} ⟶^{id} u_{∗}u^{∗} and ϵ_{u} are adjunction morphisms.

Dually, the functor u^{∗} has a natural left adjoint, u_{!}, which assigns to any object x →^{f} y of A^{2} the object y and to any morphism (1) the morphism y →^{h} y′.

2.5.1. Q-subcategories of (A^{2} ⇄ A). Let Ā be a full subcategory of the category A^{2} which contains all objects x ⟶^{id_{x}} x. Then the functor u^{∗} takes values in the subcategory Ā,

hence it induces a structure of a Q-subcategory, 𝔸 = (Ā ⇄^{u} A), of the Q-category A^{2} ⇄ A. The functor u_{!} : A^{2} ⟶ A induces a functor Ā ⟶ A left adjoint to u^{∗}.

2.5.2. Q-categories with a functor u_{!} and the Q-category of morphisms.

Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that the functor u^{∗} has a left adjoint, u_{!}. Then there is a canonical morphism u_{∗} ⟶^{𝔯_{u}} u_{!} equal to the composition u_{∗}(η_{u}^{!} ∘ ϵ_{u}) ∘ η_{u}u_{∗}. Here

IdĀ ⟶^{η_{u}^{!}} u^{∗}u_{!}, Id_{A} ⟶^{η_{u}} u_{∗}u^{∗}, and u^{∗}u_{∗} ⟶^{ϵ_{u}} IdĀ are adjunction morphisms. Thus we have a canonical morphism (Ψ_{𝔸}, id_{u_{∗}}, Id_{A}) : 𝔸 ⟶ (A^{2} ⇄ A), where the functor Ψ_{𝔸} assigns

to any object, ȳ, of Ā the canonical morphism u_{∗}(ȳ) ⟶^{𝔯_{u}(ȳ)} u_{!}(ȳ).

On the other hand, there is a canonical morphism from the Q-category 𝔸 to the Qcategory of cosieves SA ⇄ A defined in 2.3. Notice that for any ȳ ∈ Ā, the category ȳ\u^{∗} is isomorphic to the category u_{!}(ȳ)\A. Two morphisms, g_{1}, g_{2} : u_{!}(ȳ) ⟶ x define the same object of the sieve Rȳ (cf. 2.3) iff the square

\begin{matrix}u_{∗}(ȳ) & ⟶^{𝔯_{u}(ȳ)} & u_{!}(ȳ) \\ 𝔯_{u}(ȳ) ↓ & & ↓ g_{1} \\ u_{!}(ȳ) & ⟶^{g_{2}} & x \end{matrix}  (1)

is commutative. Suppose u_{!}(ȳ) ∐_{u_{∗}(ȳ)} u_{!}(ȳ) exists, and let p_{1}, p_{2} be canonical coprojections

u_{!}(ȳ) \overrightarrow{⟶} u_{!}(ȳ) ∐_{u_{∗}(ȳ)} u_{!}(ȳ). Then the commutativity of (1) means that g_{1} = g ∘ p_{1} and

g_{2} = g ∘ p_{2} for a uniquely determined morphism u_{!}(ȳ) ∐_{u_{∗}(ȳ)} u_{!}(ȳ) ⟶^{g} x.

The following is one of our main examples of a Q-category with the functor u_{!}.

2.6. The Q-category of infinitesimal algebra epimorphisms. Let A be the category Alg_{k} of associative unital k-algebras, and let Ā be the full subcategory of the category Alg_{k}^{2} of k-algebra morphisms whose objects are epimorphisms with a nilpotent kernel.

2.6.1. Note. The commutative version of 2.6 (i.e. A is the category CAlg_{k} of commutative algebras and Ā is the subcategory of CAlg_{k}^{2} whose objects are commutative algebra epimorphisms with nilpotent kernel) can be interpreted as the category of infinitesimal extensions of affine schemes over k.

2.6.2. The Q-category of thickennings of a scheme. A non-affine version of the example 2.6.1 is the Q-category of thickennings of a scheme. Fix a scheme X. Let A be the category of (Zariski) open subschemes of X, and Ā the category of thickennings: objects of the category Ā are nilpotent scheme closed immersions U ⟶ T, where U is any open subscheme of X. The fully faithful functor A ⟶^{u^{∗}} Ā, U ⟼ (U ⟶^{id_{U}} U), is left adjoint to the functor Ā ⟶^{u_{∗}} A sending an immersion U ⟶ T to U.

2.7. Q-categories of functors. Fix a category C. To any Q-category 𝔸 = (Ā ⇄^{u} A),

we assign a Q-category C^{𝔸} = (CĀ ⇄^{C^{u}} C^{A}). Here C^{A} denotes the category of functors A ⟶ C and C^{u} is a morphism with the inverse image functor

C^{u_{∗}} : C^{A} ⟶ CĀ, F ⟼ F ∘ u_{∗},

and the direct image functor C^{u^{∗}} : G ⟼ G ∘ u^{∗}. If C is the category Sets, we shall write

𝔸^{∨} = (Ā^{∨} ⇄^{u^{∨}} A^{∨}) instead of Sets^{𝔸} = (SetsĀ ⇄^{Sets^{u}} Sets^{A}).

3. Sheaves, monopresheaves and epipresheaves in a Q-category.

3.1. Definitions. Fix a Q-category 𝔸 = (Ā ⇄^{u} A).

3.1.1. 𝔸-sheaves. We call an object x of the category A an 𝔸-sheaf if the canonical map

Ā(ȳ, u^{∗}(x)) ⟶ A(u_{∗}(ȳ), x), g ⟼ η_{x}^{−1} ∘ u_{∗}(g),  (1)

is an isomorphism for any ȳ ∈ ObĀ. Here η_{u} is an adjunction (iso)morphism Id_{A} ⟶ u_{∗}u^{∗}. We denote by 𝔉𝔸 the full subcategory of the category A generated by 𝔸-sheaves.

3.1.2. 𝔸-monopresheaves. We call an object x of the category A an 𝔸-monopresheaf, or an 𝔸-separated presheaf, if the canonical map (1) is injective for any ȳ ∈ ObĀ. We denote by 𝔐𝔸 the full subcategory of A formed by 𝔸-monopresheaves.

3.1.3. The canonical morphism ρ_{u}. Let x be an object of the category A such that the functor A(u_{∗}(−), x) is representable, i.e. A(u_{∗}(−), x) ≃ Ā(−, u^{!}(x)) for some u^{!}(x) ∈ ObĀ. There is a canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) corresponding to the isomorphism η_{u}^{−1}(x) : u_{∗}u^{∗}(x) ⟶ x. It follows from the definitions that x is an 𝔸monopresheaf iff the morphism ρ_{u}(x) is a monomorphism. Note, however, that x can be a monopresheaf without the functor A(u_{∗}(−), x) being representable.

3.1.4. 𝔸-epipresheaves. We call an object x of A an 𝔸-epipresheaf if the functor A(u_{∗}(−), x) is representable and the canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) (cf. 3.1.3) is a strict epimorphism.

We denote by 𝔈𝔸 the full subcategory of the category A formed by 𝔸-epipresheaves. It follows from 3.1.3 that an object x of A is an 𝔸-sheaf iff it is an 𝔸-monopresheaf

and an 𝔸-epipresheaf.

3.1.5. Dual notions. Let 𝔸 = (Ā ⇄^{u} A) be a Q^{o}-category. We call an object x of A a sheaf (resp. a monopresheaf, resp. an epipresheaf) in 𝔸 if x is a sheaf (resp. a monopresheaf, resp. an epipresheaf) in the dual Q-category 𝔸^{op}.

3.2. Sheaves and monopresheaves in 𝔸^{∧}. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category,

u = (u^{∗}, u_{∗}). And let 𝔸^{∧} = (Ā^{∧} _{u} ; ⇄ A^{∧}) be the corresponding Q-category of presheaves of

sets. Note that since u_{∗} is a right adjoint to u^{∗}, the functor A^{∧} \underrightarrow{_{u_{∗}}} Ā^{∧}, X ⟼ X ∘ u_{∗}, is a right adjoint to u^{∗}. Thus, a presheaf A^{op} ⟶^{X} Sets is a sheaf iff the canonical morphism u^{∗}(X) ⟶ û_{∗}(X) = X ∘ u_{∗} is an isomorphism. That is for any Y ∈ ObĀ, the canonical morphism

u^{∗}(X)(Y) = Ā^{∧}(Y, u^{∗}(X)) ≃ co lim_{(V,ξ)∈A/X} Ā^{∧}(Y, u^{∗}(V)) ⟶ Xu_{∗}(Y) = A^{∧}(u_{∗}(Y), X)

(1)

is an isomorphism.

Similarly, a presheaf A^{op} ⟶^{X} Sets is a monopresheaf iff the canonical morphism u^{∗}(X) ⟶ X ∘ u_{∗} is a monomorphism. That is for any Y ∈ ObĀ, the canonical morphism (1) is a monomorphism.

It follows that an object x of A is a sheaf (resp. a monopresheaf) in the Q-category 𝔸 iff the presheaf of sets A(−, x) is a sheaf (resp. a monopresheaf) in 𝔸^{∧}.

3.3. Sheaves and monopresheaves in Q-categories of functors. Fix a Q-

category 𝔸 = (Ā ⇄^{ū} A) and a category C. For any pair of functors, Ā ⟶^{Ḡ} C and A ⟶^{F} C, we have the canonical map

CĀ(Ḡ, F ∘ u_{∗}) ⟶ C^{A}(Ḡ ∘ u^{∗}, F), ϕ ⟼ F η_{u}^{−1} ∘ ϕu^{∗},  (1)

(cf. 2.7). By definition, the functor F is a C^{𝔸}-sheaf (resp. a C^{𝔸}-monopresheaf) iff the morphism (1) is bijective (resp. injective) for any Ḡ.

3.3.1. Note. We shall usually call C^{𝔸}-sheaves (resp. C^{𝔸}-monopresheaves) sheaves (resp. monopresheaves) on 𝔸 with values in C.

3.4. Sheaves in 𝔸 and sheaves on 𝔸. The following proposition shows that these two notions are, in a sense, dual to each other.

3.4.1. Proposition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that the inverse image functor u^{∗} has a left adjoint, u_{!}. The functor A(x, −) : A ⟶ Sets is a sheaf (resp. a monopresheaf) in 𝔸^{∨} = Sets^{𝔸} if and only if x is a sheaf (resp. a monopresheaf) in the

Q^{o}-category 𝔸_{!} = (Ā ⇄^{u^{o}} A).

Proof. We denote the functor A(x, −) by F.

(i) Suppose Ḡ = Ā(ȳ, −) : Ā ⟶ Sets for some object ȳ of Ā. Then by Yoneda’s Lemma,

Ā^{∨}(Ḡ, F ∘ u_{∗}) ≃ F (u_{∗}(ȳ)) = A(x, u_{∗}(ȳ)) ≃ Ā(u^{∗}(x), ȳ).

Here A^{∨} = Sets^{A}.

(ii) Since the functor u^{∗} has a left adjoint, u_{!}, we have G ∘ u^{∗} = Ā(ȳ, u^{∗}(−)) ≃ A(u_{!}(ȳ), −), hence

A^{∨}(Ḡ ∘ u^{∗}, F) ≃ A^{∨}(A(u_{!}(ȳ), −), A(x, −))

and by Yoneda’s Lemma, A^{∨}(A(u_{!}(ȳ), −), A(x, −) ≃ A(x, u_{!}(ȳ)). Thus we have a commutative diagram

\begin{matrix}Ā^{∨}(Ḡ, F ∘ u_{∗}) & ⟶ & A^{∨}(Ḡ ∘ u^{∗}, F) \\ ≀↑ & & ↑≀ \\ Ā^{op}(ȳ, u^{∗}(x)) & ⟶ & A^{op}(u_{!}(ȳ), x) \end{matrix}  (1)

in which vertical arrows are canonical isomorphisms. This shows that if the functor A(x, −) is a sheaf (resp. a monopresheaf) of sets on the Q-category 𝔸, the object x is a sheaf in

the Q^{o}-category 𝔸_{!} = (Ā ⇄^{u^{o}} A).

(iii) Any functor Ā ⟶^{G} Sets is a colimit of the diagram hĀ/G, where hĀ denotes the Yoneda’s functor Ā^{op} ⟶ Ā^{∨}, ȳ ⟼ Ā(ȳ, −). Thus, the canonical morphism

A^{∨}(G, F ∘ u_{∗}) ⟶ A^{∨}(G ∘ u^{∗}, F)  (2)

is isomorphic to the limit

lim_{ȳ∈Ob(hĀ/G)} (Ā^{op}(ȳ, u^{∗}(x)) ⟶^{α_{ȳ,x}} A^{op}(u_{!}(ȳ), x)).  (3)

If x is a sheaf in 𝔸_{!}, then all morphisms α_{ȳ,x} in (3) are isomorphisms, hence their limit, (2), is an isomorphism.

Similarly, if x is a monopresheaf in 𝔸_{!}, then all morphisms α_{ȳ,x} in (3) are monomorphisms, hence their limit, (2), is a monomorphism.

3.5. The functors u_{C}^{!} and (u_{C})_{!}. Suppose the category C has small limits. Then the functor C^{u^{∗}} : CĀ ⟶ C^{A}, G ⟼ G ∘ u^{∗}, has a right adjoint, u_{C}^{!}, given for all ȳ ∈ ObĀ and any F : A ⟶ C by

u_{C}^{!}(F)(ȳ) = lim(F ∘ 𝔊ȳ),  (1)

where 𝔊ȳ is the functor ȳ\u^{∗} ⟶ A, (x, ȳ ⟶ u^{∗}(x)) ⟼ x. If C = Sets, we shall write u^{!} instead of u_{Sets}^{!}.

A functor F is a sheaf (resp. a monopresheaf, resp. an epipresheaf) iff the canonical morphism F ∘ u_{∗} ⟶ u_{C}^{!}(F) is an isomorphism (resp. a monomorphism, resp. a strict epimorphism), i.e. for any ȳ ∈ ObĀ, the canonical morphism

F (u_{∗}(ȳ)) ⟶ lim_{V ∈Ob(ȳ\u^{∗})} F (V)  (2)

is an isomorphism (resp. a monomorphism, resp. a strict epimorphism).

3.5.1. Lemma. (a) A functor A ⟶^{F} C is a sheaf (resp. a monopresheaf) on the Q-category 𝔸 iff the functor

C(z, F −) : A ⟶ Sets

is a sheaf (resp. a monopresheaf) for any z ∈ ObC.

(b) A functor A ⟶^{F} C is an epipresheaf on 𝔸 iff u_{C}^{!}(F) exists and for any z ∈ ObC, the functor C(F −, z) : A ⟶ Sets^{op} is an epipresheaf.

Proof. (a) (i) Suppose first that C has limits of small diagrams. Since for any z ∈ ObC, the functor C(z, −) preserves small limits, the morphism 3.5(2) is an isomorphism (resp. a monomorphism) iff

C(z, F (u_{∗}(ȳ))) ⟶ lim_{V ∈Ob(ȳ\u^{∗})} C(z, F (V))

is an isomorphism for all z ∈ ObC.

(ii) In the general case, denote by F^{∧} the composition of the functor A ⟶^{F} C with the Yoneda embedding C ⟶^{h} C^{∧}. The functor F is a sheaf in C^{𝔸} iff F^{∧} is a sheaf in (C^{∧})^{𝔸}. Since C^{∧} is a category with limits, the assertion follows from (i).

(b) We leave the argument to the reader.

3.5.2. The functor (u_{C})_{!}. Fix a Q-category 𝔸 = (Ā ⇄^{u} A) and a category C. Suppose the category C has small colimits. Then the functor

C^{u_{∗}} : C^{A} ⟶ CĀ, F ⟼ F ∘ u_{∗},

has a left adjoint, (u_{C})_{!}, given by

(u_{C})_{!}(G)(x) = colim(G ∘ 𝔉_{u_{∗}/x}),

where 𝔉_{u_{∗}/x} is the canonical functor u_{∗}/x ⟶ Ā, (ȳ, u_{∗}(ȳ) → x) ⟼ ȳ.

We shall write û_{!} instead of (u_{Sets})_{!}.

3.5.2.1. Special cases. If the functor u_{∗} has a right adjoint, u^{!}, then (u_{C})_{!} is isomorphic to the functor G ⟼ G ∘ u^{!}.

If G = Ā(ȳ, −) for some ȳ ∈ ObĀ, then u_{!}(G) ≃ A(u_{∗}(ȳ), −).

3.6. Presheaves and sheaves on a Q-category with values in Sets. Let F be a functor A ⟶ Sets. Then

u^{!}(F)(ȳ) = lim_{V ∈Ob(ȳ\u^{∗})} F (V) ≃ A^{∨}(Ā(ȳ, u^{∗}(−)), F)

and u^{∗}(F)(ȳ) = F ∘ u_{∗} ≃ A^{∨}(u_{∗}(ȳ), F). Here A^{∨} denotes the category of functors from A to Sets, and u_{∗}(ȳ) (as any other object of A) is identified with the corresponding corepresentable functor, A(u_{∗}(ȳ), −). The canonical morphism Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) induces a morphism

A^{∨}(A(u_{∗}(ȳ), −), F) ⟶ A^{∨}(Ā(ȳ, u^{∗}(−)), F).  (1)

The presheaf F is a sheaf (resp. a monopresheaf, resp. an epipresheaf) iff the morphism (1) is an isomorphism (resp. a monomorphism, resp. an epimorphism).

3.6.1. Note. The observation above extends to the case of presheaves with values in a category C having small limits. In this case, the Q-category C^{𝔸} is equivalent to the Q-subcategory C_{𝔠𝔠}^{𝔸^{∨}} of the category C^{(𝔸^{∨})^{op}} formed by presheaves on resp. Ā^{∨} and A^{∨} with values in C having a right adjoint (cf. 2.10.2). For any presheaf F^{∨} : (A^{∨})^{op} ⟶ C, the canonical morphism Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) induces a morphism

F^{∨}(A(u_{∗}(ȳ), −)) ⟶ F^{∨}(Ā(ȳ, u^{∗}(−))).  (2)

The presheaf F^{∨} is a sheaf (resp. a monopresheaf, resp. an epipresheaf) iff the morphism (2) is an isomorphism (resp. a monomorphism, resp. an epimorphism).

3.7. Presheaves and sheaves on Q-categories with a functor u_{!}. Let 𝔸 =

(Ā ⇄^{u} A) be a Q-category such that the functor u^{∗} has a left adjoint, u_{!}. For instance, 𝔸 is a full Q-subcategory of the Q-category (A^{2} ⇄ A) of morphisms (cf. 2.5 and 2.5.1). Denote by 𝔯_{u} the canonical morphism u_{∗} ⟶ u_{!}. For any category C, the functor C^{u_{!}} is a right adjoint to C^{u^{∗}}. Thus, a functor A ⟶^{F} C is a sheaf (resp. a monopresheaf, resp. an epipresheaf) on 𝔸 iff the morphism

F ∘ u_{∗} ⟶^{F (𝔯_{u})} F ∘ u_{!}  (1)

is an isomorphism (resp. a monomorphism, resp. a strict epimorphism).

Denote by Σ_{u} the family {𝔯_{u}(ȳ) : u_{∗}(ȳ) ⟶ u_{!}(ȳ) | ȳ ∈ ObĀ} of morphisms of A. It follows that the category of sheaves on 𝔸 with values in a category C is isomorphic to the category of functors Σ_{u}^{−1}A ⟶ C, where Σ_{u}^{−1}A is the quotient category by Σ_{u}.

3.7.1. Sheaves on Q-categories and localizations. Let Σ be a family of morphisms of a category A containing all isomorphisms of A (or, at least, all identical morphisms). Denote by A_{Σ} the full subcategory of A^{2} formed by morphisms of Σ. This defines a Q-subcategory, 𝔸_{Σ} = (A_{Σ} ⇄ A) of (A^{2} ⇄ A). Sheaves on 𝔸_{Σ} with values in a category C are functors A ⟶^{F} C which transform morphisms of Σ into invertible morphisms. In other words, the category of sheaves on 𝔸_{Σ} with values in C is isomorphic to the category of functors from the quotient category Σ^{−1}A to C.

3.8. Sheaves on Grothendieck sites. Below we show that (pre)sheaves on a site can be realized as (pre)sheaves on the Q-category of the form 𝔸_{Σ} (see 3.7.1) where Σ is the class of covers of a pretopology; in particular, it satisfies (left) Ore conditions.

3.8.1. A Grothendieck pretopology associated with a quasi-(co)site. Let (A, 𝔗) be a quasi-cosite. For any functor A ⟶^{X} Sets, we denote by 𝔗^{∨}(X) the set of all subfunctors, R, of X such that for any object x of A and any morphism A(x, −) ⟶ X, the subfunctor R ×_{X} A(x, −) of A(x, −) belongs to 𝔗(x). It follows that X ⟶^{id_{X}} X belongs to 𝔗^{∨}(X) and the function X ⟼ 𝔗^{∨}(X) is invariant under a base change: for any morphism Y ⟶ X and any R ∈ 𝔗^{∨}(X), the subfunctor R ×_{X} Y ⟶ Y belongs to 𝔗^{∨}(Y). Thus, the function 𝔗^{∨} is a pretopology on the category A^{∨} of functors A ⟶ Sets which we regard as a Q^{o}-category. Since any cover in this pretopology consists of one arrow, the Q^{o}-category (𝔗^{∨} ⇄ A^{∨}) is a Q^{o}-subcategory of the Q^{o}-category ((A^{∨})^{2} ⇄ A^{∨}) of morphisms of A^{∨}.

If (A, 𝔗) is a cosite, then the Yoneda embedding, A^{op} ⟶^{h^{A}} A^{∨}, determines a Q-category embedding (𝔗 ⇄ A) ⟶ (𝔗^{∨} ⇄ A^{∨})^{op}.

Dually, if (A, 𝔗) is a site, then the Yoneda embedding, A ⟶^{h_{A}} A^{∧}, induces a Q^{o}- category embedding (𝔗 ⇄ A) ⟶ (𝔗^{∧} ⇄ A^{∧}).

3.8.2. Proposition. Let (A, 𝔗) be a site, and let C be a category with small limits. Then the Q-category of presheaves on (A, 𝔗) with values in C (which is, by definition, the Q-category of functors on the Q-category (𝔗^{op} ⇄ A^{op}) with values in C) is naturally equivalent to the full Q-subcategory of the Q-category of presheaves on (𝔗^{∧} ⇄ A^{∧}) formed by those presheaves which preserve small limits.

In particular, A^{op} ⟶^{F} C is a sheaf (resp. monopresheaf, resp. epipresheaf) iff the corresponding presheaf on (𝔗^{∧} ⇄ A^{∧}) is a sheaf (resp. monopresheaf, resp. epipresheaf).

Proof. The assertion follows from Proposition II.1.3 in [GZ] (cf. 2.10.2). Details are left to the reader.

3.9. Sheaves on a Q-category and sheaves on the associated quasi-cosite.

Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let 𝔗𝔸 be the quasi-cosite associated with 𝔸 (see 2.4). For any category C, the categories of sheaves (resp. categories of monopresheaves, resp. categories of epipresheaves) on 𝔸 and on 𝔗𝔸 with values in a category C are, usually, quite different.

In fact, suppose the functor u^{∗} has a left adjoint, u_{!}. Then sheaves (resp. monopresheaves, resp. epipresheaves) on 𝔸 with values in the category C are precisely those functors A ⟶^{F} C which map morphisms of Σ_{u} = {𝔯_{u}(ȳ) : u_{∗}(ȳ) ⟶ u_{!}(ȳ) | ȳ ∈ ObĀ} to invertible morphisms (resp. to monomorphisms, resp. to strict epimorphisms) (cf. 3.7).

Suppose, in addition, that for any ȳ ∈ ObĀ, there exists a push-forward

u_{∗}(ȳ) ⟶^{𝔯_{u}(ȳ)}  u_{!}(ȳ)
𝔯_{u}(ȳ) ↓ ↓
u_{!}(ȳ) ⟶ u_{!}(ȳ) ∐_{u_{∗}(ȳ)} u_{!}(ȳ)

Then a functor A ⟶^{F} C is a sheaf on 𝔗𝔸 iff for any ȳ ∈ ObĀ, the diagram

F (u_{∗}(ȳ)) ⟶^{𝔯_{u}(ȳ)} F (u_{!}(ȳ)) \overrightarrow{⟶} F (u_{!}(ȳ) ∐_{u_{∗}(ȳ)} u_{!}(ȳ))  (1)

is exact (cf. 2.4).

3.9.1. Sheaves on the associated quasi-cosite: the general case. Suppose that the Q-category 𝔸 is arbitrary, but the category C has small limits. Then the category C^{𝔸} can be realised as the Q-subcategory, C_{𝔠𝔠}^{(𝔸^{∨})^{op}}, of the category of presheaves on 𝔸^{∨} with values in C (cf. 2.10.2). Let (A^{∨})^{op} ⟶^{F^{∨}} C denote a presheaf corresponding to a functor A ⟶^{F} C. By 3.6.1, F is a sheaf on 𝔸 iff for any ȳ ∈ ObĀ, the morphism

F^{∨}(A(u_{∗}(ȳ), −)) ⟶ F^{∨}(Ā(ȳ, u^{∗}(−))).  (2)

induced by the canonical morphism Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −) is an isomorphism. A functor A ⟶^{F} C is a sheaf on 𝔗𝔸 iff for any ȳ ∈ ObĀ, the diagram

F^{∨}(A(u_{∗}(ȳ), −)) ⟶ F^{∨}(Ā(ȳ, u^{∗}(−)) \overrightarrow{⟶} F^{∨}(Ā(ȳ, u^{∗}(−)) ∏_{A(u_{∗}(ȳ),−))} Ā(ȳ, u^{∗}(−)))

(3) is exact. If follows from the definition of the presheaf F^{∨} that F^{∨}(A(u_{∗}(ȳ), −)) ≃ F (u_{∗}(ȳ)). If the functors A(ȳ, u^{∗}(−)) and Ā(ȳ, u^{∗}(−)) ∏_{A(u_{∗}(ȳ),−))} Ā(ȳ, u^{∗}(−)) are representable, the

diagram (3) is equivalent to the diagram (1).

If C = Sets, then F^{∨}(X) = Hom_{A^{∨}}(X, F) for any X ∈ ObA^{∨}.

3.9.2. Note. Observe that the category of sheaves on 𝔸 is a (stictly full) subcategory of the category of sheaves on the associated quasi-cosite, 𝔗𝔸.

3.9.3. Sheaves in terms of covers. Let A be a category, and let τ be a function which assigns to every object x of A the family, τ_{x} of sets of arrows to x (called covers of x, cf. 2.4) which contains {id_{x}}. We call τ a quasi-pretopology if for any two covers U, V and any two elements (x_{u} →^{u} x) ∈ U and (x_{v} →^{v} x) ∈ V, the pull-back x_{u} ×_{x} x_{v} exists. We shall identify the τ with the corresponding Q^{o}-category 𝔸_{τ} = (A_{τ} ⇄ A) (cf. 2.4).

Suppose 𝔸 = 𝔸_{τ}^{op} = (A_{τ} ⇄ A)^{op} for a quasi-pretopology τ. Let ȳ = (y_{i} →^{ϕ_{i}} y| i ∈ I) be an object of A_{τ} (i.e. a cover; cf. 2.4.1) and x an object of A. Then A^{op}(u_{∗}(ȳ), x) = A(x, y) and A_{τ}^{op}(ȳ, u^{∗}(x)) = A_{τ} ((x →^{id_{x}} x), ȳ) = ∐_{i∈I} A(x, y_{i}). Here coproduct is the coproduct in Sets, i.e. the disjoint union. The canonical morphism

A_{τ}^{op}(ȳ, u^{∗}(x)) = ∐_{i∈I} A(x, y_{i}) ⟶ A^{op}(u_{∗}(ȳ), x) = A(x, y)

maps each morphism x ⟶^{f_{i}} y_{i} to the composition of f_{i} and y_{i} ⟶^{ϕ_{i}} y. The equivalence relation

( ∐_{i∈I} A(x, y_{i})) ∏_{A(x,y)} ( ∐_{i∈I} A(x, y_{i})) ⊆ ( ∐_{i∈I} A(x, y_{i})) ∏ ( ∐_{i∈I} A(x, y_{i}))

is defined correspondingly: a morphism x ⟶^{f_{i}} y_{i} is equivalent to a morphism x ⟶^{f_{j}} y_{j} iff ϕ_{i} ∘ f_{i} = ϕ_{j} ∘ f_{j}. It follows that

( ∐_{i∈I} A(x, y_{i})) ∏_{A(x,y)} ( ∐_{i∈I} A(x, y_{i})) = ∐_{i,j∈I} (A(x, y_{i}) ∏_{A(x,y)} A(x, y_{j})) = ∐_{i,j∈I} A(x, y_{i} ×_{y} y_{j}).

Thus the diagram (2) in the case of a quasi-pretopology is (isomorphic to)

∐_{i,j∈I} A(−, y_{i} ×_{y} y_{j}) \overrightarrow{⟶} ∐_{i∈I} A(−, y_{i}) ⟶ A(−, y)  (4)

Let F be any presheaf of sets on A. Then the functor A^{∧}(−, F) maps the sequence (4) to the diagram

F (y) ⟶ ∏_{i∈I} F (y_{i}) \overrightarrow{⟶} ∏_{i,j∈I} F (y_{i} ×_{y} y_{j}).  (5)

Thus a presheaf F is a sheaf on the quasi-site 𝔗𝔸_{τ} associated with 𝔸_{τ} iff the functor A^{∧}(−, F) maps the diagram (4) to an exact diagram.

4. Formally étale, formally unramified, and formally smooth functors.

4.1. Example: (pre)sheaves on the Q-category of infinitesimal epimorphisms and formally étale functors. Let A be the category CAlg_{k} of commutative associative unital k-algebras, Ā a full subcategory of A^{2} whose objects are k-algebra epimorphisms with a nilpotent kernel. We denote by C𝔸lg_{k}^{inf} the corresponding full Qsubcategory of (CAlg_{k}^{2} ⇄ CAlg_{k}). We call C𝔸lg_{k}^{inf} the Q-category of commutative infinitesimal epimorphisms.

A functor CAlg_{k} ⟶^{F} Sets is formally étale (resp. formally unramified, resp. formally smooth) if the canonical morphism F ∘ u_{∗} ⟶ F ∘ u_{!} is an isomorphism (resp. a monomorphism, resp. a strict epimorphism). Comparing with 3.7, we obtain that F is formally étale (resp. formally unramified, resp. formally smooth) iff it is a sheaf (resp. a monopresheaf, resp. an epipresheaf) of sets on the Q-category C𝔸lg_{k}^{inf}.

4.2. A noncommutative version. Let A be the category Alg_{k} of associative unital algebras over a commutative ring k, Ā a full subcategory of A^{2} whose objects are algebra epimorphisms with a nilpotent kernel. We denote by 𝔸lg_{k}^{inf} the corresponding full Q-subcategory of (Alg_{k}^{2} ⇄ Alg_{k}). We call 𝔸lg_{k}^{inf} the Q-category of infinitesimal epimorphisms.

4.3. Proposition. Let a functor Alg_{k} ⟶^{F} Sets be corepresentable by a k-algebra R.

(a) The functor F is an epipresheaf on 𝔸 iff the algebra R is quasi-free in the sense of Quillen and Cuntz [CQ1]. The latter is equivalent to the condition: the R ⊗_{k} R^{op}-module Ω_{R|k}^{1} of Keller differentials of R (which is the kernel of the multiplication R ⊗_{k} R ⟶ R) is projective.

(b) The functor F is a monopresheaf on 𝔸 iff Ω_{R|k}^{1} = 0.

Proof. A standard argument shows that F is an epipresheaf (resp. a monopresheaf) iff for any k-algebra epimorphism S ⟶^{ϕ} R such that Ker(ϕ)^{2} = 0, there exists a splitting (resp. at most one splitting), that is a k-algebra morphism R ⟶^{ψ} S such that ϕ ∘ ψ = id_{R}.

(a) Thus F is an epipresheaf iff Ext_{R^{e}}^{2}(R, M) = 0 for any R^{e}-module M. Here R^{e} denote the k-algebra R ⊗_{k} R^{op}. Consider the long exact sequence

... ⟶ Ext_{R^{e}}^{i}(R, M) ⟶ Ext_{R^{e}}^{i}(R^{e}, M) ⟶ Ext_{R^{e}}^{i}(Ω_{R|k}^{1}, M) ⟶ Ext_{R^{e}}^{i+1}(R, M) ⟶ ...

corresponding to the short exact sequence 0 ⟶ Ω_{R|k}^{1} ⟶ R^{e} ⟶ R ⟶ 0. Since Ext_{R^{e}}^{i}(R^{e}, M) = 0 for all i ≥ 1 and all R^{e}-modules M, Ext_{R^{e}}^{i}(Ω_{R|k}^{1}, M) ≃ Ext_{R^{e}}^{i+1}(R, M) for all i ≥ 1 and all R^{e}-modules M. In particular, Ext_{R^{e}}^{2}(R, M) = 0 for all M iff Ext_{R^{e}}^{1}(Ω_{R|k}^{1}, M) = 0 for all M. The latter means precisely that Ω_{R|k}^{1} is a projective R^{e}-module.

(b) Let R ⟶^{ψ} S be a k-algebra morphism such that ϕ ∘ ψ = id_{R}. It gives a decomposition of S into a semidirect product of R and an R-bimodule, M, with multiplication

defined by (r, m)(r′, m′) = (rr′, r · m′ + m · r′). Any other splitting, R ⟶^{ψ′} S, is (id_{R}, d), where R ⟶^{d} M is a derivation sending k to zero. Thus, the set of splittings of ϕ is in one-to-one correspondence with Der_{R|k}(M). But Der_{R|k}(M) ≃ Hom_{R^{e}}(Ω_{R|k}^{1}, M). Hence ϕ is unramified iff Ω_{R|k}^{1} = 0.

4.4. Formally 𝔸-smooth functors. Examples 4.1 and 4.2 suggest the following interpretation of epipresheaves, monopresheaves and sheaves on a Q-category:

4.4.1. Definition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. We say that a functor A ⟶^{F} Sets is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) if it is an epipresheaf (resp. a monopresheaf, resp. a sheaf) on 𝔸; i.e. the canonical morphism

u^{∗}(F) = F ∘ u_{∗} ⟶ u^{!}(F) = lim_{(V,ξ)∈Ob(ȳ\u^{∗})} F (V)  (1)

is a strict epimorphism (resp. a monomorphism, resp. an isomorphism).

4.4.2. A reformulation. One can, using isomorphisms

u^{!}(F)(ȳ) ≃ A^{∨}(Ā(ȳ, u^{∗}(−)), F) and u^{∗}(F)(ȳ) = F ∘ u_{∗} ≃ A^{∨}(u_{∗}(ȳ), F)

(see 3.6), reformulate the notion of the formal 𝔸-smoothness as follows.

A functor A ⟶^{F} Sets is formally 𝔸-smooth iff the canonical map

A^{∨}(u_{∗}(ȳ), F) ⟶ A^{∨}(Ā(ȳ, u^{∗}(−)), F), g ⟼ g ∘ αȳ,  (2)

is surjective for all ȳ ∈ ObĀ. Here u_{∗}(ȳ) is identified with its image in A^{∨}, i.e. the functor A(u_{∗}(ȳ), −) : A ⟶ Sets, and αȳ denotes the canonical morphism

Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −), h ⟼ η_{u}(−)^{−1} ∘ u_{∗}(h).  (3)

A functor A ⟶^{F} Sets is formally 𝔸-unramified (resp. formally 𝔸-étale) iff the map (2) is injective (resp. bijective) for all ȳ ∈ ObĀ.

4.5. Formally 𝔸-smooth, formally 𝔸-unramified, and formally 𝔸-étale objects. We say that an object x of the category A is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) if the functor A(x, −) : A ⟶ Sets is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-smooth).

4.5.1. 𝔸-smooth, 𝔸-unramified, and 𝔸-étale objects. We say that an object x of A is 𝔸-smooth (resp. 𝔸-unramified) if it is formally 𝔸-smooth (resp. formally 𝔸unramified) and finitely presented. The latter means that the functor A(x, −) preserves colimits of filtered diagrams. We call an object x 𝔸-étale if it is both 𝔸-smooth and 𝔸-unramified.

4.5.2. Example. Let A be the category of associative unital algebras over a commutative ring k, Ā a full subcategory of A^{2} formed by all algebra epimorphisms. Then for any projective k-module V, the tensor algebra, T_{k}(V), of V is formally 𝔸-smooth. It is 𝔸-smooth iff the projective k-module V is of finite type.

Note that there is only one (up to isomorphism) 𝔸-étale algebra: the ring k.

4.6. Quasi-free and separable algebras. Let A be the category 𝔄ss_{k} 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, R \overrightarrow{⟶}^{f}_{g} S, are equivalent if they are conjugated, i.e. g(−) =

tf(−)t^{−1} for an invertible element t of S. Let Ā be the full subcategory of A^{2} formed by the equivalence classes of algebra epimorphisms with a nilpotent kernel. We denote by 𝔄ss_{k}^{inf} the corresponding full Q-subcategory of (A^{2} ⇄ A).

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) which induces a strict fully faithful morphism C𝔸lg_{k}^{inf} ↪ 𝔄ss_{k}^{inf} of Q-categories (cf. 4.1).

Recall that a k-algebra R is called separable if R is a projective left R^{e}-module, R^{e} = R ⊗_{k} R^{o}. It follows from the exact sequence of R^{e}-modules

0 ⟶ Ω_{R|k}^{1} ⟶ R^{e} ⟶ R ⟶ 0

that if R is separable, then Ω_{R|k}^{1} is a projective R^{e}-module, i.e. R is quasi-free [CQ1].

4.6.1. Proposition. Let R be an associative k-algebra.

(a) The following conditions are equivalent:

(i) The algebra R is formally 𝔄ss_{k}^{inf}-smooth.

(ii) The left R^{e}-module of Keller differentials, Ω_{R/k}^{1} = Ker(R^{e} → R), is projective.

(b) The following conditions are equivalent:

(iii) R is formally 𝔄ss_{k}^{inf}-étale.

(iv) R is formally 𝔄ss_{k}^{inf}-unramified.

(v) R is separable.

Proof. (a) Let S ⟶^{ϕ} R be a k-algebra morphism such that there exists a k-algebra morphism R ⟶^{ψ} S right inverse to ϕ in the category 𝔄ss_{k}. The latter means, in particular, that ϕ ∘ ψ is conjugate to id_{R}; i.e. there exists an invertible element t of R such that for any r ∈ R, ϕ ∘ ψ(r) = trt^{−1}. The composition, ψ_{t}, of ψ with the inner automorphism r ⟼ t^{−1}rt is a right inverse to ϕ in the category Alg_{k}. This shows that R is formally 𝔄ss_{k}^{inf}-smooth iff it is formally 𝔸lg_{k}^{inf}-smooth (cf. 4.2). The assertion follows from 4.3 (or [CQ1]).

(b) The implication (iii)⇒(iv) is true by definition.

(iv)⇒(v). Let M be an R^{e}-module, S a semiproduct of R and M, S ⟶^{ϕ} R the canonical epimorphism. It follows from (a) that any right inverse to ϕ in the sense of 𝔄ss_{k} is conjugate to a right inverse, R ⟶^{ψ} S to ϕ in the sense of Alg_{k}. The morphism ψ is of the form r ⟼ r + D(r) for some (any) derivation R ⟶^{D} M which sends k to zero. If R is 𝔄ss_{k}^{inf}-unramified, the morphism ψ is equivalent to the morphism R ⟶ S, r ⟼ r. This means that there exists an invertible element u of S such that ψ(r) = uru^{−1} for all r ∈ R. The element u can be written as t(1_{R} + z), where 1_{R} is the unit of R, t is an invertible element of R, and z ∈ M. Then

uru^{−1} = trt^{−1} + (tzt^{−1})(trt^{−1}) − (trt^{−1})(tzt^{−1})  (1)

In particular, ϕ ∘ ψ(r) = trt^{−1} for all r ∈ R. But ϕ ∘ ψ = id_{R}, hence the element t is central. Thus ψ(r) = r + z_{t}r − rz_{t}, where z_{t} = tzt^{−1}, i.e. D is an inner derivation. It is known [CQ1] (and easy to prove) that R is a separable k-algebra iff any derivation of R in any R^{e}-module M is inner, hence the implication.

(v)⇒(iii). Let R be a separable k-algebra. Let T ⟶^{ϕ} S be a k-algebra morphism with a nilpotent kernel and R ⟶^{f} S an arbitrary algebra morphism. It follows from the argument in [CQ1] that any two liftings of f to a morphism R ⟶ T are conjugate by an element t of T such that 1 − t belongs to the kernel of ϕ, in particular it is nilpotent. Conversely, such a lifting property implies that R is separable.

5. Formally 𝔸-smooth, formally 𝔸-unramified, and formally 𝔸-étale mor-

phisms. Fix a Q-category 𝔸 = (Ā ⇄^{u} A). Let X, Y be functors A ⟶ Sets.

5.1. Definition. We call a morphism X ⟶^{f} Y formally 𝔸-smooth if for any ȳ ∈ ObĀ

and for any pair of morphisms A(u_{∗}(ȳ), −) ⟶^{g} Y, Ā(ȳ, u^{∗}(−)) ⟶^{g′} X such that the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) & ⟶^{g′} & X \\ αȳ ↓ & & ↓ f \\ A(u_{∗}(ȳ), −) & ⟶^{g} & Y \end{matrix}  (1)

commutes, there exists a morphism A(u_{∗}(ȳ), −) ⟶^{γ} X such that γ ∘αȳ = g′ and f ∘γ = g. In other words, the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) ⟶^{g′} X \\ αȳ ↓ ↗_{γ} ↓ f \\ A(u_{∗}(ȳ), −) ⟶^{g} Y \end{matrix}  (2)

commutes.

5.2. Definition. We call a morphism X ⟶^{f} Y formally 𝔸-unramified if for any

ȳ ∈ ObĀ and any pair of morphisms, A(u_{∗}(ȳ), −) ⟶^{g} Y, Ā(ȳ, u^{∗}(−)) ⟶^{g′} X, making the diagram 5.1(1) commute, there exists at most one morphism, A(u_{∗}(ȳ), −) ⟶^{γ} X, such that the diagram 5.1(2) commute.

5.2.1. Note. Any monomorphism X ⟶^{f} Y is formally 𝔸-unramified.

5.3. Definition. We call a morphism X ⟶^{f} Y formally 𝔸-étale if it is both formally 𝔸-smooth and formally 𝔸-unramified.

5.3.1. A reformulation. Consider the diagram 5.1(1). We have canonical, functorial in ȳ and X isomorphisms

A^{∨}(Ā(ȳ, u^{∗}(−)), X) = A^{∨}(û_{∗}(Ā(ȳ, −)), X) ≃ Ā^{∨}(Ā(ȳ, −), u^{!}(X) ≃ u^{!}(X)(ȳ)

and

A^{∨}(u_{∗}(ȳ), −), Y) ≃ Y (u_{∗}(ȳ)) = u^{∗}(Y)(ȳ).

The commutative diagram

\begin{matrix}u^{∗}(X) & ⟶^{u^{∗}(f)} & u^{∗}(Y) \\ α_{u}(X) ↓ & & ↓ α_{u}(Y) \\ u^{!}(X) & ⟶^{u^{!}(f)} & u^{!}(Y) \end{matrix}  (1)

induces a morphism

u^{∗}(X) ⟶ u^{∗}(Y) ×_{u^{!}(Y)} u^{!}(X)  (2)

5.3.1.1. Proposition. (a) A morphism X ⟶^{f} Y is formally 𝔸-unramified (resp. formally 𝔸-étale) iff (2) is a monomorphism (resp. an isomorphism).

(b) A morphism X ⟶^{f} Y is formally 𝔸-smooth iff for any ȳ ∈ ObĀ, the map

u^{∗}(X)(ȳ) ⟶ (u^{∗}(Y) ×_{u^{!}(Y)} u^{!}(X))(ȳ)

is surjective. In particular, (2) is an epimorphism.

Proof is left to the reader.

5.3.1.2. Corollary. Suppose Y ∈ ObA^{∨} is formally 𝔸-étale. Then a morphism X ⟶^{f} Y is formally 𝔸-étale iff X is formally 𝔸-étale.

Proof. The morphism f is formally 𝔸-étale iff the square (1) is cartesian. In particular, if the right vertical arrow of (1) is an isomorphism (which means exactly that Y is formally 𝔸-étale), then the left vertical arrow of (1) is an isomorphism too, i.e. X is formally 𝔸-étale.

Conversely, if both vertical arrows of the square (1) are isomorphisms (i.e. both X and Y are formally 𝔸-étale), then the square (1) is cartesian.

5.3.2. Formally étale morphisms and 𝔸-sheaves. Let ∗ denote the functor which assigns to all objects of A a one element set – a final object of the category A^{∨}. For any X ∈ ObA^{∨}, denote by π_{X} the unique map X ⟶ ∗. It follows that X is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) iff the map X ⟶^{π_{X}} ∗ is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale).

Thus, formally 𝔸-étale morphisms might be viewed as relative versions of sheaves of sets on 𝔸 .

5.4. Proposition. (a) The composition of formally 𝔸-smooth (resp. formally 𝔸unramified, resp. formally 𝔸-étale) morphisms is formally 𝔸-smooth (resp. formally 𝔸unramified, resp. formally 𝔸-étale).

(b) Let X, Y, Z be functors A ⟶ Sets, and let X ⟶^{f} Y and Y ⟶^{h} Z be functor morphisms.

(i) Suppose h ∘ f is formally 𝔸-unramified. Then f is formally 𝔸-unramified.

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

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

Proof. (a) Suppose X ⟶^{h∘f} Z is formally 𝔸-smooth. Let

\begin{matrix}Ā(ȳ, u^{∗}(−)) & ⟶^{g′} & X \\ αȳ ↓ & & ↓ h ∘ f \\ A(u_{∗}(ȳ), −) & ⟶^{g} & Z \end{matrix}  (1)

be a commutative diagram. Since the morphism h is formally 𝔸-smooth, there exists a

morphism A(u_{∗}(ȳ), −) ⟶^{γ′} Y such that the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) ⟶^{f∘g′} Y \\ αȳ ↓ ↗_{γ′} ↓ f \\ A(u_{∗}(ȳ), −) ⟶^{g} Z \end{matrix}  (2)

commutes. In particular, γ′ ∘ αȳ = f ∘ g′, i.e. the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) & ⟶^{g′} & X \\ αȳ ↓ & & ↓ f \\ A(u_{∗}(ȳ), −) & ⟶^{γ′} & Y \end{matrix}  (3)

commutes. Since f is formally 𝔸-smooth, there exists a morphism γ : A(u_{∗}(ȳ), −) ⟶ X such that the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) ⟶^{g′} X \\ αȳ ↓ ↗_{γ} ↓ f \\ A(u_{∗}(ȳ), −) ⟶^{γ′} Y \end{matrix}  (4)

commutes. Combining (3) and (4), we obtain the commutative diagram

Ā(ȳ, u^{∗}(−)) ⟶^{g′} X

αȳ ↓ ↗_{γ} ↓ h ∘ f

A(u_{∗}(ȳ), −) ⟶^{g} Z

Hence h ∘ f is formally 𝔸-smooth.

We leave to the reader the checking that if f and h are formally 𝔸-unramified (resp. formally 𝔸-étale), then h ∘ f is formally 𝔸-unramified (resp. formally 𝔸-étale).

(b) (i) Suppose h ∘ f is formally 𝔸-unramified, and let

Ā(ȳ, u^{∗}(−)) ⟶^{g′} X

αȳ ↓ ↗_{γ} ↓ f

A(u_{∗}(ȳ), −) ⟶^{g} Y

be a commutative diagram. Then the diagram

Ā(ȳ, u^{∗}(−)) ⟶^{g′} X

αȳ ↓ ↗_{γ} ↓ h ∘ f

A(u_{∗}(ȳ), −) ⟶^{h∘g} Z

commutes. Since h ∘ f is formally 𝔸-unramified, the morphism γ is uniquely defined. (ii) Suppose h is formally 𝔸-unramified and h ∘ f is formally 𝔸-smooth. Let

Ā(ȳ, u^{∗}(−)) ⟶^{g′} X
αȳ ↓  ↓ f
A(u_{∗}(ȳ), −) ⟶^{g} Z

be a commutative diagram. Since h ∘ f is formally 𝔸-smooth, there exists a commutative diagram

Ā(ȳ, u^{∗}(−)) ⟶^{g′} X

αȳ ↓ ↗_{γ} ↓ h ∘ f

A(u_{∗}(ȳ), −) ⟶^{h∘g} Z

Since h is formally 𝔸-unramified, g = f ∘ γ.

(c) The proof of the assertion (c) is left to the reader.

5.5. Corollary. Let X, Y, Z be functors A ⟶ Sets, and let X ⟶^{f} Y, Y ⟶^{h} Z be functor morphisms. Suppose h is formally 𝔸-étale. Then h ∘ f is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) iff f belongs to the same class.

5.6. Proposition. Let 𝔸 be a Q-category. Let X, Y, Y′ be functors A ⟶ Sets and Y′ ⟶^{h} Y ⟵^{f} X functor morphisms. If X ⟶^{f} Y is formally 𝔸-smooth (resp. formally

𝔸-nonramified, resp. formally 𝔸-étale), then the canonical projection X ×_{Y} Y′ ⟶^{f′} Y′ is formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale).

Proof. Set X′ = X ×_{Y} Y′ and denote by h′ the canonical projection X′ ⟶ X. Consider a commutative diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) & ⟶^{g′} & X′ & ⟶^{h′} & X \\ αȳ ↓ & & ↓ f′ & & ↓ f \\ A(u_{∗}(ȳ), −) & ⟶^{g} & Y′ & ⟶^{h} & Y \end{matrix}  (1)

(a) Suppose f is formally 𝔸-smooth. Then there exists a morphism A(u_{∗}(ȳ), −) ⟶^{γ} X such that the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) ⟶^{h′∘g′} X \\ αȳ ↓ ↗_{γ} ↓ f \\ A(u_{∗}(ȳ), −) ⟶^{h∘g} Y \end{matrix}  (2)

commutes. By the universal property of the fiber product, there exists a unique morphism

Ā(ȳ, u^{∗}(−)) ⟶^{γ′} X′ such that the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) ⟶^{g′} X′ \\ αȳ ↓ ↗_{γ′} ↓ f′ \\ A(u_{∗}(ȳ), −) ⟶^{g} Y′ \end{matrix}  (3)

commutes. This shows that the projection f′ is formally smooth.

(b) Suppose the morphism X ⟶^{f} Y is formally 𝔸-unramified. And suppose that

there exists a morphism Ā(u_{∗}(ȳ), −) ⟶^{γ′} X′ such that the diagram (3) commutes. This implies that the diagram (2), with γ = h′ ∘ γ′, is commutative. Since f is formally 𝔸unramified, the commutativity of (2) determines γ uniquely. By the universal property of fiber product, the morphisms γ′ is uniquely determined by the morphisms γ and h, hence f′ is formally 𝔸-unramified.

(c) It follows from (a) and (b) that if f is formally 𝔸-étale, then f′ is formally 𝔸-étale too.

5.7. Definition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let R, S be objects of A. We call a morphism R ⟶^{ϕ} S formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) iff the corresponding functor morphism, A(ϕ, −) : A(S, −) ⟶ A(R, −), has the respective property.

5.7.1. A special case. If the functor u^{∗} has a left adjoint, u_{!}, one can define formally 𝔸-smooth, formally 𝔸-unramified and formally 𝔸-étale morphisms of A in terms of the category A itself:

A morphism R ⟶^{f} S of A is formally 𝔸-smooth iff any ȳ ∈ ObĀ and a commutative diagram of the form

\begin{matrix}u_{!}(ȳ) & ⟵^{g′} & S \\ α′ ; ȳ ↑ & & ↑ f \\ u_{∗}(ȳ) & ⟵^{g} & R \end{matrix}  (1)

(where α′ ; ȳ is the canonical functorial morphism) extends to a commutative diagram

\begin{matrix}u_{!}(ȳ)) ⟵^{g′} S \\ α′ ; ȳ ↑ ↙_{γ} ↑ f \\ u_{∗}(ȳ) ⟵^{g} R \end{matrix}  (2)

A morphism R ⟶^{f} S is formally 𝔸-unramified iff for any commutative diagram of the form (1) there exists at most one morphism S ⟶^{γ} u_{∗}(ȳ) such that the diagram (2) commutes.

5.8. Formally smooth and formally étale morphisms of algebras. If 𝔸 is the Q-category Alg_{k}^{inf} of 4.2, we call formally 𝔸-smooth (resp. formally 𝔸-unramified, resp. formally 𝔸-étale) morphisms simply formally smooth (resp. formally unramified, resp. formally étale). We have the following relative analogue of Proposition 4.6.2:

5.8.1. Proposition. Let R, S be associative k-algebras, and let R ⟶^{ϕ} S be a k-algebra morphism.

(a) The morphism ϕ is formally unramified iff the morphism S⊗_{R}S ⟶ S, s⊗t ⟼ st, is an isomorphism, or, equivalently, Ω_{S|R}^{1} = Ker(S ⊗_{R} S ⟶ S) = 0.

(b) Suppose the k-algebra R is separable. Then the morphism R ⟶^{ϕ} S is formally smooth iff Ω_{S|R}^{1} is a projective left S^{e}-module.

Proof. A standard argument shows that R ⟶^{ϕ} S is formally smooth (resp. formally unramified) iff for any R-ring epimorphism T ⟶^{α} S such that Ker(α)^{2} = 0, there exists an R-ring morphism (resp. at most one R-ring morphism) S ⟶^{β} T such that α ∘ β = id_{S}.

(a) Let S ⟶^{β} T be an R-ring morphism such that α∘β = id_{S}. It gives a decomposition of T into a semidirect product of S and an S-bimodule, M, with multiplication defined by (s, m)(s′, m′) = (ss′, s · m′ + m · s′). Any other right inverse to α, is of the form (id_{S}, D), where S ⟶^{D} M is a derivation sending R to zero. The latter means precisely that D is an R^{e}-module morphism, R^{e} = R ⊗_{k} R^{o}. Thus, the set of splittings of α is in one-to-one correspondence with the set Der_{S|R}(M) of derivations S ⟶^{D} M which are R^{e}-module morphisms. But Der_{S|R}(M) is naturally isomorphic to Hom_{S^{e}}(Ω_{S|R}^{1}, M). Hence ϕ is unramified iff Ω_{S|R}^{1} = 0.

(b) Suppose the k-algebra R is separable, i.e. R is a projective R^{e}-module. Then the S^{e}-bimodule S ⊗_{R} S is projective.

In fact, for any S^{e}-module M, there is a functorial isomorphism Hom_{S^{e}}(S ⊗_{R} S, M) ≃ Hom_{R^{e}}(R, ϕ_{∗}(M)). Here ϕ_{∗} is the pull-back functor S^{e} − mod ⟶ R^{e} − mod induced by the morphism ϕ. Since R is a projective R^{e}-module and the functor ϕ_{∗} is exact, the functor M ⟼ Hom_{R^{e}}(R, ϕ_{∗}(M)) is exact. Therefore the functor M ⟼ Hom_{S^{e}}(S ⊗_{R} S, M) is exact, i.e. S ⊗_{R} S is a projective S^{e}-module.

By 4.6.1, the algebra R is separable iff it is 𝔄ss^{inf}-étale. The latter means that the morphism k ⟶ R is formally 𝔄ss^{inf}-étale. If follows from 5.4(ii), that the morphism R ⟶^{ϕ} S is formally 𝔄ss^{inf}-smooth iff the composition of k ⟶ R and ϕ is formally 𝔄ss^{inf}-smooth, i.e. the k-algebra S is formally 𝔄ss^{inf}-smooth. By 4.3 and 4.6.1, the k-algebra S is formally 𝔄ss^{inf}-smooth iff it is formally smooth (i.e. Alg_{k}^{inf}-smooth). On the other hand, the algebra S is formally smooth iff Ext_{S^{e}}^{2}(S, M) = 0 for any S^{e}-module M. Consider the long exact sequence

... ⟶ Ext_{S^{e}}^{i}(S, M) ⟶ Ext_{S^{e}}^{i}(S ⊗_{R} S, M) ⟶ Ext_{S^{e}}^{i}(Ω_{S|R}^{1}, M) ⟶ Ext_{S^{e}}^{i+1}(S, M) ⟶ ...

(1)

corresponding to the short exact sequence 0 ⟶ Ω_{S|R}^{1} ⟶ S ⊗_{R} S ⟶ S ⟶ 0. Since S ⊗_{R} S is a projective S^{e}-module, Ext_{S^{e}}^{i}(S ⊗_{R} S, M) = 0 for all i ≥ 1 and all S^{e}-modules M. Therefore Ext_{S^{e}}^{i}(Ω_{S|R}^{1}, M) ≃ Ext_{S^{e}}^{i+1}(S, M) for all i ≥ 1 and all S^{e}-modules M. In particular, Ext_{S^{e}}^{2}(S, M) = 0 for all M iff Ext_{S^{e}}^{1}(Ω_{S|R}^{1}, M) = 0 for all M. The latter means precisely that Ω_{S|R}^{1} is a projective S^{e}-module.

5.8.1.1. Corollary. Suppose R is a separable k-algebra. Then a k-algebra morphism R ⟶^{ϕ} S is formally unramified iff it is formally étale.

Proof. By 5.8.1(a), R ⟶^{ϕ} S is unramified iff Ω_{S|R}^{1} = 0. By 5.8.1(b), ϕ is formally smooth iff Ω_{S|R}^{1} is a projective S^{e}-module. In particular, ϕ is formally smooth (hence étale), if Ω_{S|R}^{1} = 0.

5.8.2. Proposition. Let R, S be associative k-algebras, and let R ⟶^{ϕ} S be a k-algebra morphism. The following conditions are equivalent:

(i) ϕ is formally unramified and flat.

(ii) ϕ is a flat monomorphism. (iii) ϕ^{∗} is an exact localization.

If the conditions above hold, then (iv) ϕ is formally étale.

Proof. (ii)⇒(i), because every monomorphism is formally unramified.

(i)⇒(iii). By 5.8.1(a), the canonical morphism S ⊗_{R} S ⟶ S, s ⊗ t ⟼ st, is an isomorphism. Since ϕ^{∗}ϕ_{∗} ≃ (S ⊗_{R} S) ⊗_{S} − and Id_{S−mod} ≃ S ⊗_{S} −, this means precisely that the adjunction morphism ϕ^{∗}ϕ_{∗} ⟶ Id_{S−mod} is an isomorphism. The latter is equivalent to the full faithfulness of the direct image functor ϕ_{∗}. By [GZ], Proposition I.1.3, ϕ^{∗} is a localization.

(iii)⇒(ii) follows from the fact that any morphisms R ⟶^{ϕ} S such that its inverse image functor, ϕ^{∗}, is a localization, is an algebra epimorphism.

In fact, let S \overrightarrow{⟶}^{f_{1}}_{f_{2}} T be a pair of algebra morphisms such that f_{1} ∘ ψ = f_{2} ∘ ϕ, i.e. we

have the diagram of algebra morphisms over R:

\begin{matrix}S \overrightarrow{⟶}^{f_{1}}_{f_{2}} T \\ _{ϕ} ↖ ↗_{γ} \\ R \end{matrix}  (1)

Here γ = f_{1} ∘ ϕ. Applying to (1) first scalar restriction functor and then the functor ϕ_{∗}ϕ^{∗}, we obtain the diagram ϕ_{∗}ϕ^{∗}(R) ⟶ ϕ_{∗}ϕ^{∗}ϕ_{∗}(S) \overrightarrow{⟶} ϕ_{∗}ϕ^{∗}γ_{∗}(T) which is isomorphic to the diagram

ϕ_{∗}ϕ^{∗}(R) ⟶ ϕ_{∗}(S) \overrightarrow{⟶}γ_{∗}(T),  (2)

because, due to the fact that ϕ^{∗} is a localization, ϕ_{∗} is a fully faithful functor, or, equivalently, ϕ^{∗}ϕ_{∗} ≃ Id_{S−mod}. Notice that the morphism ϕ_{∗}ϕ^{∗}(R) ⟶ ϕ_{∗}(S) in (2) is an isomorphism. Since it equalizes the pair ϕ_{∗}(S) \overrightarrow{⟶}γ_{∗}(T), this pair is trivial. Hence the initial pair of morphisms is trivial: f_{1} = f_{2}.

{(iii),(i)} ⇒(iv). It suffices to show that if R ⟶^{ϕ} S is an exact localization, then ϕ is formally smooth. A standard argument shows that a morphism R ⟶^{ϕ} S is smooth iff any R-ring strict epimorphism (i.e. a surjection) T ⟶^{g} S such that the square of the kernel of g is zero, has right inverse. Denote the kernel of g by J. Thus we have an exact sequence of R-bimodules

0 ⟶ J ⟶ T ⟶ S ⟶ 0.  (3)

Denote by Φ^{∗} the functor

R^{e} − mod ⟶ S^{e} − mod, M ⟼ S ⊗_{R} M ⊗_{R} S.

Notice that this functor is an exact localization having a (necessarily fully faithful) right adjoint, Φ_{∗}. In particular, it maps the exact sequence (3) into exact sequence. Applying the functor Φ^{∗} to the diagram

\begin{aligned}0 ⟶ J ⟶ & T ⟶ S & ⟶ 0 \\ & ↖ ↗ \\ & R \end{aligned}  (4)

we obtain the diagram

\begin{aligned}0 ⟶ Φ^{∗}(J) ⟶ & Φ^{∗}(T) ⟶ Φ^{∗}(S) & ⟶ 0 \\ & ↖  ↗ \\ & Φ^{∗}(R) \end{aligned}  (5)

Since Φ^{∗} is a localization, the natural morphism S ⟶ Φ_{∗}Φ^{∗}(S) is an isomorphism, Φ^{∗}(R) = S ⊗_{R} S ≃ S, and the k-algebra morphism Φ^{∗}(ϕ) : Φ^{∗}(R) ⟶ Φ^{∗}(S) is an isomorphism.

Note that J is an S-bimodule. This implies that Φ_{∗}Φ^{∗}(J) ≃ J. Thus we have a commutative diagram

\begin{aligned}0 & ⟶ & J & ⟶ & T & ⟶ & S & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & Φ_{∗}Φ^{∗}(J) & ⟶ & Φ_{∗}Φ^{∗}(T) & ⟶ & Φ_{∗}Φ^{∗}(S) & ⟶ & 0 \\ & & & & & ↖  ↗ \\ & & & & & S \end{aligned}  (6)

whose left and right vertical arrows are isomorphisms. Since both rows are exact sequences, it follows that the adjunction morphism T ⟶ Φ_{∗}Φ^{∗}(T) is an isomorphism too, hence the assertion.

5.9. Formally 𝔄ss_{k}^{inf}-unramified and formally 𝔄ss_{k}^{inf}-étale morphisms. The following assertion is a relative version of 4.6.1.

5.9.1. Proposition. Let R, S be associative k-algebras, and let R ⟶^{ϕ} S be a k-algebra morphism.

1) The following conditions are equivalent:

(i) The morphism ϕ is formally 𝔄ss_{k}^{inf}-unramified.

(ii) Any derivation S ⟶^{D} M which is an R^{e}-module morphism is inner.

(iii) The canonical S^{e}-module epimorphism S ⊗_{R} S ⟶ S has a right inverse.

2) Suppose that the k-algebra R is separable. Then

(a) The morphism R ⟶^{ϕ} S is formally 𝔄ss_{k}^{inf}-smooth iff Ω_{S|R}^{1} is a projective S-bimodule.

(b) The following conditions are equivalent:

(iv) The morphism ϕ is formally 𝔄ss_{k}^{inf}-unramified.

(v) ϕ is formally 𝔄ss_{k}^{inf}-étale.

(vi) S is a separable k-algebra (i.e. S is a projective S^{e}-module).

Proof. 1) (i)⇔(ii). Let T be a semidirect product of S and an S^{e}-bimodule M, and let T ⟶^{α} S the natural projection, (s, z) ⟼ s. Any k-algebra morphism S ⟶ T which is right inverse to α in category 𝔄ss_{k} is conjugate to a k-algebra morphism, S ⟶^{β} T, which is right inverse to α in Alg_{k}. Any such morphism β is of the form s ⟼ (s, D(s), where S ⟶^{D} M is an S|R-derivation. If R ⟶^{ϕ} S is 𝔄ss_{k}^{inf}-unramified, β is of the form s ⟼ usu^{−1}. The argument of 4.6.1 shows that this (together with the equality α∘β = id_{S}) implies that D is an inner derivation.

Conversely, if the morphism S ⟶^{β} T is given by s ⟼ (s, D(s)), where D is an inner derivation, i.e. D(s) = s·z−z·s for some element z of M and all s ∈ S, then β(s) = usu^{−1}, where u = 1_{S} − z).

(ii)⇒(iii). The functor Der_{S|R} : S^{e} − mod ⟶ Sets, M ⟼ Der_{S|R}(M), is representable by the S^{e}-module Ω_{S|R}^{1} = Ker(S ⊗_{R} S ⟶ S). The canonical monomorphism

Ω_{S|R}^{1} ⟶^{i_{ϕ}} S ⊗_{R} S induces a map

Hom_{S^{e}}(S ⊗_{R} S, M) ⟶ Hom_{S^{e}}(Ω_{S|R}^{1}, M)  (1)

Notice that Hom_{S^{e}}(S ⊗_{R} S, M) ≃ Hom_{R^{e}}(R, ϕ_{∗}(M)), and Hom_{R^{e}}(R, ϕ_{∗}(M)) is naturally isomorphic to the center, 𝔷(ϕ_{∗}(M)) = {v ∈ M| r · v = v · r for all r ∈ R}, of the R^{e}-module ϕ_{∗}(M). The composition of the bijection 𝔷(ϕ_{∗}(M)) ⟶ Hom_{S^{e}}(S ⊗_{R} S, M) and the map (1) assigns to each central element, z, of ϕ_{∗}(M) the corresponding inner derivation, s ⟼ s · z − z · s. Thus, each derivation of Der_{S|R}(M) is inner iff the map (1) is surjective. In the case M = Ω_{S|R}^{1}, this implies the existence of an S^{e}-module morphism S⊗_{R}S ⟶^{p} Ω_{S|R}^{1} such that p∘i_{ϕ} = id. Or, equivalently, the canonical S^{e}-module morphism S ⊗_{R} S ⟶ S has a right inverse.

The implication (iii)⇒(ii) follows from the argument above.

2) (a) A k-algebra morphism R ⟶^{ϕ} S is formally 𝔄ss_{k}^{inf}-smooth iff it is formally Alg_{k}^{inf}-smooth. By 5.8.1, if R is a separable k-algebra, then ϕ is formally Alg_{k}^{inf}-smooth iff Ω_{S|R}^{1} is a projective S^{e}-module.

(b) By the argument of 5.8.1, if R is a separable k-algebra, then the S^{e}-module S ⊗_{R} S is projective. By 1), the morphism R ⟶^{ϕ} S is 𝔄ss_{k}^{inf}-unramified iff the S^{e}- module morphism S ⊗_{R} S ⟶ S has a right inverse. Since S ⊗_{R} S is projective, the latter implies that S is a projective S^{e}-module, hence (equivalently) Ω_{S|R}^{1} is a projective

S^{e}-module, i.e. the morphism ϕ is formally 𝔄ss_{k}^{inf}-smooth. This proves the implications (iv)⇒(v)⇒(vi)⇒(v). The implication (v)⇒(iv) is true by definition.

5.9.2. Corollary. The following conditions on a k-algebra morphism R ⟶^{ϕ} S are equivalent:

(a) ϕ is formally 𝔄ss_{k}^{inf}-étale.

(b) The adjunction morphism ϕ^{∗}ϕ_{∗} ⟶^{ϵ_{ϕ}} Id_{S−mod} has a right inverse.

Proof. (a)⇒(b). By 5.9.1, the canonical S^{e}-module epimorphism S ⊗_{R} S ⟶^{μ} S has

a right inverse, S ⟶^{tau′} S ⊗_{R} S. The morphism τ′ defines a morphism, Id_{S−mod} ⟶^{τ} ϕ^{∗}ϕ_{∗}. The equality μ ∘ τ = id_{S} implies that the composition of τ with the adjunction morphism, ϕ^{∗}ϕ_{∗} ⟶^{ϵ_{ϕ}} Id_{S−mod} is the identity morphism.

(b)⇒(a). Conversely, any morphism, Id_{S−mod} ⟶^{τ′} ϕ^{∗}ϕ_{∗}, is induced by an S^{e}-module morphism, S ⟶^{τ} S ⊗_{R} S. The morphism τ′ is a right inverse to the adjunction morphism ϕ^{∗}ϕ_{∗} ⟶^{ϵ_{ϕ}} Id_{S−mod} iff the composition of the bimodule morphism τ with the canonical morphism S ⊗_{R} S ⟶ S equals to id_{S}.

5.10. Another description of the category 𝔄ss_{k}.

5.10.1. Proposition. Two k-algebra morphisms, R \overrightarrow{⟶}^{ϕ}_{ψ} S, are conjugate iff the

corresponding 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 ⊗_{k} M & ⟶^{·t} & S ⊗_{k} 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 S ⊗_{R,ψ} M ⟶^{λ_{t}} 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 k-algebra 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 unital ring, u(R) equals to s ⟼^{·t} 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 sψ(r)t = stϕ(r) for any s ∈ S, hence ψ(r) = tϕ(r)t^{−1}.

5.10.2. Corollary. The category 𝔄ss_{k} is isomorphic to the category whose objects are associative k-algebras; morphisms are equivalence classes of k-algebra morphisms with respect to the following relation: two k-algebra morphisms ϕ, ψ : R ⟶ S are equivalent iff the inverse image functors ϕ^{∗}, ψ^{∗} : R − mod ⟶ S − mod are isomorphic.

5.11. Formally 𝔸-open immersions. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, X, Y functors A ⟶ Sets. We call a morphism X ⟶^{f} Y a formally 𝔸-open immersion if it is a formally 𝔸-smooth monomorphism, or, equivalently, a formally 𝔸-étale monomorphism.

5.11.1. Proposition. (a) The composition of formally 𝔸-open immersions is a formally 𝔸-open immersion.

(b) Let X ⟶ Y be a formally 𝔸-open immersion. For any morphism T ⟶ Y, the canonical projection T ×_{Y} X ⟶ T is a formally 𝔸-open immersion.

(c) Let S ∈ ObA^{∨}, and let (X, ξ) ⟶^{f} (Y, μ), (X′, ξ′) ⟶^{f′} (Y′, μ′) be morphisms of objects over S. The morphisms f, f′ are formally 𝔸-open immersions iff the morphism f ×_{S} f′ : X ×_{S} X′ ⟶ Y ×_{S} Y′ is a formally 𝔸-open immersion.

Proof. The assertions (a), (b), (c) follow from the fact that both monomorphisms and formally 𝔸-étale morphisms are stable under composition, base change, and fiber product (see 5.4, 5.6).

5.11.2. Proposition. Let X \overrightarrow{⟶}^{f_{1}}_{f_{2}} Y ⟶^{h} Z be a diagram of presheaves of sets on

A^{op} (i.e. functors A ⟶ Sets) such that h∘f_{1} = h∘f_{2}. If the morphism h is 𝔸-unramified, then the morphism Ker(f_{1}, f_{2}) ⟶^{i} X is a formally 𝔸-open immersion.

Proof. Consider the diagram

\begin{matrix}Ā(ȳ, u^{∗}(−)) & ⟶^{g′} & Ker(f_{1}, f_{2}) \\ αȳ ↓ & & ↓ i \\ A(u_{∗}(ȳ), −) & ⟶^{g} & X \overrightarrow{⟶}^{f_{1}}_{f_{2}} Y ⟶^{h} Z \end{matrix}  (1)

with commutative square. The diagram

Ā(ȳ, u^{∗}(−)) ⟶^{f_{1}∘i∘g′} Y
αȳ ↓  ↓ h
A(u_{∗}(ȳ), −) ⟶^{h∘f_{1}∘g} Z

commutes; and since h is unramified, there is at most one morphism A(u_{∗}(ȳ), −) ⟶^{γ} Y such that h∘γ = h∘f_{1} ∘g. Since h∘f_{1} = h∘f_{2}, this uniqueness implies that f_{1} ∘g = f_{2} ∘g. Therefore, there exists a unique morphism A(u_{∗}(ȳ), −) ⟶^{λ} X such that g = i ∘ λ. This shows that the monomorphism i is formally 𝔸-smooth, hence the assertion.

5.11.3. Corollary. Any section of a formally 𝔸-unramified morphism is a formally 𝔸-open immersion.

Proof. Let Y ⟶^{s} X be a section of a formally 𝔸-unramified morphism X ⟶^{f} Y. Then the morphism s induces an isomorphism Y ⟶ Ker(sf, id_{X}). The assertion follows now from 5.11.2.

5.11.4. Formally open immersions of affine schemes.

5.11.4.1. Proposition. Let 𝔸 be the Q-category Alg_{k}^{inf} of 4.2. Then the following

conditions on a flat k-algebra morphism R ⟶^{ϕ} S are equivalent:

(i) ϕ is formally 𝔸-unramified. (ii) ϕ is formally 𝔸-étale. (iii) ϕ^{∗} is a localization. (iv) ϕ is a formally open 𝔸-immersion.

Proof. The equivalence of the first three conditions is the content of 5.8.2. By the definition of a formally 𝔸-open immersion, (iv) ⇒ (ii).

{(iii), (ii)} ⇒ (iv). By definition, R ⟶^{ϕ} S is a formally open 𝔸-immersion iff the morphism of functors Alg_{k}(ϕ, −) : Alg_{k}(S, −) ⟶ Alg_{k}(R, −) is a formally étale monomorphism, or, equivalently, ϕ is a formally étale algebra epimorphism. It is étale by (ii). And ϕ^{∗} ≃ S ⊗_{R} − being a localization, implies that ϕ is an epimorphism (see the argument of 5.8.2, (iii) ⇒ (i)).

5.11.4.2. Proposition. A flat k-algebra morphism R ⟶^{ϕ} S is a formally 𝔄ss_{k}^{inf}- open immersion iff the functor ϕ^{∗} is a localization.

Proof. Suppose ϕ^{∗} is a localization. Then the adjunction morphism ϕ^{∗}ϕ_{∗} ⟶ Id_{S−mod}

is an isomorphism. Therefore, by 5.9.2, ϕ is formally 𝔄ss_{k}^{inf}-étale. Let S \overrightarrow{⟶}^{f_{1}}_{f_{2}} T be a pair

of algebra morphisms such that f_{1} ∘ ϕ is equivalent to f_{2} ∘ ϕ (i.e. both define the same morphism in 𝔄ss_{k}). Then, by 5.10.1, (f_{1}∘ϕ)^{∗} ≃ (f_{2}∘ϕ)^{∗}, or, equivalently, ϕ_{∗}∘f_{1∗} ≃ ϕ_{∗}∘f_{2∗}. Since ϕ_{∗} is a fully faithful functor, this implies that f_{1∗} ≃ f_{2∗}, hence f_{1} and f_{2} define the same morphism in 𝔄ss_{k}.

Conversely, suppose ϕ is a formally 𝔄ss_{k}^{inf}-open immersion. In particular, it is a formally 𝔄ss_{k}^{inf}-smooth morphism. It follows from 4.3 and 4.6.1 that formal 𝔄ss_{k}^{inf}- smoothness is the same as formal Ass_{k}^{inf}-smoothness.

5.12. Smooth, étale, and unramified morphisms. Open immersions.

5.12.1. Finitely presentable and locally finitely presentable morphism. Let C be a category, C^{∧} the category of presheaves of sets on C (i.e. functors C^{op} ⟶ Sets). Fix an object, Y, of C^{∧}. We call a morphism X ⟶^{f} Y locally finitely presentable (resp. locally of finite type) if for any filtered projective system D ⟶^{𝔇} C/Y, the canonical map

colim C^{∧}/Y (𝔇, (X, f)) ⟶ C^{∧}/Y (lim 𝔇, (X, f))  (1)

is bijective (resp. injective). Here C/Y denotes the full subcategory of the category C^{∧}/Y whose objects are pairs (V, V → Y) with representable V.

A morphism X ⟶^{f} Y is called finitely presentable (resp. of finite type) if for any filtered projective system D ⟶^{𝔇} C^{∧}/Y, the canonical map

colim C^{∧}/Y (𝔇, (X, f)) ⟶ C^{∧}/Y (lim 𝔇, (X, f))  (2)

is bijective (resp. injective).

In order to avoid repetitions, we introduce intermediate notions. Fix a full subcategory E of the category C^{∧} containing all representable functors. Let X, Y be objects of C^{∧}. We call a morphism X ⟶^{f} Y of E-finite type (resp. E-finitely presentable) if for any filtered projective system D ⟶^{𝔇} E/Y, the canonical map

colim C^{∧}/Y (𝔇, (X, f)) ⟶ C^{∧}/Y (lim 𝔇, (X, f))

is injective (resp. bijective).

5.12.2. Proposition. Let Σ_{E}^{1} (resp. Σ_{E}^{0}) denote the class of all E-finitely presentable (resp. of E-finite type) morphisms of the category C^{∧}.

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

(b) If the morphism f in the cartesian square

X′ ⟶^{g′} X
f′ ↓  ↓ f
Y′ ⟶^{g} Y

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

(c) Suppose that X ⟶^{f} Y and Z ⟶^{h} W are morphisms over S which belong to Σ_{E}^{i}.

Then X ×_{S} Z ⟶^{f×_{S}h} Y ×_{S} W belongs to Σ_{E}^{i}, i = 0, 1.

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

Proof. (a) Let morphisms X ⟶^{f} Y ⟶^{g} Z belong to Σ_{E}^{i}. We claim that the composi-

tion X ⟶^{gf} Z belongs to the same class; i.e. for any filtered projective system D ⟶^{𝔇} E/Z, the canonical map

colim C^{∧}/Z(𝔇, (X, gf)) ⟶ C^{∧}/Z(lim 𝔇, (X, gf))

is injective if i = 0 and bijective if i = 1. First, consider the case i = 0.

Let (u_{ν}) and (u′_{ν}) be two inductive systems of arrows 𝔇(ν) ⟶ (X, gf), ν ∈ ObD, (i.e. gfu_{ν} = gfu′_{ν} for all ν) such that the compositions of u_{ν} and u′_{ν} with the canonical morphism lim 𝔇 ⟶^{p_{ν}} 𝔇(ν) are equal. With more reason, (fu_{ν})p_{ν} = (fu′_{ν})p_{ν}. Since Y ⟶^{g} Z is of E-finite type, fu_{μ} = fu′_{μ} for an appropriate μ. Replacing 𝔇 by the composition, 𝔇_{μ}, of 𝔇 with the canonical functor μ\D ⟶ D, we can regard (u_{ν}) and (u′_{ν}) as inductive systems of arrows 𝔇_{μ}(ν) ⟶ (X, f), ν ∈ Obμ\D which equalize the

canonical morphism lim 𝔇 = lim 𝔇_{μ} ⟶^{p_{ν}} 𝔇_{μ}(ν). Since X ⟶^{f} Y belongs to Σ_{E}^{0}, there exists λ such that u_{λ} = u′_{λ}; i.e. the systems (u_{ν}) and (u′_{ν}) define the same element of colim C^{∧}/Z(𝔇, (X, gf)).

Suppose now that the morphisms f and g belong to Σ_{E}^{1}. Let D ⟶^{𝔇} E/Z be a filtered projective system, and let lim 𝔇 ⟶^{h} (X, gf) be an arbitrary morphism. Consider the morphism lim 𝔇 ⟶^{fh} (Y, g). Since Y ⟶^{g} Z belongs to Σ_{E}^{1}, there exists a unique element, 𝔲, of colim C^{∧}/Z(𝔇, (Y, g)) whose image in C^{∧}/Z(lim 𝔇, (Y, g)) coincides with fh. Let (u_{ν}) be an inductive system of arrows {𝔇_{μ}(ν) ⟶ (Y, g)} representing the element 𝔲; i.e. the diagrams

lim 𝔇_{μ} ⟶^{h} X
p_{ν} ↓  ↓ f
𝔇_{μ}(ν) ⟶^{u_{ν}} Y

commute (here 𝔇_{μ} has the same sense as above). Since X ⟶^{f} Y belongs to Σ_{E}^{1}, there is a unique element of colim C^{∧}/Y (𝔇_{μ}, (X, f)) whose image in C^{∧}/Y (lim 𝔇_{μ}, (X, f)) is given by h. Here we use that lim 𝔇_{μ} = lim 𝔇.

(b) Suppose that X ⟶^{f} Y belongs to Σ_{E}^{1}. Let D ⟶^{𝔇} E/Y′ be a filtered projective system, and let lim 𝔇 = (V, V →^{v} Y′). Fix a morphism (V, v) ⟶^{h} (X′, f′). Since f

belongs to Σ_{E}^{1}, the morphism (V, gv) ⟶^{g′h} (X, f) is the image of a unique element, 𝔲, of colim C^{∧}/Y (g_{∗}𝔇, (X, f)). Here g_{∗}𝔇 : D ⟶ C^{∧}/Y is the diagram obtained by composition 𝔇 with g. Let (u_{ν}) be an inductive system of arrows {g_{∗}𝔇(ν) ⟶ (X, f)} representing the element 𝔲. Then the diagrams

(V, gv) ⟶^{h} (X′, fg′)
p_{ν} ↓  ↓ g′
g_{∗}𝔇(ν) ⟶^{u_{ν}} (X, f)

commute. By the universal property of cartesian squares, there exists a unique morphism

𝔇(ν) ⟶^{u′_{ν}} (X′, f′) such that u_{ν} = g′u′_{ν}.

The proof of remaining assertions follows a similar routine. We leave the arguments to the reader.

5.12.3. Definitions. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. Let X, Y be presheaves of sets on A^{op} (i.e. functors A ⟶ Sets). We call a morphism X ⟶^{f} Y smooth (resp. étale, resp. unramified) if it is locally finitely representable and formally smooth (resp. formally étale, resp. formally unramified).

We call an 𝔸-smooth monomorphism an 𝔸-open immersion.

6. Formally smooth and formally infinitesimal morphisms. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category. We call a morphism U ⟶^{ϕ} T of presheaves of sets on A^{op} formally 𝔸-infinitesimal if for any formally 𝔸-smooth morphism X ⟶^{f} Y and for any commutative diagram

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

there exists a morphism T ⟶^{γ} X such that the diagram

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

commutes.

6.1. Proposition. (a) Any split monomorphism (in particular, any isomorphism) is formally 𝔸-infinitesimal.

(b) The composition of formally 𝔸-infinitesimal morphisms is formally 𝔸-infinitesimal.

(c) Let U ⟶^{ϕ} T be a formally 𝔸-infinitesimal morphism and U ⟶^{ψ} V any morphism. Then the canonical morphism V ⟶ V ∐_{U} V is formally 𝔸-infinitesimal.

Proof. (a) Obvious.

(b) Let U ⟶^{ϕ} T and T ⟶^{ψ} S be formally 𝔸-infinitesimal morphisms. And let

U ⟶^{g′} X
ψϕ ↓  ↓ f
S ⟶^{g} Y

be a commutative diagram with f formally 𝔸-smooth. Since ϕ is formally 𝔸-infinitesimal, there exists a morphism T ⟶^{γ} X such that the diagram

U ⟶^{g′} X

ϕ ↓ ↗_{γ} ↓ f

T ⟶^{gψ} Y

commutes. Since ψ is formally 𝔸-infinitesimal, there exists a morphism S ⟶^{γ′} X such that the diagram

T ⟶^{γ} X

ψ ↓ ↗′_{γ} ↓ f

S ⟶^{g} Y

commutes.

(c) Consider a commutative diagram

U ⟶^{ψ}  V  ⟶^{g′} X
ϕ ↓  ↓ p_{1}  ↓ f
T ⟶^{p_{2}} T ∐_{U} V ⟶^{g} Y

in which p_{1}, p_{2} are canonical projections. Since ϕ is formally 𝔸-infinitesimal, there exists a morphism T ⟶^{γ} X such that the diagram

U ⟶^{g′ψ} X

ϕ ↓ ↗_{γ} ↓ f

T ⟶^{gp_{2}} Y

commutes. Due to the universal properties of fiber coproducts, the commutative diagram

U ⟶^{ψ} V
ϕ ↓  ↓ g′
T ⟶^{γ} X

determines uniquely a morphism T ∐_{U} V ⟶^{γ′} X such that the diagram

V  ⟶^{g′} X
ϕ ↓  ↓ f
T ∐_{U} V ⟶^{g} Y

commutes.

6.2. The Q-category 𝔸_{𝔸}^{∨}. We denote by 𝔸_{𝔸}^{∨} the full Q-subcategory, (A_{𝔸}^{∨} ⇄ A^{∨}), of the Q-category ((A^{∨})^{2} ⇄ A^{∨}) of morphisms of A^{∨} (cf. 2.5), where A_{𝔸}^{∨} denotes the full subcategory of (A^{∨})^{2} whose objects are formally 𝔸-infinitesimal morphisms. It follows from 6.1 that 𝔸_{𝔸}^{∨} is a Q-category corresponding to a pretopology.

6.3. General pattern. Fix a category C and a family, 𝔐, of morphisms of C containing all identical morphisms.

(i) We call a morphism X ⟶^{f} Y in C formally 𝔐-smooth if any commutative diagram

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

such that ϕ ∈ 𝔐 extends to a commutative diagram

\begin{matrix}T ⟶^{g′} X \\ ϕ ↓ ↗_{γ} ↓ 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.

6.4. 𝔑-infinitesimal morphisms. On the other hand, given a class 𝔑 of morphisms of C, denote by 𝔑_{inf} the class of all morphisms T ⟶^{ϕ} S of C such that any commutative

diagram (1) such that X ⟶^{f} Y belongs to 𝔑 extends to a diagram (2). Morphisms of 𝔑_{inf} will be called 𝔑-infinitesimal morphisms.

6.5. Remarks. (a) Given a Q-category 𝔸 = (Ā ⇄^{u} A), take as C the category A^{∨} of functors A ⟶ Sets, and set 𝔐 = 𝔐_{𝔸} to be the family of morphisms {Ā(ȳ, u^{∗}(−)) ⟶^{αȳ} A(u_{∗}(ȳ), −) | ȳ ∈ ObĀ}. It follows from definitions that a morphism X ⟶ Y is formally 𝔸-smooth (resp. formally 𝔸-unramified) iff it is formally 𝔐_{𝔸}-smooth (resp. formally 𝔐_{𝔸}-unramified).

Similarly, the notion of an 𝔸-infinitesimal morphism is a special case of a 𝔑-infinitesimal morphism for an obvious choice of the category C and the family of morphisms 𝔑; namely, C = A^{∨}, and 𝔑 is the class of formally 𝔸-smooth morphisms.

(b) The main reason to introduce this setting here is the natural ”duality”

𝔐 ⟼ 𝔐_{fsm}, 𝔑 ⟼ 𝔑_{inf}.

If follows from definitions that 𝔐 ⊆ 𝔑_{inf} iff 𝔑 ⊆ 𝔐_{fsm}. In a more symmetric way, the latter relations can be expressed as follows:

Any commutative diagram

T ⟶^{g} X
ϕ ↓  ↓ f
S ⟶^{g′} Y

such that ϕ ∈ 𝔐 and f ∈ 𝔑 extends to a commutative diagram

T ⟶^{g′} X

ϕ ↓ ↗_{γ} ↓ f

S ⟶^{g} Y

6.6. Proposition. Let 𝔐 be a family of arrows of a category C.

(a) The class 𝔐_{fsm} (resp. 𝔐_{fnr}, resp. 𝔐_{fet}) of formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) morphisms is closed under composition and contains all isomorphisms of the category C.

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

(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).

(c) 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.

(d) Let X ⟶^{f} Y ⟵^{h} S 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} T ⟶^{f′} T is formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale).

Proof. The argument follows the lines of the proofs of 5.4 and 5.6.

6.6.1. Corollary. Let X ⟶^{f} Y, Y ⟶^{h} Z be morphisms of C. Suppose h is formally 𝔐-étale. Then h ∘ f is formally 𝔐-smooth (resp. formally 𝔐-unramified, resp. formally 𝔐-étale) iff f belongs to the same class.

6.7. Proposition. Let 𝔑 be a family of arrows of C.

(a) Any split monomorphism (in particular, any isomorphism) belongs to 𝔑_{inf}.

(b) The class 𝔑_{inf} of 𝔑-infinitesimal morphisms is closed under composition.

(c) Let T ⟵^{ϕ} U ⟶^{ψ} S be morphisms such that there exists T ∐_{U} S. If ϕ belongs to 𝔑_{inf}, then the canonical morphism S ⟶ T ∐_{U} S belongs to 𝔑_{inf}.

Proof. The argument is similar to the proof of 6.1.

6.8. Example: separated, universally closed, and proper morphisms. Let A be the category CAlg_{k} of commutative k-algebras. Let Ā be the category of faithfully flat

k-algebra morphisms, 𝔸 = (Ā ⇄^{u} A) the corresponding full Q-subcategory of (A^{2} ⇄ A). Spaces in the sense of Grothendieck, in particular schemes and algebraic spaces, are sheaves of sets on the Q-category 𝔸.

Let 𝔐′_{v} be the family of canonical injections of valuation rings to their fields of fractions; and let 𝔐_{v} denote the image of 𝔐′_{v} in the category A^{∨} of functors A ⟶ Sets.

6.8.1. Proposition. Let X ⟶^{f} Y be a quasi-separated scheme morphism. Then (a) The morphism f is separated iff it is formally 𝔐_{v}-unramified.

(b) The morphism f is universally closed iff it is formally 𝔐_{v}-smooth.

(c) The morphism f is proper iff it is formally 𝔐_{v}-étale.

Proof. The assertions (a) and (c) are equivalent resp. to the Grothendieck’s criterion of separatedness and properness (see EGA, Ch.II, 7.2.3 and 7.2.8). A proof of the assertion (b) can be extracted from the argument of Theorem 7.2.8, EGA, Ch.II.

Standard properties of separated and proper morphisms become special cases of assertions on formally 𝔐-unramified and formally 𝔐-étale morphisms (cf. 5.4 and 5.6):

6.8.2. Corollary. (a) Any monomorphism is a separated morphism.

(b) A composition of two separated (resp. proper) morphisms is separated (resp. proper).

(c) Separated (resp. proper) morphisms are stable under base change.

(d) If X ⟶^{f} Y and Y ⟶^{g} Z are two morphisms such that g ∘ f is separated, then f is separated.

(e) If X ⟶^{f} Y and Y ⟶^{g} Z are two morphisms such that g is separated and g ∘ f is proper, then f is proper.

(f) If X ⟶^{f} Y and X′ ⟶^{f′} Y′ are separated (resp. proper) morphisms over S, then their product, f ×_{S} f′ : X ×_{S} X′ ⟶ Y ×_{S} Y′, is also separated (resp. proper).

6.8.3. Remarks. (a) One can introduce the notions of formally separated and formally proper morphisms by omitting the condition that the morphism in question is quasicompact. In terms of the family 𝔐_{v}, a morphism is formally separated (resp. formally proper) iff they are formally 𝔐_{v}-unramified (resp. formally 𝔐_{v}-étale). It follows that the assertions obtained from 6.8.1 and 6.8.2 by dropping the quasi-compactness condition and inserting ’formally’ at appropriate places, are corollaries of 6.6.

(b) The notions of a (formally) proper morphism and a (formally) separated morphism make sense for morphisms of arbitrary presheaves of sets on the category A^{op}, not only for scheme morphisms, because the notions of a (formally) 𝔐-smooth and (formally) 𝔐unramified morphisms make sense for morphisms of presheaves of sets on A^{op}.

(c) At the moment, it is not clear what might be an adequate noncommutative version of the family 𝔐_{v}.

7. Affine morphisms. Closed immersions. Separated morphisms and separated presheaves.

7.1. Representable morphisms. Let P be a class of morphisms of the category A having the following properties:

(a) A composition of a morphism from P with any isomorphism belongs to P.

(b) If X ⟶^{f} Y is a morphism from P, then for any Z ⟶^{g} Y, there exists a fiber product X ×_{Y} Z and the projection X ×_{Y} Z ⟶ Z belongs to P.

Let F, G be presheaves of sets. A morphism F ⟶ G is called representable by a morphism of P if for any h_{X} ⟶ G, the projection F ×_{G} h_{X} ⟶ h_{X} is of the form h_{u} for a morphism u ∈ P. In particular, the functor F ×_{G} h_{X} is representable.

Denote by P^{∧} the class of all morphisms of A^{∧} representable by morphisms of P. Clearly a morphism h_{X} ⟶ h_{Y} belongs to P^{∧} iff it is of the form h_{w} with w ∈ P.

7.1.1. Lemma. The class P^{∧} is invariant under the base change: if F ⟶ G belongs to P^{∧} and H ⟶ G is an arbitrary morphism, then the projection H ×_{G} F ⟶ H belongs to P^{∧}.

Proof is left to the reader.

7.1.2. Lemma. Let P and Q be classes of morphisms of the category A satisfying the conditions (a), (b). Then

(i) The intersection P ∩ Q has the properties (a) and (b).

(ii) If P is closed under the composition, then P^{∧} has the same property.

Proof is left to the reader.

7.1.3. Standard examples. 1) The class 𝔐 = 𝔐(A) of all monomorphisms has the property 7.1(b) and is closed under the composition.

2) Same holds for the class E^{u} = E^{u}(A) of universal epimorphisms. Recall that a morphism X ⟶^{f} Y is called a universal epimorphism if for any morphism V ⟶ Y, there exists a fiber product X ×_{Y} V and the canonical projection X ×_{Y} V ⟶ V is an epimorphism.

7.2. Affine morphisms. Let A be a category with fiber products, and let P be the class of all morphisms of A. In this case we shall call P-representable morphisms affine.

It follows that a presheaf morphism F ⟶ G is affine iff for any object X of A and any morphism h_{X} ⟶ G, the presheaf F ×_{G} h_{X} is representable.

7.2.1. Lemma. Let A be a category with finite limits, and let G be a presheaf of sets on a category A. The following conditions are equivalent:

(a) For any object X of A, any morphism h_{X} ⟶ G is affine.

(b) The diagonal morphism G ⟶ G × G is affine.

Proof. (a)⇒(b). Let X be an object of A and h_{X} ⟶^{f} G × G an arbitrary morphism. Taking compositions of f with projections G × G ⇉ G, we obtain a pair of morphisms

h_{X} \overrightarrow{⟶}^{f_{1}}_{f_{2}} G. Their fiber product, h_{X} ×_{G} h_{X}, is a part of the cartesian square

G  ⟶^{Δ_{G}} G × G
↑  ↑ f_{1} × f_{2}
h_{X} ×_{G} h_{X} ⟶ h_{X} × h_{X}

The condition (a) implies that h_{X} ×_{G} h_{X} is affine. Since h_{X} ⟶ G × G is arbitrary, the diagonal morphism Δ_{G} : G ⟶ G × G is affine.

(b)⇒(a). Let X, Y be objects of A, and let h_{X} ⟶ G ⟵ h_{Y} be arbitrary morphisms. Consider the cartesian square

G  ⟶^{Δ_{G}} G × G
↑  ↑
h_{X} ×_{G} h_{Y} ⟶ h_{X} × h_{Y}

Since A has finite products, h_{X}×h_{Y} is representable, h_{X}×h_{Y} ≃ h_{X×Y}. Since by hypothesis (b) the diagonal morphism Δ_{G} is affine, the presheaf h_{X} ×_{G} h_{Y} is representable too, hence the assertion.

7.3. Strict monomorphisms and closed immersions. For a morphism Y ⟶^{f} X of a category A, denote by Λ_{f} the class of all pairs of morphisms u_{1}, u_{2} : X ⇉ V equalizing

f. A morphism Y ⟶^{f} X is called a strict monomorphism if any morphism g : Z ⟶ X such that Λ_{f} ⊆ Λ_{g} has a unique decomposition g = f ∘ g′. It follows that any strict monomorphism is a monomorphism. We denote the class of strict monomorphisms of the category A by 𝔐_{s}(A), or by 𝔐_{s}. The class 𝔈_{s} = 𝔈_{s}(A) of strict epimorphisms is defined dually.

Clearly the composition of a strict monomorphism with an isomorphism is a strict monomorphism. If the category A has fiber products, then the class 𝔐_{s} = 𝔐_{s}(A) of strict monomorphisms of the category A satisfies the condition 7.1(b) too.

In fact, consider the diagram

\begin{matrix}X ×_{Y} V & ⟶^{p_{2}} & V \\ p_{1} ↓ & & ↓ g \\ X & ⟶^{f} & Y & \overrightarrow{⟶} & Z \end{matrix}  (1)

where Y ⇉ Z is an arbitrary pair of arrows from the class Λ_{f} of arrows equalizing f. But then p_{2} is a universal arrow equalizing all pairs Λ_{f} ∘ g = {(u_{1}g, u_{2}g)| (u_{1}, u_{2}) ∈ Λ_{f}}.

7.3.1. Note. Suppose a morphism Y ⟶^{f} X is such that there exists a fiber coproduct X ∐_{Y} X. Then f is a strict monomorphism iff it is a kernel of the coprojections Y \overrightarrow{⟶} Y ∐_{X} Y. In particular, if the category A has fiber coproducts, then strict monomorphisms can be defined as morphisms X ⟶ Y such that the diagram X ⟶ Y \overrightarrow{⟶} Y ∐_{X} Y is exact.

7.3.2. Lemma. (a) If the composition, gf, of morphisms X ⟶^{f} Y ⟶^{g} Z is a strict monomorphism, then f is a strict monomorphism.

(b) Any retraction is a strict monomorphism.

Proof. (a) If gf is a universal morphism with respect to the class of arrows

Λ_{gf} = \{Z \overrightarrow{⟶}^{u_{1}}_{u_{2}} V | u_{1}gf = u_{2}gf\},

then f is universal for the class of arrows Λ_{gf} ∘ g = {(u_{1}g, u_{2}g)| (u_{1}, u_{2}) ∈ Λ_{gf}}. (b) Let X ⟶^{p} Y is a retraction, i.e. there exists a morphism Y ⟶^{e} X such that

ep = id_{X}. Then p is a kernel of the pair X \overrightarrow{⟶}^{id_{X}}_{pe} X.

In fact, if Y ⟶^{f} X is a morphism equalizing the pair (id_{X}, pe), then f = p ∘ (ef); and this decomposition is unique because p is a monomorphism.

7.3.3. Closed immersions of presheaves of sets. Let F, G be presheaves of sets on A. We call a morphism F ⟶ G a closed immersion if it belongs to 𝔐_{s}^{∧}, i.e. if it is representable by a strict monomorphism. In particular, a closed immersion h_{X} ⟶ h_{Y} of representable functors is of the form h_{u}, where u is a strict monomorphism.

7.3.4. Example. Let A be the category CAff/k of commutative affine schemes over Spec(k). Then strict monomorphisms are exactly closed immersions of affine schemes. Let X and Y be arbitrary schemes identified with the corresponding sheaves of sets on the category C = CAff/k. Then a morphism X ⟶ Y is a closed immersion in the sense of the definition 7.3.1 iff it is a closed immersion of schemes in the conventional sense.

This example shows in particular that a strict monomorphism of (pre)sheaves is not necessarily a closed immersion. For instance, if X ⟶^{f} Y is a scheme morphism, the diagonal morphism X ⟶^{Δ_{f}} X ×_{Y} X is a kernel of the natural pair of arrows X ×_{Y} X ⇉ X, hence it is a strict monomorphism of sheaves of sets. But Δ_{f} is a closed immersion (in the sense of 7.3.2) only if the scheme morphism f is separated. Note that, in general, Δ_{f} is not even affine.

7.3.5. Example. Let R be an associative k-algebra and A the category (R\Alg_{k})^{op} of noncommutative affine schemes over R. Let (S, R ⟶^{s} S) and (T, R ⟶^{t} T) be R-rings and (S, s) ⟶^{f} (T, t) and R-ring morphism. The corresponding morphism of affine schemes is a strict monomorphism iff the diagram T ∏_{S} T ⇉ T ⟶^{f} S is exact. The latter means that S is the quotient of T by the two-sided ideal Ker(f).

7.4. Separated morphisms and separated presheaves. Let X, Y be presheaves of sets on a category A. We call a morphism X ⟶^{f} Y separated if the canonical morphism X ⟶^{Δ_{f}} X ×_{Y} X is a closed immersion. We say that a presheaf X on A is separated if the diagonal morphism X ⟶ X × X is a closed immersion.

Let • denote the constant presheaf A^{op} ⟶ Sets with values in a one point set. Since • is a final object of the category of presheaves of sets on A, a presheaf X is separated iff the (unique) morphism X ⟶ • is separated.

II. Locally affine spaces and schemes. Grassmannians.

8. Locally affine spaces.

8.1. Spaces and covers. Fix a Q-category 𝔸 = (Ā ⇄^{u} A). Let 𝔗𝔸 be the associated quasi-cosite (cf. 2.4). We call (the category of) sheaves of sets on 𝔗𝔸 (the category of) 𝔸-spaces, or simply spaces if it is clear what is 𝔸. We denote by Esp_{𝔸} the full subcategory of A^{∨} formed by 𝔸-spaces, and by u^{+} the sheafification functor A^{∨} ⟶ Esp_{𝔸} (a left adjoint to the inclusion functor Esp_{𝔸} ⟶ A^{∨}).

If 𝔗𝔸 is a cosite, then (and only then) the sheafification functor is exact. Otherwise, it is only right exact, as any functor having a left adjoint.

We call a space X affine, if X is corepresentable, i.e. X ≃ A(x, −) for some x ∈ ObA. We are particularly interested in the case when the Q-category 𝔸 is subcanonical, i.e. all corepresentable functors are 𝔸-spaces.

8.2. Remark. In what follows, the Q-category 𝔸, or ruther the associated quasicosite 𝔗𝔸, is just a device serving to define the subcategory of spaces, Esp_{𝔸}. We might start with choosing somehow a strictly full subcategory, 𝔈, of the category A^{∨} such that the inclusion functor 𝔈 ↪ A^{∨} has a right adjoint, and declare 𝔈 the category of Spaces (cf. A1.10). The requirement that 𝔸 should be subcanonical means that Ob𝔈 contains all corepresentable functors. Note, however, that this setting is not more general than the one we started with: if we take as 𝔸 the Q-category A^{∨} ⇄ 𝔈, the category Esp_{𝔸} of 𝔸-spaces coincides with 𝔈.

8.3. Affine τ-covers, locally affine τ-spaces, and τ-schemes. Let τ be a quasitopology on the category Esp_{𝔸} of 𝔸-spaces.

We call a τ-cover {X_{u} →^{u} X | u ∈ U} affine, if all X_{u} are affine (i.e. corepresentable). We call an 𝔸-space X τ-locally affine if it has an affine τ-cover. We call the cover {X_{u} →^{u} X | u ∈ U} 2-affine if it is affine and for any u, v ∈ U, the

space X_{u} ×_{X} X_{v} has an affine τ-cover.

A τ-cover {X_{u} →^{u} X | u ∈ U} will be called a Zariski affine τ-cover if it is affine and consists of monomorphisms.

We call an 𝔸-space X a τ-scheme if it has a Zariski affine τ-cover.

A τ-cover {X_{u} →^{u} X | u ∈ U} will be called Zariski 2-affine if it is Zariski affine and for any u, v ∈ U, the space X_{u} ×_{X} X_{v} has a Zariski affine τ-cover.

We denote by LA_{𝔸,τ} the full subcategory of Esp_{𝔸} whose objects are τ-locally affine 𝔸-spaces, and by Sch_{𝔸,τ} the full subcategory of Esp_{𝔸} whose objects are τ-schemes.

8.3.1. Note. Let {X_{u} →^{u} X | u ∈ U} be a 2-cover, and let {X_{u,v}^{ν} ⟶^{ν_{u,v}} X_{u}×_{X}X_{v} | u, v ∈

U, ν ∈ J_{u,v}} be an affine cover of X_{u} ×_{X} X_{v}, u, v ∈ U. We can associate with this data the following diagram formed by affine spaces and their morphisms:

\begin{matrix}X_{u,v}^{ν} \\ ↙  ↘ \\ X_{u}  X_{v} & u, v ∈ U, ν ∈ J_{u,v} \end{matrix}  (1)

Suppose the quasi-pretopology τ is such that all covers {X_{u} →^{u} X | u ∈ U} are strictly epimorphic families of arrows, i.e. ∐_{u∈U} X_{u} ⟶ X is a strict epimorphism. Then the space X is a colimit of the diagram (1). In particular, the space X can be reconstructed from the local affine data given by the diagram (1).

8.4. Semiseparated and weakly separated covers and spaces. We call a cover

{X_{u} →^{u} X | u ∈ U}

– weakly separated if X_{u} and the pull-back X_{u} ×_{X} X_{v} are affine (i.e. corepresentable) for all u, v ∈ U,

– semiseparated if the space X_{u} and the morphism X_{u} →^{u} X are affine for all u.

Clearly semiseparated covers are weakly separated, and weakly separated covers are 2-covers; i.e. a space which has a weakly separated cover is locally affine. We call a space which has a semiseparated (resp. weakly separated) affine cover semiseparated (resp. weakly separated).

8.4.1. Proposition. Suppose that 𝔸 = (Ā ⇄^{u} A) is subcanonical (i.e. all corepresentable functors A ⟶ Sets are 𝔸-spaces), and A has products. Then every separated locally affine 𝔸-space is τ-semiseparated.

Proof. Let X be any separated 𝔸-space, not necessarily locally affine. By definition, X is separated iff the diagonal morphism Δ_{X} : X ⟶ X × X is a closed immersion, i.e. it is representable by strict monomorphisms. The latter means that for any pair of

morphisms T \overrightarrow{⟶}^{p_{1}}_{p_{2}} X with T affine, the canonical morphism Ker(p_{1}, p_{2}) ⟶ T is a strict

monomorphism of affine spaces (i.e. corepresentable functors; cf. 7.4). In particular, the diagonal morphism Δ_{X} is affine. Let T, V be affine 𝔸-spaces and T ⟶^{f} X ⟵^{g} V arbitrary morphisms. Consider the cartesian square

\begin{matrix}X & ⟶^{Δ_{X}} & X × X \\ ↑ & & ↑ f × g \\ T ×_{X} V & ⟶^{𝔧} & T × V \end{matrix}  (2)

Since A has products, the product of corepresentable functors T and V is a corepresentable functor. Since 𝔸 is subcanonical, this product is an affine 𝔸-space. In particular, it coincides with the product T × V in the diagram (2) (taken in the category of 𝔸-spaces).

Since T × V is affine, the morphism T ×_{X} V ⟶^{𝔧} T × V is a strict monomorphism of affine 𝔸-spaces. In particular, T ×_{X} V is affine. This shows that any morphism from an affine 𝔸-space to X is affine, hence the assertion.

8.4.2. Example: semiseparated schemes and algebraic spaces. Let X be a (commutative) scheme, or an algebraic space. An affine cover {U_{i} →^{u_{i}} X | i ∈ J} is called semiseparated if each morphism U_{i} →^{u_{i}} X is affine. A scheme (or algebraic space) is called semiseparated if it has a semiseparated cover.

8.5. Natural quasi-topologies on Esp_{𝔸}. Let P be a class of morphisms in Esp_{𝔸} which contains all identical arrows and is closed under the composition (that is P is a subcategory of Esp_{𝔸} having same objects as Esp_{𝔸}). We call a set of arrows {X_{u} →^{u} X | u ∈ U} from a P-cover of X if

(i) it is strictly epimorphic; (ii) all arrows of U belong to P. ’Strictly epimorphic’ means that the corresponding morphism ∐_{u∈U} X_{u} ⟶ X is a

strict epimorphism. This defines a quasi-topology, τ^{P}, on Esp_{𝔸}. It remains to choose the class P.

Let 𝔸_{1} = (Ā_{1} ⇄ A) be another Q-category with the same underlying category A (thought as the Q-category of thickennings). Then we have the following choices:

– the class Pét (resp. P_{fét}) of (resp. formally) 𝔸_{1}-étale morphisms,

– the class P_{sm} (resp. P_{fsm}) of (resp. formally) 𝔸_{1}-smooth morphisms,

– the class P_{zar} (resp. P_{fzar}) of (resp. formally) 𝔸_{1}-open immersions (cf. 5.12.3). We denote the corresponding quasi-topologies resp. by τét, τ_{fét}, τ_{sm}, τ_{fsm}, τ_{zar}, and τ_{fzar} and call them resp. étale, formally étale, smooth, formally smooth, Zariski and formally Zariski quasi-topology.

We call τ_{zar}-locally affine 𝔸-spaces 𝔸_{1}-schemes.

8.6. Remark. Each of the classes, Pét, P_{sm}, and P_{zar} is stable under a base change. But, strict epimorphisms fail, in general, to be invariant under a base change, hence τét, τ_{sm}, and τ_{zar} are not topologies usually. Same holds for formal versions of these quasi-pretopologies.

Let P be closed under base change. We define a τ_{c}^{P}-cover as a universally strictly epimorphic set of arrows {X_{u} →^{u} X | u ∈ U} contained in P. This means that for any morphism Y ⟶ X, the set of arrows {X_{u} ×_{X} Y → Y | u ∈ U} is a τ^{P}-cover (i.e. is strictly epimorphic). It follows that τ_{c}^{P}-covers form a pretopology, and the corresponding topology is the topology coinduced by τ, i.e. it is the strongest among those topologies which are coarser than τ.

9. Commutative and noncommutative schemes and algebraic spaces.

9.1. Commutative schemes and algebraic spaces. Let A be the category CAlg_{k} of commutative k-algebras, Ā_{1} the full subcategory of A^{2} formed by k-algebra epimorphisms with nilpotent kernels, Ā the full subcategory of A^{2} formed by faithfully flat morphisms. Then 𝔸-spaces are sheaves of sets on affine schemes endowed with the flat topology, i.e. spaces in the Grothendieck’s sense. And 𝔸_{1}-schemes are usual commutative schemes. If τ = τét, then locally affine (resp. separated) spaces are Artin’s (resp. separated) algebraic spaces.

9.2. Noncommutative schemes. Let A be the category Alg_{k} of associative kalgebras, Ā_{1} the full subcategory of A^{2} formed by k-algebra epimorphisms with nilpotent kernels, Ā the full subcategory of A^{2} formed by faithfully flat morphisms. The notion of an 𝔸_{1}-scheme is a natural noncommutative version of a scheme. If τ = τét (cf. 8.5), then, by 5.11.4.1, locally affine 𝔸-spaces are, precisely, schemes (in the sense of 8.5).

9.3. Noncommutative algebraic spaces. Let A be the category 𝔄ss_{k} of associative k-algebras with morphisms defined up to conjugation (cf. 4.6). Let Ā be the full subcategory of A^{2} formed by equivalence classes of faithfully flat algebra morphisms, and let Ā_{1} be the full subcategory of A^{2} formed by equivalence classes (with respect to conjugation, see 4.6) of k-algebra epimorphisms with nilpotent kernels. If τ is the étale quasi-topology (cf. 8.5), then locally affine 𝔸-spaces in this setting seem to be an adequate noncommutative version of Artin’s algebraic spaces.

Schemes and locally affine spaces with respect to the smooth quasi-topology are same in the settings of 9.2 and 9.3.

9.4. Remark. The definition of Artin’s algebraic space obtained in 9.1 is more general than the ones usually used. Usually some finiteness restrictions are imposed. For instance, Knutson considers only quasi-compact quasi-separated algebraic spaces. And in [A1], algebraic spaces are separated.

Similarly, in the noncommutative case, one can impose finiteness conditions, for example consider quasi-compact and quasi-separated locally affine spaces and schemes (see definitions in 8.2 and Appendix 2).

10. Vector fibers and Grassmannians.

Fix an associative unital k-algebra R. Let A be the category R\Alg_{k} 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) which we call for convenience R-rings. We denote by R^{e} the k-algebra R⊗_{k} R^{o}.

10.1. Vector fiber associated with a bimodule. Let M be a left R^{e}-module. We denote by 𝕍_{R}(M) the ’spectrum’ of the tensor algebra T_{R}(M) = ⊕_{n≥0}M^{⊗n} of the R^{e}-module M. Here M^{⊗0} = R and M^{⊗(n+1)} = M ⊗_{R} M^{⊗n} for n ≥ 0.

10.1.1. Lemma. For any unital ring morphism R ⟶^{φ} S, there is a natural isomorphism

SpecS ∏_{SpecR} 𝕍_{R}(M) ⟶^{∼} 𝕍_{S}(φ̄^{∗}(M))

over SpecS. Here φ̄^{∗}(M)) = S ⊗_{R} M ⊗_{R} S.

Proof. Consider arbitrary commutative square

SpecA ⟶ 𝕍_{R}(M) ↓  ↓ SpecS ⟶ SpecR ,

or, the corresponding commutative square of k-algebra morphisms

\begin{matrix}T_{R}(M) & ⟶ & A \\ ↑ & & ↑ \\ R & ⟶^{φ} & S \end{matrix}  (1)

The algebra morphism T_{R}(M) ⟶ A is uniquely determined by an R^{e}-module morphism M ⟶ A. The commutativity of the diagram (1) implies that the pair of morphisms M ⟶ A ⟵ S defines an S^{e}-module morphism φ̄^{∗}(M) = S^{e}⊗_{R^{e}}M ⟶ A which, in turn, uniquely determines a k-algebra morphism T_{S}(φ̄^{∗}(M)) ⟶ A. Therefore T_{S}(φ̄^{∗}(M)) ≃ T_{R}(M) ⋆_{R} S as algebras over S; hence the assertion.

10.1.2. Proposition. Let M be a left R^{e}-module. The space 𝕍_{R}(M) is locally of finite type (resp. locally finitely presentable) over R iff the R^{e}-module M is of finite type (resp. locally finitely presentable).

Proof. Let A = R\Alg_{k}, and let D ⟶^{𝔇} A be a filtered inductive system. Then we have a commutative diagram of canonical morphisms

colim Hom_{A}(T_{R}(M), 𝔇) ⟶^{λ} Hom_{A}(T_{R}(M), colim 𝔇)
≀↓  ↓≀
colim Hom_{R^{e}}(M, Φ_{R}𝔇) \underrightarrow{_{λ}} Hom_{R^{e}}(M, Φ_{R}(colim 𝔇))

(2)

Here A ⟶^{Φ_{R}} R^{e}−mod is the functor which maps a R-ring (S, R ⟶ S) to the left R^{e}-module S. The functor Φ_{R} is a right adjoint to the functor

R^{e} − mod ⟶^{T_{R}} A, M ⟼ (T_{R}(M), R → T_{R}(M).

The functor Φ_{R} preserves colimits of filtered inductive systems, i.e. the canonical morphism colim Φ_{R}𝔇 ⟶ Φ_{R}(colim 𝔇) is an isomorphism. Thus, the map λ̃ in (2) is the composition of a canonical map

colim Hom_{R^{e}}(M, Φ_{R}𝔇) \underrightarrow{_{λ}} Hom_{R^{e}}(M, colim Φ_{R}𝔇)

and an isomorphism Hom_{R^{e}}(M, colim Φ_{R}𝔇) ⟶^{∼} Hom_{R^{e}}(M, Φ_{R}(colim 𝔇)). Together with the commutativity of (2), this means that λ is injective (resp. bijective) iff λ̂ is injective (resp. bijective). Therefore, the morphism 𝕍_{R}(M) ⟶ Spec(R) is locally of finite type (resp. locally finitely presentable) if the R^{e}-module M is of finite type (resp. locally finitely presentable).

Let now D̃ \underrightarrow{_{𝔇}} R^{e} − mod be a filtered inductive system. Let Ex_{R} denote the functor R^{e} − mod ⟶ A = R\Alg_{k} denote the functor which assigns to every left R^{e}-module L the pair (L_{R}, R → L_{R}), where L_{R} is the extension R by L. Then for every R^{e}-module L, the composition Φ_{R} ∘ Ex_{R} transfers L into R ⊕ L. Taking in (2) 𝔇 = Ex_{R} ∘ 𝔇̃, we obtain (from the following (2) discussion) a commutative diagram

colim Hom_{A}(T_{R}(M), 𝔇) ⟶^{λ} Hom_{A}(T_{R}(M), colim 𝔇)
≀↓  ↓≀
colim Hom_{R^{e}}(M, Φ_{R}𝔇) \underrightarrow{_{λ}} Hom_{R^{e}}(M, colim Φ_{R}𝔇)

(3)

If 𝕍_{R}(M) is locally of finite type (resp. locally finitely presentable) over R, then the map λ in (3) is injective (resp. bijective). And λ is injective (resp. bijective) iff λ̂ has the same property. Since there are natural isomorphisms

colim Hom_{R^{e}}(M, Φ_{R}𝔇) ≃ colim Hom_{R^{e}}(M, R ⊕ 𝔇̃)
≃ Hom_{R^{e}}(M, R) ⊕ colim Hom_{R^{e}}(M, 𝔇̃)

and

Hom_{R^{e}}(M, colim Φ_{R}𝔇) ≃ Hom_{R^{e}}(M, colim (R ⊕ 𝔇̃)
≃ Hom_{R^{e}}(M, R) ⊕ Hom_{R^{e}}(M, colim 𝔇̃)

compatible with the map λ̂ in (3), λ̂ is injective (resp. bijective) iff the canonical map colim Hom_{R^{e}}(M, 𝔇̃) ⟶ Hom_{R^{e}}(M, colim 𝔇̃) is injective (resp. bijective). This shows that if 𝕍_{R}(M) is locally of finite type (resp. locally finitely presentable) over R, then the R^{e}-module M is of finite type (resp. locally finitely presentable).

10.2. A note on a base change. Fix an object S of a category E. Let E/S ⟶^{f_{S}} E

and E^{∧}/S \underrightarrow{_{f_{S}^{∗}}} E^{∧} be the forgetful functors. The functor f_{S} induces a functor

f_{S}^{∗} : E^{∧} ⟶ (E/S)^{∧}, X ⟼ X ∘ f_{S},  (1)

The functor f̂_{S}^{∗} has a right adjoint

f̂_{S_{∗}} : E^{∧} ⟶ E^{∧}/S, X ⟼ (X × S, X × S → S).  (2)

For every presheaf X on E, there is a natural isomorphism

f_{S}^{∗}(X) ≃ Hom_{E^{∧}/S}(h^{S}(−), f̂_{S_{∗}}(X)),  (3)

where the functor E/S ⟶^{h^{S}} E^{∧}/S is induced by the Yoneda embedding. In other words, the functor f_{S}^{∗}(X) is the restriction to E/S of a functor representable in E^{∧}/S.

If a presheaf X on E is representable by an object X̄, and there exists a product X̄ ×S in E, then the presheaf f_{S}^{∗}(X) on E/S is representable by the object (X̄ × S, X̄ × S → S). This situation is illustrated by 10.1.1. In the general case, we shall identify E/S with a full subcategory of E^{∧}/S and omit the embedding h^{S}.

10.3. Inner hom. Let M, V be left R-modules. Consider the functor

H_{R}(M, V) : R\Alg_{k} ⟶ Sets

which assigns to each algebra R ⟶^{ϕ} S over R the set Hom_{S}(ϕ^{∗}(M), ϕ^{∗}(V)).

10.3.1. Proposition. Let M and V be left R-modules. For any unital ring morphism R ⟶^{φ} S, there is a natural isomorphism

SpecS ∏_{SpecR} H_{R}(M, V) ⟶^{∼} H_{S}(φ^{∗}(M), φ^{∗}(V))

over SpecS (cf. 10.2(5)).

Proof. Consider a commutative square

SpecA ⟶^{ξ} H_{M,V}
γ ↓  ↓
SpecS ⟶^{ϕ} SpecR

The morphism ξ corresponds to an element of H_{M,V} (R → A), i.e. to an A-module epimorphism A ⊗_{R} M ⟶ A ⊗_{R} V. Since A ⊗_{R} − ≃ A ⊗_{S} (S ⊗_{R} −), this epimorphism defines an element of H_{ϕ^{∗}(M),ϕ^{∗}(V )}(S → A) which uniquely determines a morphism

SpecA ⟶^{ξ̄} H_{ϕ^{∗}(M),ϕ^{∗}(V )} over SpecS. This implies the assertion.

10.3.2. Lemma. If V is a projective R-module of finite type, then the functor H_{R}(M, V) is representable.

Proof. In fact, for any algebra R ⟶^{ϕ} S over R, we have:

Hom_{S}(ϕ^{∗}(M), ϕ^{∗}(V) ≃ Hom_{R}(M, ϕ^{∗}(V)) = Hom_{R}(M, S ⊗_{R} V).

If V is a projective R-module of finite type, then S ⊗_{R} V ≃ Hom^{R}(V^{∨}, S), where V^{∨} is the right R-module dual to V, i.e. V^{∨} = Hom_{R}(V, R); and Hom^{R}(−, −) denotes the functor of right R-module morphisms. Thus,

Hom_{R}(M, S ⊗_{R} V) ≃ Hom_{R}(M, Hom^{R}(V^{∨}, S)) ≃ Hom_{R^{e}} (M ⊗_{k} V^{∨}, S)

and

Hom_{R^{e}} (M ⊗_{k} V^{∨}, S) ≃ R\Alg_{k}(T_{R}(M ⊗_{k} V^{∨}), S),

hence the assertion.

10.3.3. Corollary. Let M be a left R-module and V a projective left R-module of finite type. Then, for any unital ring morphism R ⟶^{φ} S, there is a natural isomorphism

S ⋆_{R} T_{R}(M ⊗_{k} V^{∨}) ⟶^{∼} T_{S}(φ^{∗}(M) ⊗_{k} φ^{∗}(V)^{∨})

over S. Here φ^{∗}(V)^{∨} = Hom_{S}(φ^{∗}(V), S) ≃ Hom_{R}(V, S).

Proof. By 10.3.2, the functor H_{R}(M, V) is representable by the tensor algebra T_{R}(M⊗_{k} V^{∨}) of the R^{e}-module M ⊗_{k} V^{∨}. In particular, the functor H_{S}(φ^{∗}(M), φ^{∗}(V)) is represented by the tensor algebra of S^{e}-module

φ^{∗}(M) ⊗_{k} φ^{∗}(V^{∨}) = S ⊗_{R} M ⊗_{k} V^{∨} ⊗_{R} S ≃ S^{e} ⊗_{R^{e}} (M ⊗_{k} V^{∨}).

The assertion follows now from 10.1.1 (see also 10.2.1).

10.3.4. Corollary. Let M be a left R-module and V a projective left R-module of finite type. If the R-module M is locally of finite type (resp. locally finitely presentable), then the functor H_{R}(M, V) is locally of finite type (resp. locally finitely presentable) over R. If V is a generator of the category R − mod, then the converse holds; i.e. the functor H_{R}(M, V) is of finite type (resp. finitely presentable) over R iff the R-module M is of finite type (resp. finitely presentable).

Proof. By 10.3.2, the functor H_{R}(M, V) is representable by the tensor algebra T_{R}(M⊗_{k} V^{∨}) of the R^{e}-module M ⊗_{k} V^{∨}, i.e. H_{R}(M, V) ≃ 𝕍_{R}(M ⊗_{k} V^{∨}). If the R-module M is of finite type (resp. finitely presentable), then the R^{e}-module M ⊗_{k} V^{∨} is of finite type (resp. finitely presentable). If V is a generator of the category R − mod (that is Hom_{R}(V, −) is a faithful functor), then the R^{e}-module M ⊗_{k} V^{∨} is of finite type (resp. finitely presentable) iff the R-module M has this property. The assertion follows now from 10.1.2.

10.4. The functor Gr_{M,V}. Let A = R\Alg_{k}. Let M be a left R-module and V a projective left R-module. Consider the functor Gr_{M,V} : A ⟶ Sets which assigns to any R-ring (S, R →^{s} S) (an object of A) the set of isomorphism classes of S-module 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.

10.4.1. The functor G_{M,V}. Denote by G_{M,V} the functor A ⟶ 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} (S, s) ⟶ Gr_{M,V} (S, s), (v, u) ⟼ [u],  (1)

is a (strict) functor epimorphism.

10.4.2. 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} : s^{∗}(M) ⟶ s^{∗}(V) are equivalent. The latter means that there exists an isomorphism φ : s^{∗}(V) ⟶ 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.

10.4.3. 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. (a) Suppose the R-module V is finite. For any algebra morphism ϕ : R ⟶ S, we have the following functorial isomorphisms:

Hom_{S}(ϕ^{∗}(M), ϕ^{∗}(V)) ≃ Hom_{R}(M, ϕ_{∗}ϕ^{∗}(V)) = Hom_{R}(M, S ⊗_{R} V) ≃
Hom_{R}(M, Hom^{R}(V^{∨}, S)) ≃ Hom_{R^{e}}(M ⊗_{k} V^{∨}, S) ≃ R\Alg_{k}(T_{R}(M ⊗_{k} V^{∨}), S)

Here Hom^{R}(V^{∨}, S) is the (left) R-module of right R-module morphisms from V^{∨} to S, R^{e} = R ⊗_{k} R^{op}, and T_{R}(M ⊗_{k} V^{∨}) is the tensor algebra of the R-bimodule M ⊗_{k} V^{∨}.

(b) The set G_{M,V} (S) is the kernel of the pair of morphisms

Hom_{S}(ϕ^{∗}(M), ϕ^{∗}(V)) × Hom_{S}(ϕ^{∗}(V), ϕ^{∗}(M)) \overrightarrow{⟶} Hom_{S}(ϕ^{∗}(V), ϕ^{∗}(V)) (5)

where one arrow assigns to each pair (u, v) the composition, u ∘ v, of morphisms u and v, and the other one maps each pair (u, v) to the identity morphism, id_{ϕ^{∗}(V)}. Since the modules M and V are finite, we have canonical functorial isomorphisms:

Hom_{S}(ϕ^{∗}(M), ϕ^{∗}(V)) × Hom_{S}(ϕ^{∗}(V), ϕ^{∗}(M)) ≃
Hom_{R^{e}}(M ⊗_{k} V^{∨}, S) × Hom_{R^{e}}(V ⊗_{k} M^{∨}, S) ≃
Hom_{R^{e}}(M ⊗_{k} V^{∨} ⊕ V ⊗_{k} M^{∨}, S) ≃ R\Alg_{k}(T_{R}(M ⊗_{k} V^{∨} ⊕ V ⊗_{k} M^{∨}), S)

and

Hom_{S}(ϕ^{∗}(V), ϕ^{∗}(V)) ≃ R\Alg_{k}(T_{R}(V ⊗_{k} V^{∨}), S)

Thus, to the diagram (1), there corresponds a diagram

T_{R}(V ⊗_{k} V^{∨}) \overrightarrow{⟶} T_{R}(M ⊗_{k} V^{∨} ⊕ V ⊗_{k} M^{∨})  (6)

of algebra morphisms. The cokernel, G_{M,V}, of the pair of morphisms (6) corepresents the kernel of the pair of morphisms (5). This proves the corepresentability of G_{M,V}.

(c) A similar argument proves the corepresentability of ℜ_{M,V}. Details are left to the reader.

10.4.3.1. Proposition. If M and V are projective R-modules of a finite type, then the functors G_{M,V}, ℜ_{M,V}, and Gr_{M,V} are locally finitely presentable over R.

Proof. It follows from the argument of 10.4.3 that G_{M,V} is isomorphic to the kernel of a pair of arrows

𝕍_{R}(M ⊗_{k} V^{∨} ⊕ V ⊗_{k} M^{∨}) \overrightarrow{⟶} 𝕍_{R}(V ⊗_{k} V^{∨})  (7)

(see (6)). Since the R-modules M and V are projective of finite type, the R^{e}-modules V ⊗_{k}V^{∨} and M ⊗_{k}V^{∨}⊕V ⊗_{k}M^{∨} are projective of finite type; in particular, they are finitely presentable. Therefore, by 10.1.2, both functors in (7) are locally finitely presentable over R. The kernel of a pair of arrows between locally finitely presentable over R functors is locally finitely presentable over R; hence G_{M,V} is locally finitely presentable over R. By a similar reason the functor of relations ℜ_{M,V} is locally finitely presentable over R. Since Gr_{M,V} is a cokernel of a pair of arrows between locally finitely presentable over R functors (see 10.4.2(4)), it is locally finitely presentable too.

10.4.4. Universality with respect to the base change.

10.4.4.1. Proposition. Let M, V be R-modules. For any unital k-algebra morphism R ⟶^{ϕ} S, there is a natural isomorphism between the diagram

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

and the diagram

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

In particular, SpecS ∏_{SpecR} Gr_{M,V} is isomorphic to Gr_{ϕ^{∗}(M),ϕ^{∗}(V )}.

Proof. Consider a commutative square

SpecA ⟶^{ξ} Gr_{M,V}
γ ↓  ↓
SpecS ⟶^{ϕ} SpecR

The morphism ξ corresponds to an element of Gr_{M,V} (R → A), i.e. to the equivalence class of an A-module epimorphism A ⊗_{R} M ⟶ A ⊗_{R} V. Since A ⊗_{R} − ≃ A ⊗_{S} (S ⊗_{R} −), this epimorphism defines an element of Gr_{ϕ^{∗}(M),ϕ^{∗}(V )}(S → A) which corresponds to a

morphism SpecA ⟶^{ξ̄} Gr_{ϕ^{∗}(M),ϕ^{∗}(V )} over SpecS. The latter means that the diagram

SpecA ⟶^{ξ̄} Gr_{ϕ^{∗}(M),ϕ^{∗}(V )}

γ ↘ ↙

SpecS

commutes. This implies that SpecS ∏_{SpecR} Gr_{M,V} is isomorphic to Gr_{ϕ^{∗}(M),ϕ^{∗}(V )}. Simi-

larly, one can show that SpecS ∏_{SpecR} G_{M,V} is isomorphic to G_{ϕ^{∗}(M),ϕ^{∗}(V )}. It follows from

the universality of these constructions that the isomorphisms can be chosen in such a way that the diagram

SpecS ∏_{SpecR} G_{M,V} ⟶ SpecS ∏_{SpecR} Gr_{M,V}
↓  ↓
G_{ϕ^{∗}(M),ϕ^{∗}(V )}  ⟶^{π}  Gr_{ϕ^{∗}(M),ϕ^{∗}(V )}

(3)

commutes. Notice that the functor SpecS ∏_{SpecR} − preserves fibered products. Since

ℜ_{M,V} = G_{M,V} ∏_{Gr_{M,V}} G_{M,V} , the diagram (3) induces an isomorphism

SpecS ∏_{SpecR} ℜ_{M,V} ⟶ ℜ_{ϕ^{∗}(M),ϕ^{∗}(V )}.

Hence the assertion.

10.4.4.2. Proposition. Let M and V be projective left R-modules of finite type. And let G_{M,V} be a k-algebra representing the functor G_{M,V} and R_{M,V} a k-algebra representing

the functor ℜ_{M,V}. Then, for any unital k-algebra morphism R ⟶^{ϕ} S, there is a natural isomorphism between the k-algebras

S ⋆_{R} G_{M,V} ⟶ G_{ϕ^{∗}(M),ϕ^{∗}(V )} and S ⋆_{R} R_{M,V} ⟶ R_{ϕ^{∗}(M),ϕ^{∗}(V )}.  (4)

Proof. By the part (b) of the argument of 10.4.3, the functor G_{M,V} is the kernel of a pair of arrows

H_{R}(M, V) × H_{R}(V, M) \overrightarrow{⟶} H_{S}(V, V).  (5)

By 10.4.4.3.2 and 10.4.4.3.3, all functors in this diagram are representable and satisfy the desired property with respect to a base change. This implies the first isomorphism (4). We leave to the reader establishing the second isomorphism.

10.5. Smoothness.

10.5.1. Proposition. Let M, V be projective R-modules of finite type. Then all functors and all morphisms of the canonical diagram

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

are smooth.

Proof. By 10.4.3.1, all functors in the diagram (1) are locally finitely presentable over R. It follows from 5.12.2(d) that all morphisms of the diagram (1) are locally finitely presentable. It remains to show that all functors and all morphisms of the diagram (1) are formally smooth.

Fix an R-ring epimorphism T ⟶ S with a nilpotent kernel.

(a) By Yoneda’s lemma, a morphism Spec(S) ⟶ G_{M,V} is uniquely defined by an element of G_{M,V} (S), i.e. by a pair of S-module morphisms

S ⊗_{R} V ⟶^{g} S ⊗_{R} M ⟶^{h} S ⊗_{R} V  (2)

such that h ∘ g = id. Since M and V are projective modules and the algebra morphism T ⟶ S is an epimorphism, the diagram (2) can be lifted to a commutative diagram

T ⊗_{R} V ⟶^{g′} T ⊗_{R} M ⟶^{h′} T ⊗_{R} V
↓  ↓  ↓
S ⊗_{R} V ⟶^{g} S ⊗_{R} M ⟶^{h} S ⊗_{R} V

Since V is a module of finite type and the kernel of the morphism T ⟶ S is nilpotent, in particular it is contained in the Jacobson’s radical of T, the fact that the composition h ∘ g is an isomorphism implies (by Nakayama’s Lemma) that h′ ∘ g′ is an isomorphism. Set ḡ = g′ and h̄ = (h′ ∘ g′)^{−1} ∘ h′. It follows that h̄ ∘ ḡ = id_{T ⊗_{R}V}. Hence G_{M,V} is formally smooth.

(b) A morphism Spec(S) ⟶ Gr_{M,V} is given by an element, ξ, of Gr_{M,V} (S). Since the map G_{M,V} (S) ⟶ Gr_{M,V} (S) is surjective, the element ξ is the image of an element, ξ′, of G_{M,V} (S). By (a), the element ξ′ can be lifted to an element, ξ′_{T}, of G_{M,V} (T). The image of ξ′_{T} in Gr_{M,V} (T) is a preimage of ξ.

(c) A morphism Spec(S) ⟶ ℜ_{M,V} is given by a pair of elements, (u_{1}, v_{1}), (u_{2}, v_{2}) of G_{M,V} (S) satisfying the following relations:

u_{2} = (u_{2} ∘ v_{1}) ∘ u_{1}, u_{1} = (u_{1} ∘ v_{2}) ∘ u_{2}  (3)

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

u_{1} ∘ v_{1} = id = u_{2} ∘ v_{2}  (4)

(cf. 10.4.2). By (a), each of the pairs can be lifted to an element of G_{M,V} (T), resp. (u′_{1}, v′_{1}) and (u′_{2}, v′_{2}). Set ϕ′ = u′_{2} ∘ v′_{1} : T ⊗_{R} V ⟶ T ⊗_{R} V. It follows that u′_{2} = ϕ′ ∘ u′_{1}. Since S ⊗_{T} ϕ′ = u_{2} ∘ v_{1} is invertible and the kernel of T ⟶ S is nilpotent, ϕ′ is invertible too. This shows that the functor of relations, ℜ_{M,V}, is formally smooth.

(d) Consider the commutative diagram

G_{M,V}  ⟶^{π} Gr_{M,V}
g ↑  ↑ g_{1}
SpecS ⟶^{ϕ̄} SpecT

(5)

in which ϕ̄ is the morphism corresponding to a ring epimorphism ϕ : T ⟶ S with a nilpotent kernel. The morphism g_{1} in (5) is uniquely defined by an element of Gr_{M,V} (R → T), i.e. by a T -module epimorphism T ⊗_{R} M ⟶^{u} T ⊗_{R} V. By the same Yoneda’s lemma, the morphism g in (5) is uniquely determined by an element of G_{M,V} (S, s), i.e. a pair of S-module morphisms

S ⊗_{R} V ⟶^{v′} S ⊗_{R} M ⟶^{u′} S ⊗_{R} V

such that u′ ∘ v′ = id. The commutativity of the diagram (5) is equivalent to the commutativity of the diagram

\begin{matrix}T ⊗_{R} V & & T ⊗_{R} M & ⟶^{u} & T ⊗_{R} V \\ ϕ_{V} ↓ & & ϕ_{M} ↓ & & ↓ ϕ_{V} \\ S ⊗_{R} V & ⟶^{v′} & S ⊗_{R} M & ⟶^{u′} & S ⊗_{R} V \end{matrix}  (6)

in which the vertical arrows correspond to the ring epimorphism T ⟶^{ϕ} S, hence they are epimorphisms. Since T ⊗_{R} V is a projective T -module and ϕ_{M} is a T -module epimorphism, there exists a T -module morphism T ⊗_{R} V ⟶^{w} T ⊗_{R} M such that the diagram

\begin{matrix}T ⊗_{R} V & ⟶^{w} & T ⊗_{R} M & ⟶^{u} & T ⊗_{R} V \\ ϕ_{V} ↓ & & ϕ_{M} ↓ & & ↓ ϕ_{V} \\ S ⊗_{R} V & ⟶^{v′} & S ⊗_{R} M & ⟶^{u′} & S ⊗_{R} V \end{matrix}  (6′)

commutes. Since T ⊗_{R}V is a projective T module of finite type, it follows from Nakayama’s Lemma that u ∘ w is an isomorphism. Set v = w ∘ (u ∘ w)^{−1}. Then u ∘ v = id and the diagram

\begin{matrix}T ⊗_{R} V & ⟶^{v} & T ⊗_{R} M & ⟶^{u} & T ⊗_{R} V \\ ϕ_{V} ↓ & & ϕ_{M} ↓ & & ↓ ϕ_{V} \\ S ⊗_{R} V & ⟶^{v′} & S ⊗_{R} M & ⟶^{u′} & S ⊗_{R} V \end{matrix}  (6′′)

commutes. The pair of arrows

T ⊗_{R} V ⟶^{v} T ⊗_{R} M ⟶^{u} T ⊗_{R} V  (7)

is an element of G_{M,V} (R → T) which corresponds to a morphism SpecT ⟶^{γ} G_{M,V}.

Since the pair (7) is a preimage of the element T ⊗_{R} M ⟶^{u} T ⊗_{R} V of Gr_{M,V} (R → T) corresponding to the morphism SpecT ⟶^{g_{1}} Gr_{M,V} (by definition of the morphism π), we have the equality: π ∘ γ = g_{1}. The commutativity of the diagram (6”) means exactly that the diagram

G_{M,V}
g ↗ ↖^{γ}
SpecS ⟶^{ϕ̄} SpecT

commutes.

(e) Since the morphism π in the cartesian square

ℜ_{M,V}  ⟶^{p_{1}} G_{M,V}
p_{2} ↓  ↓ π
G_{M,V}  ⟶^{π} Gr_{M,V}

is formally smooth, the morphisms p_{1} and p_{2} are formally smooth (see 5.6).

We need a slightly stronger version of a part of Proposition 10.5.1:

10.5.2. Proposition. All morphisms of the canonical diagram (1) are coverings for the smooth topology.

Proof. Let T ⟶^{ξ} Gr_{M,V} be a morphism with affine T. Since the presheaf morphism

G_{M,V} ⟶^{π} Gr_{M,V} is surjective, there exists a morphism T ⟶^{ξ′} Gr_{M,V} such that π ∘ ξ′ =

ξ. This implies that the canonical projection T ×_{Gr_{M,V}} G_{M,V} ⟶^{π′} T has a splitting; in particular, it is surjective. Since by 10.5.1, π is a smooth morphism, the projection π′ is smooth too, hence the assertion.

10.6. 𝔸-Grassmannians. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, where A is the category R\Alg_{k} of R-rings. We denote by Gr𝔸_{M,V} the 𝔸-space associated with Gr_{M,V}. We call the functor Gr𝔸_{M,V} an 𝔸-Grassmannian of the type (M, V). In particular, Gr_{M,V} is the 𝔸-Grassmannian, where 𝔸 corresponds to the discrete cotopology.

Suppose M and V are projective modules of finite type. Let G_{M,V} and R_{M,V} be kalgebras corepresenting the functors resp. G_{M,V} and ℜ_{M,V}. And let 𝔭_{i}, i = 1, 2, be the morphisms G_{M,V} \overrightarrow{⟶} R_{M,V} corresponding to the projections ℜ_{M,V} \overrightarrow{⟶} G_{M,V}.

If 𝔸 is subcanonical, then the exact diagram (4) induces an exact diagram of 𝔸-spaces

ℜ_{M,V} \overrightarrow{⟶} G_{M,V} ⟶ Gr𝔸_{M,V} .  (7)

10.6.1. Note. If 𝔸 is a Q-category corresponding to a Grothendieck (pre)topology, Gr𝔸_{M,V} (S, s) might be described as the set of equivalence classes of quotients of S ⊗_{R} M which are locally isomorphic to S ⊗_{R} V. Here equivalence is also understood as a local isomorphism. If 𝔸 does not correspond to a (pre)topology (more precisely, it is not stable under an arbitrary base change), a naive interpretation of this description produces a map from R-rings to Sets which, in general, is not functorial.

10.6.2. The Grassmannians Gr+_{M,V}. Let M, V be projective left R-modules of finite type. Let A be the category R\Alg_{k}; and suppose that Ā ⊆ A^{2} consists of all faithfully flat algebra morphisms. We shall write in this case Gr+_{M,V} instead of Gr𝔸_{M,V}. The canonical morphisms 𝔭_{i} : G_{M,V} ⟶ R_{M,V} , i = 1, 2, are faithfully flat. One can show that the 𝔸-Grassmannian Gr+_{M,V} is a locally affine 𝔸-space.

10.6.3. Noncommutative projective spaces. We shall write Nℙ_{M} instead of Gr+_{M,R^{1}} and call it the noncommutative projective space of the R-module M. Here R^{1} denotes the free R-module of the rank one. This space was introduced and described in [KR1] in the case when R is a field, hence M is a finite-dimensional vector space over R.

10.7. Smooth topology and Grassmannians. We denote by Gr\begin{matrix}𝔰𝔪 \\ M,V \end{matrix} the associated with Gr_{M,V} sheaf of sets for the smooth topology (cf. 8.6). It follows from 10.5.1 and 10.5.2 that the exact diagram (1) in 10.5.1 induces an exact diagram of sheaves

ℜ_{M,V} \overrightarrow{⟶}^{p_{1}}_{p_{2}} G_{M,V} ⟶^{π̃} Gr\begin{matrix}𝔰𝔪 \\ M,V \end{matrix}  (1)

whose arrows are covers in the smooth topology (see 10.5.2). In particular, Gr\begin{matrix}𝔰𝔪 \\ M,V \end{matrix} is a locally affine space with respect to the smooth pretopology.

10.8. Affine Zariski subschemes of a Grassmannian. Noncommutative Grassmannians are not schemes. But they have affine Zariski subschemes (constructed below) which being restricted to commutative algebras produce a Zariski affine cover of the corresponding commutative Grassmannian (when it is not empty).

Fix an R-module morphism V ⟶^{ϕ} M. For any R-ring (S, R ⟶^{s} S), consider the set F_{ϕ;M,V} (S, s) of equivalence classes of all morphisms s^{∗}(M) ⟶^{v} V′ such that v ∘ s^{∗}(ϕ) is

an isomorphism. Here two morphisms, s^{∗}(M) ⟶^{v} V′ and s^{∗}(M) ⟶^{v′} V′′, are equivalent iff v′ = ψ ∘ v for some S-module isomorphism V′ ⟶^{ψ} V′′.

10.8.1. Proposition. (a) The map (S, s) ⟼ F_{ϕ;M,V} (S, s) is naturally extended to a subfunctor, F_{ϕ;M,V} : R\Alg_{k} ⟶ Sets, of the functor Gr_{M,V}.

(b) Suppose that the R-module V is projective of finite type. Then the functor F_{ϕ;M,V} is representable by an affine scheme and the morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is affine.

Proof. (a) (i) Fix an object (S, R →^{s} S) of R\Alg_{k}. If s^{∗}(M) ⟶^{v} V′ belongs to F_{ϕ;M,V} (S, s), i.e. v ∘ s^{∗}(ϕ) is an isomorphism, then for any morphism (S, s) ⟶^{h} (T, t), the composition h^{∗}(v) ∘ h^{∗}s^{∗}(ϕ) is an isomorphism, and h^{∗}s^{∗}(ϕ) ≃ t^{∗}(ϕ). There is a natural morphism F_{ϕ;M,V} ⟶ Gr_{M,V}.

(ii) Note that one can identify F_{ϕ;M,V} (S, s) with the set of all S-module epimorphisms

s^{∗}(M) ⟶^{v} s^{∗}(V) such that v ∘ s^{∗}(ϕ) = id_{s^{∗}(V)}. In fact, if s^{∗}(M) ⟶^{v′} V′ is such that w = v′ ∘ s^{∗}(ϕ) : s^{∗}(V) ⟶ V′ is an isomorphism, then v = w^{−1} ∘ v′ has the required property.

(iii) One of the consequences of the observation (ii) is that the canonical morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is a monomorphism.

(b) There are two maps,

Hom_{S}(s^{∗}(M), s^{∗}(V)) \overrightarrow{⟶}^{α_{S}}_{β_{S}} Hom_{S}(s^{∗}(V), s^{∗}(V)),

defined by v ⟼^{α_{S}} v ∘ s^{∗}(ϕ), v ⟼^{β_{S}} id_{s^{∗}(V)}. The maps α_{S} and β_{S} are functorial in (S, s), hence they define morphisms, resp. α and β, from the functor

S ⟼ Hom_{S}(s^{∗}(M), s^{∗}(V)) ≃ Hom_{R}(M, s_{∗}s^{∗}(V)) = Hom_{R}(M, S ⊗_{R} V)

to the functor

S ⟼ Hom_{S}(s^{∗}(V), s^{∗}(V)) ≃ Hom_{R}(V, s_{∗}s^{∗}(V)) = Hom_{R}(V, S ⊗_{R} V).

(iv) Suppose now that V is a projective R-module of finite type. Then, by 10.3.2, the first functor is representable by the vector fiber 𝕍_{R}(M ⊗_{k} V^{∨}) of the left R^{e}-module M ⊗_{k} V^{∨}, and the second one is representable by the vector fiber 𝕍_{R}(V ⊗_{k} V^{∨}) of the projective R^{e}-module of finite type V ⊗_{k} V^{∨}. Let α′ and β′ be morphisms from 𝕍_{R}(M ⊗_{k} V^{∨}) to 𝕍_{R}(V ⊗_{k} V^{∨}) corresponding to resp. α and β. The functor F_{ϕ;M,V} is the kernel of the pair (α, β), hence it is representable by the kernel, F_{ϕ;M,V}, of the pair (α′, β′).

(v) The functor morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is representable by an affine morphism; i.e. for any R-ring (S, R →^{s} S) and any functor morphism h_{(S,s)} ⟶ Gr_{M,V}, the functor

R\Alg_{k} ⟶ Sets, (T, t) ⟼ F_{ϕ;M,V} (T, t) ∏_{Gr_{M,V} (T,t)} h_{(S,s)}(T, t)

is representable by an affine subscheme of SpecS.

In fact, by the Yoneda’s lemma, any morphism h_{(S,s)} ⟶ Gr_{M,V} is uniquely determined by an element of Gr_{M,V} (S, s), i.e. by the equivalence class, [v], of a locally split epimorphism s^{∗}(M) ⟶^{v} V′. The corresponding map h_{(S,s)}(T, t) ⟶ Gr_{M,V} (T, t)

sends any morphism (S, s) ⟶^{f} (T, t) to the equivalence class [f^{∗}(v)]. The fiber product

F_{ϕ;M,V} (T, t) ∏_{Gr_{M,V} (T,t)} h_{(S,s)}(T, t) consists of all pairs (w, γ), where γ ∈ h_{(S,s)}(T, t) and

[T ⊗_{k} M ⟶^{w} T ⊗_{k} V] are such that w ∘ (T ⊗_{k} ϕ) = id_{T ⊗_{k}V} and w = γ^{∗}(v). Since v and ϕ here are fixed, the fiber product F_{ϕ;M,V} (T, t) ∏_{Gr_{M,V} (T,t)} h_{(S,s)}(T, t) can be identified with the set of all morphisms (S, s) ⟶^{γ} (T, t) of R\Alg_{k} (i.e. k-algebra morphisms S ⟶^{γ} T satisfying t = γ ∘s) such that γ^{∗}(v ∘(T ⊗_{k} ϕ)) = id_{T ⊗_{k}V}. In other words, this fiber product is identified with the kernel of the pair of morphisms

h_{(S,s)}(T, t) \overrightarrow{⟶}^{α_{(T,t)}}_{β_{(T,t)}} Hom_{T} (T ⊗_{R} V, T ⊗_{R} V)

defined by

β_{(T,t)} : γ ⟼ id_{t^{∗}(V)} = id_{T ⊗_{k}V} , α_{(T,t)} : γ ⟼ γ^{∗}(v) ∘ (T ⊗_{k} ϕ).

The morphisms β_{(T,t)}, α_{(T,t)} are functorial in (T, t), and Hom_{T} (T ⊗_{k} V, T ⊗_{k} V) ≃ 𝕍_{R}(V ⊗_{k} V^{∨})(T, t). Hence the morphisms β = (β_{(T,t)}), α = (α_{(T,t)}) define a pair of arrows

SpecS \overrightarrow{⟶}^{α′}_{β′} 𝕍_{R}(V ⊗_{k} V^{∨}), and the functor (T, t) ⟼ F_{ϕ;M,V} (T, t) ∏_{Gr_{M,V} (T,t)} h_{(S,s)}(T, t)

is representable by the kernel of the pair (α′, β′).

10.8.2. Proposition. Let M ⟶^{ϕ} V be an R-module morphism, and let V be a projective R-module of finite type. If M is a finitely presentable R-module (resp. an Rmodule of finite type), then F_{ϕ;M,V} is locally finitely presentable (resp. locally of finite type) over R.

Proof. By the part (iv) of the argument of 10.8.1, the functor F_{ϕ;M,V} is isomorphic to the kernel of a pair of arrows 𝕍_{R}(M ⊗_{k} V^{∨}) \overrightarrow{⟶} 𝕍_{R}(V ⊗_{k} V^{∨}) over R. By 10.3.4, 𝕍_{R}(V ⊗_{k} V^{∨}) is locally finitely presentable over R, and 𝕍_{R}(M ⊗_{k} V^{∨}) is locally finitely presentable (resp. locally of finite type), if the R-module M is finitely presentable (resp. of finite type). The kernel of a pair of morphisms between locally finitely presentable functors (resp. functors locally of finite type) is locally finitely presentable (resp. locally of finite type); hence the assertion.

10.8.2.1. Corollary. Suppose M and V are projective R-modules of finite type. Then the canonical morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is locally finitely presentable.

Proof. By 10.4.3.1, Gr_{M,V} is locally finitely presentable over R, and by 10.8.2, F_{ϕ;M,V} has the same property. By 5.12.2(d), the morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is locally finitely presentable.

10.8.3. Proposition. Let M be a projective R-module and V a projective Rmodule of finite type. Then F_{ϕ;M,V} is formally smooth over R and the canonical morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is a formally open immersion.

If M is a projective module of finite type, then F_{ϕ;M,V} is smooth over R and the morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is an open immersion.

Proof. (a) Let M be a projective R-module. Since by 10.5.1, Gr_{M,V} is formally smooth over R and the composition of formally smooth morphisms is formally smooth, the formal smoothness of F_{ϕ;M,V} over R is a consequence of the formal smoothness of the canonical morphism F_{ϕ;M,V} ⟶ Gr_{M,V}.

Fix an R-ring epimorphism (T, t) ⟶^{α} (S, s) with a nilpotent kernel which is a part of a commutative diagram

\begin{matrix}h_{(S,s)} & ⟶ & F_{ϕ;M,V} \\ α ↓ & & ↓ \\ h_{(T,t)} & ⟶ & Gr_{M,V} \end{matrix}  (1)

By Yoneda’s lemma, the morphism h_{(S,s)} ⟶ F_{ϕ;M,V} in (1) is uniquely defined by an

element of F_{ϕ;M,V}, i.e. by an S-module morphism s^{∗}(M) ⟶^{u′} s^{∗}(V) such that u′ ∘ s^{∗}(ϕ) = id_{s^{∗}(V)} and morphism, and the morphism h_{(T,t)} ⟶ Gr_{M,V} is uniquely defined by an element of Gr_{M,V} (T, t). The commutativity of (1) is equivalent to the commutativity of the diagram

\begin{matrix}T ⊗_{R} V & ⟶^{t^{∗}(ϕ)} & T ⊗_{R} M & ⟶^{u} & T ⊗_{R} V \\ α_{V} ↓ & & α_{M} ↓ & & ↓ α_{V} \\ S ⊗_{R} V & ⟶^{s^{∗}(ϕ)} & S ⊗_{R} M & ⟶^{u′} & S ⊗_{R} V \end{matrix}  (2)

whose vertical arrows are induced by the ring epimorphism T ⟶^{α} S, hence they are epimorphisms. Since T ⊗_{R} V is a projective T module of finite type and the kernel of T ⟶^{α} S is nilpotent, it follows from Nakayama’s lemma that u ∘ t^{∗}(ϕ) is an isomorphism. Set ũ = (u ∘ t^{∗}(ϕ))^{−1} ∘ u. Then ũ ∘ t^{∗}(ϕ) = id_{t^{∗}(V)} and the diagram

\begin{matrix}T ⊗_{R} V & ⟶^{t^{∗}(ϕ)} & T ⊗_{R} M & \underrightarrow{_{u}} & T ⊗_{R} V \\ α_{V} ↓ & & α_{M} ↓ & & ↓ α_{V} \\ S ⊗_{R} V & ⟶^{s^{∗}(ϕ)} & S ⊗_{R} M & ⟶^{u′} & S ⊗_{R} V \end{matrix}  (3)

commutes. The pair of arrows

T ⊗_{R} V ⟶^{t^{∗}(ϕ)} T ⊗_{R} M \underrightarrow{_{u}} T ⊗_{R} V  (4)

is an element of F_{ϕ;M,V} (T, t) which corresponds to a morphism h_{(T,t)} ⟶^{γ} F_{ϕ;M,V}. It follows from the construction that adjoining the morphism γ to the diagram (1) makes a commutative diagram. This shows that the canonical monomorphism F_{ϕ;M,V} ⟶ Gr_{M,V} is formally smooth, hence a formally open immersion.

(b) Suppose now that M is a projective R-module of a finite type. Then by 10.8.2, the functor F_{ϕ;M,V} is locally finitely presentable over R, and by 10.8.2.1, the morphism F_{ϕ;M,V} ⟶ Gr_{M,V} is locally finitely presentable. Therefore, F_{ϕ;M,V} is smooth over R and F_{ϕ;M,V} ⟶ Gr_{M,V} is an open immersion.

10.8.4. Projective completion of a vector bundle. Let M′ = M ⊕ V, and let V ⟶^{j_{V}} M′ be the canonical morphism. The functor F_{j_{V} ;M′,V} is isomorphic to the functor which assigns to any R-ring (S, R →^{s} S) the set Hom_{S}(s^{∗}(M), s^{∗}(V) (cf. (ii) and (b) in the argument of 10.8.1). The latter functor is representable by the vector bundle 𝕍_{R}(M ⊗_{k} V^{∨}). If the R-modules M and V (hence M′) are projective of finite type, then, by 10.8.3, we have a canonical affine open immersion 𝕍_{R}(M ⊗_{k} V^{∨}) ⟶ Gr_{M′,V}.

In particular, if R = k, than taking V = R^{1} = k^{1}, we obtain a canonical immersion 𝕍(M) ⟶ ℙ_{M⊕R^{1}}. The projective space ℙ_{M⊕R^{1}} can be regarded (as in the commutative case) as the projective completion of the vector bundle 𝕍_{R}(M).

10.8.5. Zero section and the hyperplane at infinity. Let M, V be R-modules. Set M′ = M ⊕ V, and let V ⟵^{p_{V}} M ⊕ V ⟶^{p_{M}} M be canonical projections. The projection p_{V} determines a canonical section SpecR ⟶ Gr_{M′,V} which (following the commutative tradition) will be called the zero section. The projection M′ ⟶^{p_{M}} M induces a closed embedding Gr_{M,V} ⟶ Gr_{M′,V} called the hyperplane at infinity.

10.9. Grassmannians are separated. Recall that a presheaf of sets X on a category C is separated if the canonical morphism Δ_{X} : X ⟶ X ×X is a closed immersion (cf. 7.3, 7.4). Here C is the category Aff_{k}/SpecR of affine k-schemes over R for some associative k-algebra R. In other words, C = (R\Alg_{k})^{op}.

10.9.1. Proposition. For any pair M, V of projective R-modules of finite type, the presheaf Gr_{M,V} : (Aff_{k}/SpecR)^{op} = R\Alg_{k} ⟶ Sets is separated.

Proof. Let (S, R →^{s} S) be an R-ring, and let h_{(S,s)} \overrightarrow{⟶}^{u_{1}}_{u_{2}} Gr_{M,V} be a pair of

morphisms over R. The claim is that the kernel of the pair (u_{1}, u_{2}) is representable by a closed immersion of affine schemes.

Let s^{∗}(M) ⟶^{ξ_{i}} s^{∗}(V) be an epimorphism corresponding to u_{i}, i = 1, 2. Since s^{∗}(V) is a projective S-module, there exists an S-module morphism s^{∗}(V) ⟶^{ν_{i}} s^{∗}(M) such that ξ_{i}ν_{i} = id_{s^{∗}(V)}. Set p_{i} = ν_{i}ξ_{i}. Then the diagram

s^{∗}(M) \overrightarrow{⟶}^{id}_{p_{i}} s^{∗}(M) ⟶^{ξ_{i}} s^{∗}(V)

is exact. Consider the pairs of morphisms

s^{∗}(M) \overrightarrow{⟶}^{ξ_{1}}_{ξ_{1}p_{2}} s^{∗}(V) and s^{∗}(M) \overrightarrow{⟶}^{ξ_{2}}_{ξ_{2}p_{1}} s^{∗}(V).  (1)

There exists a universal R-ring morphism (S, s) ⟶^{ψ} (T, t) such that the image by ψ^{∗} of each of the pairs (1) belongs to the diagonal. We leave to the reader arguing that the morphism ψ is a closed immersion.

Appendix 1: Complements on Q-categories.

A1.1. A Q-category and a Q^{o}-category associated with a fully faithful functor. Let G : A ⟶ B be a fully faithful functor.

A1.1.1. Denote by Ā the full subcategory of B objects of which are all y ∈ ObB such that the functor B(G(−), y) : A^{op} ⟶ Sets is representable. Denote by u^{∗} the corestriction of G to the subcategory Ā. Assigning to each y ∈ ObĀ an object, u_{∗}(y), which represents the functor B(G(−), y), determines a functor u_{∗} : Ā ⟶ A right adjoint to u^{∗}.

A1.1.2. Dually, let \underline{A} be the full subcategory of B whose objects are all z ∈ ObB such that the functor B(z, G(−)) : A ⟶ Sets is corepresentable. Let \underline{u}^{∗} denote the corestriction of G to \underline{A}. The functor \underline{u}^{∗} has a left adjoint, \underline{u}_{!}. The pair of functors \underline{u}^{∗}, \underline{u}_{!} defines a Q^{o}-category A ⇄ \underline{A}.

A1.1.3. Example. Fix a small category D. Let B be the category D^{A} of functors D ⟶ A. For any x ∈ ObA, denote by i_{x}^{D} the constant functor D ⟶ A, i_{x}^{D}(α) = id_{x} for all morphisms α of D. The map x ⟼ i_{x}^{D} defines a fully faithful functor, i^{D}, from A to D^{A}. The category Ā of A1.1.1 is the full subcategory of D^{A} formed by all functors D : D ⟶ A such that lim(D) exists. The functor u_{∗} : Ā ⟶ A, a right adjoint to u^{∗} = i^{D}|Ā, assigns to each functor D of Ā its limit.

Dually, the category \underline{A} of A1.1.2 is the full subcategory of D^{A} formed by all functors D′ : D ⟶^{D′} A such that colim(D′) exists. The functor \underline{u}_{!}, a left adjoint to \underline{u}^{∗} = i^{D}|\underline{A}, assigns to each functor D′ of \underline{A} its colimit.

If the category A has limits and colimits of functors D ⟶ A, then Ā = D^{A} = \underline{A}. In this case, we write u_{!} instead of \underline{u}_{!}. The canonical morphism 𝔯_{u} : u_{∗} ⟶ u_{!} (cf. 2.5) assigns to any functor D : D ⟶ A the natural morphism lim D ⟶ colim D.

The simplest example of such situation is D = (· → ·) and A an arbitrary category. We recover the Q-category (A^{2} ⇄ A) of morphisms of A (cf. 2.5).

A1.1.4. Example. Let A be a category. Denote by A^{·⇉·} the category of functors (· ⇉ ·) ⟶ A. Let u^{∗} denote the functor A ⟶ A^{·⇉·}, x ⟼ (x ⇉ x), where both arrows are identical. Suppose that the category A has kernels of pairs of morphisms. Then the functor u^{∗} has a right adjoint, u_{∗} : (x ⇉ y) ⟼ Ker(x ⇉ y). If the category A has cokernels of pairs of arrows, u^{∗} has a left adjoint, u_{!} : (x ⇉ y) ⟼ Cok(x ⇉ y).

A1.2. Fully faithful functors and presheaves of sets. For a category A, let A^{∧} denote the category of presheaves of sets on A, i.e. the category of contravariant functors from A to Sets. Let g^{∗} : A ⟶ B be a functor. The functor g^{∗∧} : B^{∧} ⟶ A^{∧}, X ⟼ X ∘ g^{∗}, admits a right and a left adjoint, resp. ĝ_{∗} and g^{∗}.

(i) The composition of g^{∗} with canonical embedding A ↪ A^{∧} is isomorphic to g^{∗}. For any presheaf X ∈ ObA^{∧}, we have:

g^{∗}(X)(Y) ≃ colim_{(V,ξ)∈A/X}B(Y, g^{∗}(V)).  (1)
If the functor g^{∗} has a left adjoint, g_{!}, then g^{∗} ≃ g_{!}^{∧} : Y ⟼ Y ∘ g_{!}.

(ii) The functor g^{∗} is fully faithful iff the functor g^{∗} is fully faithful.

Thus, if (and only if) g^{∗} is fully faithful, the pair of functors A^{∧} \underrightarrow{g^{∗}} B^{∧} \underrightarrow{_{g_{∗}}} A^{∧} forms a Q-category.

(iii) Note that if the functor g^{∗} is fully faithful and has a left adjoint, g_{!}, we recover a special case of A1.1, when C = Sets.

A1.2.1. The dual picture. Let g^{∗} : A ⟶ B be a fully faithful functor. Applying to the dual functor, g^{∗op} : A^{op} ⟶ B^{op} the considerations of A1.1, we obtain a Q^{o}-category (Ā^{∨} ⇄ A^{∨}) = (Ā^{∨} ⇄ A^{∨}).

A1.3. The Q-category of presheaves of sets. Let 𝔸 = (Ā ⇄^{u} A) be a Qcategory, u = (u^{∗}, u_{∗}). It follows from A1.2 that the functor u^{∗} determines a Q-category

𝔸^{∧} = (Ā^{∧} _{u} ; ⇄ A^{∧}), where û = (u^{∗}, û_{∗}). We call 𝔸^{∧} the Q-category of presheaves of sets on 𝔸. The canonical (Yoneda) embeddings A ↪ A^{∧}, Ā ↪ Ā^{∧} define a Q-category embedding, h_{𝔸} : 𝔸 ⟶ 𝔸^{∧}, which we call Yoneda embedding too.

A1.3.1. The Q-category of presheaves of sets and the Q-category of functors. Let C be a category with small colimits. For any category, B, the composition with the Yoneda embedding, h_{B} : B ⟶ B^{∧}, induces an equivalence between the category C^{B} of functors B ⟶ C and the category C_{𝔠}^{B^{∧}} of functors B^{∧} ⟶ C having a right adjoint

(see [GZ], Proposition II.1.3). In particular, for any Q-category 𝔸 = (Ā ⇄^{u} A), the Yoneda embedding h_{𝔸} : 𝔸 ⟶ 𝔸^{∧} induces an equivalence between the Q-category of functors C^{𝔸} and the Q-category C_{𝔠}^{𝔸^{∧}} = (CĀ^{∧} ; 𝔠 ⇄ C_{𝔠}^{A^{∧}}) formed by functors resp. Ā^{∧} ⟶ C and A^{∧} ⟶ C having a right adjoint.

A1.3.2. The dual realization of the Q-category of functors. Suppose C is a category with small limits. The category C^{B} of functors B ⟶ C is isomorphic to the category (C^{op})^{B^{op}}. By (the dual version of) Proposition II.1.3 in [GZ], the composition with the Yoneda embedding h^{B} : B^{op} ⟶ (B^{op})^{∧} = B^{∨} induces an equivalence between the category C^{B} and the category (C^{op})_{𝔠}^{B^{∨}} of functors (B^{∨}) ⟶ C^{op} having a right adjoint. The category (C^{op})_{𝔠}^{B^{∨}} is naturally isomorphic to the category C_{𝔠𝔠}^{(B^{∨})^{op}} of functors (B^{∨})^{op} ⟶ C (i.e. presheaves on B^{∨} with values in the category C) having a left adjoint.

It follows that for any Q-category 𝔸 = (Ā ⇄^{u} A), the Yoneda embedding h^{𝔸} : 𝔸^{op} ⟶ 𝔸^{∨} induces an equivalence between the Q-category of functors C^{𝔸} and the Q-category C_{𝔠𝔠}^{(𝔸^{∨})^{op}} = (C_{𝔠𝔠}^{(Ā^{∨})^{op}} ⇄ C_{𝔠𝔠}^{(A^{∨})^{op}}) formed by presheaves on resp. Ā^{∨} and A^{∨} with values in C having a left adjoint.

A1.4. The functor u^{!} and the Q-category 𝔸^{!}. Denote by A^{!} the full subcategory of the category A whose objects are all x ∈ ObA such that the category u_{∗}/x has a final object, u^{!}(x). The map x ⟼ u^{!}(x) is extended to a functor, u^{!} : A^{!} → Ā, defined uniquely up to isomorphism. If A^{!} = A, then u^{!} is a right adjoint to the functor u_{∗}.

In the general case, let Ā^{!} denote the full subcategory of Ā whose objects are all x̄ ∈ Ā such that u_{∗}(x̄) ∈ ObA^{!}. Since the adjunction arrow η_{u} : Id_{A} → u_{∗}u^{∗} is an isomorphism,

u^{∗}(A^{!}) ⊆ Ā^{!}. Thus we have a Q-subcategory, 𝔸^{!} = (Ā^{!} ⇄^{𝔲} A^{!}), of the Q-category 𝔸 such that the functor 𝔲^{∗} has a right adjoint, 𝔲^{!} = u^{!}.

A1.4.1. Proposition. The functor u^{!} is fully faithful, or, equivalently, a canonical morphism ϵ_{u}^{!} : u_{∗}u^{!} → J_{A^{!}} is an isomorphism. Here J_{A^{!}} denotes the inclusion functor A^{!} → A.

Proof. Passing to the Q-subcategory 𝔸^{!}, we can assume that A^{!} = A, hence J_{A^{!}} is the identical functor A → A. The functor u_{∗} is a localization (since it has a fully faithful left adjoint). Therefore the functor u^{!}, being a right adjoint to a localization, is fully faithful ([GZ], I.4).

A1.5. Sheaves in 𝔸^{!}. There is a canonical morphism ρ_{u} : u^{∗}|_{A^{!}} → u^{!} corresponding to the isomorphism η_{u}^{−1} : u_{∗}u^{∗} → Id_{A}. By definition, 𝔉𝔸 is the full subcategory of A^{!} whose objects are all x ∈ ObA^{!} such that the canonical morphism u^{∗}(x) → u^{!}(x) is an isomorphism. In particular, the category 𝔉𝔸 coincides with the category 𝔉𝔸^{!} of sheaves in the Q-subcategory 𝔸^{!} of 𝔸.

A1.5.1. Lemma. The following conditions are equivalent:

(i) 𝔉𝔸 = A. (ii) The functor u_{∗} has a right adjoint, u^{!}, and the canonical morphism u^{∗} → u^{!} is an

isomorphism.

(iii) The functor u^{∗} has a left adjoint, u_{!}, which is isomorphic to u_{∗}.

Proof. The equivalence of (i) and (ii) follows from definitions. (ii) ⇔ (iii). The isomorphism u^{∗} → u^{!} induces an isomorphism of the corresponding

left adjoint functors, u_{∗} → u_{!} (in particular, it induces the existance of a left adjoint, u_{!}, of u^{∗}). Conversely, if u^{∗} has a left adjoint, u_{!}, and the latter is isomorphic to u_{∗}, then, obviously, u^{∗} is a right adjoint to u_{∗}, i.e. u^{∗} ≃ u^{!}.

A1.6. Note. Set (𝔉̄𝔸 = u_{∗}^{−1}(𝔉𝔸). The Q-subcategory, (𝔉̄𝔸 ⇄ 𝔉𝔸), of 𝔸 induced by the inclusion functor 𝔉𝔸 → A enjoys the equivalent properties of A1.5.1.

A1.7. The functor H_{u}. Fix a Q-category 𝔸 = (Ā ⇄^{u} A) such that the functor u^{∗} has a left adjoint, u_{!}, and the functor u_{∗} has a right adjoint, u^{!}. Denote by H_{u} the functor u_{!}u^{!} : A ⟶ A. Let τ_{u} : Id_{A} ⟶ H_{u} be the composition of the isomorphism Id_{A} ⟶ u_{!}u^{∗}, the inverse to the adjunction isomorphism, η_{!}^{u} : u_{!}u^{∗} ⟶ Id_{A}, and the morphism u_{!}(ρ_{u}) : u_{!}u^{∗} ⟶ u_{!}u^{!} = H_{u}, where ρ_{u} : u^{∗} ⟶ u^{!} is the canonical morphism (cf. A1.5).

A1.7.1. Proposition. The following conditions on x ∈ ObA are equivalent:

(a) x is an 𝔸-sheaf.

(b) The canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) is an isomorphism.

(c) u^{!}(x) ≃ u^{∗}(y) for some y ∈ ObA.

(d) The morphism τ(x) : x ⟶ H_{u}(x) is an isomorphism.

Proof. (a)⇔(b) follow from definitions. (b)⇒(c) is obviously true. (c)⇒(b). If u^{!}(x) ≃ u^{∗}(y), then u_{∗}u^{!}(x) ≃ u_{∗}u^{∗}(y) ≃ y. By A1.4.1, the adjunction

morphism Id_{A} ⟶ u_{∗}u^{!} is an isomorphism. In particular, x ≃ u_{∗}u^{!}(x), hence x ≃ y. (b)⇒(d) by definition of the morphism τ_{u}.

(d)⇒(b). One can check that u^{∗}(τ_{u}) = λ_{u} ∘ ρ_{u} and u^{!}τ_{u} = ρ_{u}H_{u} ∘ λ_{u}u^{!}. Here λ_{u} : IdĀ → u^{!}u_{∗} is an adjunction morphism. Since τ_{u}(x) is an isomorphism, it follows from the first equality that ρ_{u}(x) is a strict monomorphism, and from the second one, that ρ_{u}H_{u}(x), hence ρ_{u}(x), is a strict epimorphism. Thus ρ_{u}(x) is an isomorphism.

A1.7.2. Proposition. 1) Suppose u_{∗} has a right adjoint, u^{!}. Then the following conditions on x ∈ ObA are equivalent:

(a) x is an 𝔸-monopresheaf.

(b) The canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) is a monomorphism.

(c) There exists a monomorphism, ξ : u^{∗}(x) ⟶ u^{!}(y), for some y ∈ ObA.

2) Suppose both functors, u^{!} and u_{!}, exist. If the morphism τ(x) : x ⟶ H_{u}(x) is a monomorphism, then x is an 𝔸-monopresheaf. The converse is true under the condition that the functor u_{!} maps monomorphisms to monomorphisms.

Proof. 1) (a)⇔(b)⇒(c) follow from definitions.

(c)⇒(b). The morphism ξ decomposes uniquely as the canonical morphism ρ_{u}(x) : u^{∗}(x) ⟶ x and a morphism g : x ⟶ y. Since the composition g ∘ ρ_{u}(x) is a monomorphism, ρ_{u}(x) is a monomorphism.

2) Since u^{∗} is left exact (as a functor having a left adjoint), u^{∗}(τ_{u}(x)) is a monomorphism, if τ_{u}(x) is a monomorphism. But u^{∗}(τ_{u}(x)) = λ_{u}(x) ∘ ρ_{u}(x), hence ρ_{u}(x) is a monomorphism.

By definition, τ_{u} = (η_{!}^{u})^{−1} ∘ u_{!}(ρ_{u}) (cf. A1.7). Thus, if ρ_{u}(x) is a monomorphism and u_{!} maps monoarrows to monoarrows, τ_{u}(x) is a monomorphism too.

A1.7.3. Lemma. If x is an 𝔸-epipresheaf (i.e. ρ_{u}(x) : u^{∗}(x) ⟶ u^{!}(x) is a strict epimorphism, cf. 3.1.4), then τ_{u}(x) : x ⟶ H_{u}(x) is a strict epimorphism.

Proof. The functor u_{!} maps strict epimorphisms to strict epimorphisms (as any functor having a right adjoint). In particular, if ρ_{u}(x) is a strict epimorphism, then τ_{u}(x) = η_{!}^{u}(x)^{−1} ∘ u_{!}(ρ_{u}(x)) is a strict epimorphism.

A1.8. Sheafification functors. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, 𝔉𝔸 the category of 𝔸-sheaves. A sheafification functor is a left adjoint to the inclusion functor 𝔉𝔸 ↪ A (if any). We are interested here in the case of Q-categories C^{𝔸} and mostly for C = Sets. We shall denote by 𝔉(𝔸, C) the category of sheaves on a Q-category 𝔸 with values in C. And we shall write 𝔉𝔸^{∨} in the case C = Sets.

A1.8.1. Existence.

A1.8.1.1. Lemma. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that u^{∗} preserves limits of certain type. Then the subcategory 𝔉𝔸 of sheaves in 𝔸 is closed under limits of this type (taken in A).

Proof. This follows from the characterization of objects of 𝔉𝔸 as those x ∈ ObA for which the canonical morphism u^{∗}(x) ⟶ u^{!}(x) is an isomorphism and the fact that the functor u^{!} (defined on a full subcategory, A^{!}, of A) preserves all limits (taken in A) of all small diagrams D ⟶ A^{!}.

A1.8.1.2. Lemma. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that A has small limits and the functor u^{∗} preserves small limits. Suppose, in addition that for any object x of A, there exists an 𝔸-sheaf, y, and a morphism x → y. Then the inclusion functor 𝔉𝔸 ↪ A has a left adjoint.

Proof. The assertion follows from A1.8.1.1 and the Freyd’s criterion of the existence of a left adjoint functor to an inclusion functor (cf. [GZ]).

A1.8.1.3. Corollary. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that the functor u^{∗} has a left adjoint, u_{!}, and A has small limits and a final object (the latter follows from the existence of small limits if A is equivalent to a small category). Then there exists a left adjoint to the inclusion functor 𝔉𝔸 ↪ A.

Proof. The existence of a left adjoint, u_{!} to the functor u^{∗} garantees that u^{∗} preserves limits. Let • denote the final object to the category A. Notice that any functor having a left adjoint maps a final object to a final object. In particular, u^{∗}(•) is a final objects of Ā. This implies that u^{∗}(•), η_{u}(•) is a final object of the category u_{∗}/•, i.e. • is an 𝔸-sheaf. Since, by definition, any object of A has a morphism to •, the conditions of A1.8.1.2 are fulfilled, hence the assertion.

A1.8.1.4. Proposition. Let C be a category with small limits and a final object (for instance, C = Sets, or C = k − mod for some ring k). Then for any Q-category

𝔸 = (Ā ⇄^{u} A), the inclusion functor 𝔉(𝔸, C) ↪ C^{𝔸} has a left adjoint.

Proof. Let • denote the final object of the category C. For any category B, the constant functor with value in • is a final object of the category B^{C} of functors from B to C. The functor u^{∗} : F ⟼ F ∘ u_{∗} preserves small limits and maps the final object of the category C^{A} to the final object of CĀ. This implies, in particular, that the final object of C^{A} is a sheaf on 𝔸. The assertion follows from A1.8.1.2 and (the argument of) A1.8.1.3.

A1.8.2. Construction. The functor H_{u} is used to construct a sheafification functor. We need the following technical fact.

A1.8.2.1. Proposition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, such that both functors, u_{!} and u^{!} exist. Let τ_{u} : Id_{A} ⟶ H_{u} = u_{!}u^{!} be the canonical morphism (cf. A1.7.1). Then H_{u}τ_{u} = τ_{u}H_{u}.

Proof. There is a commutative diagram of canonical morphisms

\begin{matrix}u_{!}u^{∗} & ⟶^{u_{!}η_{u}^{∗}u^{∗}} & H_{u}u_{∗}u^{∗} \\ η_{u}^{!} ↓ & & ↑ H_{u}η_{u} \\ Id_{A} & ⟶^{τ_{u}} & H_{u} \\ ϵ_{u}^{∗} ↑ & & ↓ η_{u}H_{u} \\ u_{∗}u^{!} & ⟶^{u_{∗}𝔯_{u}u^{!}} & u_{∗}u^{∗}H_{u} \end{matrix}  (1)

The upper diagram follows from the definition of τ_{u} (in A1.7.1), τ_{u} = u_{!}ρ_{u} ∘ (η_{u}^{!})^{−1}, and the equality ρ_{u} = u^{!}η_{u}^{−1} ∘ η_{u}^{∗}u^{∗}. The diagram (1) provides two formulas for τ_{u}:

(a) τ_{u} = η_{u}^{−1}H_{u} ∘ u_{∗}λ_{u}u^{!} ∘ (ϵ_{u}^{∗})^{−1}

(b) τ_{u} = H_{u}η_{u}^{−1} ∘ u_{!}η_{u}^{∗}u^{∗} ∘ (η_{u}^{!})^{−1}.

Applying H_{u} from the left to the first expression and from the right to the second one, we obtain respectively:

(a’) H_{u}τ_{u} = H_{u}η_{u}^{−1}H_{u} ∘ u_{!}(u^{!}u_{∗}𝔯_{u}u^{!} ∘ u^{!}ϵ_{u}^{∗−1}) = H_{u}η_{u}^{−1}H_{u} ∘ u_{!}(u^{!}u_{∗}λ_{u}u^{!} ∘ η_{u}^{∗}u^{!} =

H_{u}η_{u}^{−1}H_{u} ∘ u_{!}(u^{!}u_{∗}λ_{u} ∘ η_{u}^{∗})u^{!} since u^{!}ϵ_{u}^{∗−1} = η_{u}^{∗}u^{!}.

(b’) τ_{u}H_{u} = H_{u}η_{u}^{−1}H_{u} ∘ (u_{!}η_{u}^{∗}u^{∗}u_{!} ∘ η_{u}^{!−1}u_{!})u^{!} = H_{u}η_{u}^{−1}H_{u} ∘ (u_{!}η_{u}^{∗}u^{∗}u_{!} ∘ u_{!}𝔯_{u})u^{!} = H_{u}η_{u}^{−1}H_{u} ∘ u_{!}(η_{u}^{∗}u^{∗}u_{!} ∘ λ_{u})u^{!} since η_{u}^{!−1}u_{!} = u_{!}𝔯_{u}. Thus the right hand sides of (a’)

and (b’) coincide as claimed.

A1.8.2.2. Corollary. Under conditions of A1.8.2.1, H_{u}^{n}τ_{u}H_{u}^{m} = τ_{u}H_{u}^{n+m} for any nonnegative integers m, n.

We call functors which map monomorphisms to monomorphisms monofunctors.

A1.8.2.3. Corollary. (a) Suppose H_{u} is a monofunctor and x ∈ ObA is such that τ_{u}(x) is a monomorphism. Then τ_{u}(H_{u}(x)) is a monomorphism.

(b) Let u_{!} be a monofunctor. Then the subcategory 𝔐𝔸 ↪ A of 𝔸-monopresheaves is H_{u}(x)-invariant.

Proof. The assertion (a) is a consequence of the equality H_{u}τ_{u} = τ_{u}H_{u}.

(b) Since u_{!} is a monofunctor, H_{u} = u_{!}u^{!} is a monofunctor too. By A1.7.2, an object x is an 𝔸-monopresheaf iff τ_{u}(x) is a monomorphism, hence the assertion.

We set τ_{u}^{1} = τ_{u}, and define morphisms τ_{u}^{n} : Id_{A} ⟶ H_{u}^{n}, n ≥ 2, by τ_{u}^{n} = H_{u}τ_{u}^{n−1}. Let H_{u}^{∼} denote the functor ℕ ⟶ A, n ⟼ H_{u}^{n}, (n → m) ⟼ H_{u}^{n}τ_{u}^{m−n}. The colimit of H_{u}^{∼}(x) (when it exists) will be denoted by H_{u}^{∞}.

A1.8.2.4. Proposition. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category such that there exist functors u_{!} and u^{!}.

(a) Suppose u^{!} commutes with colimits of functors ℕ → A. Then object x of A has an associate 𝔸-sheaf iff the object H_{u}^{∞}(x) = colimH_{u}^{∼} exists. In the latter case, H_{u}^{∞} is an 𝔸-sheaf associated to x.

(b) Let u^{!} commute with colimits of monofunctors ℕ → A. Then an 𝔸-monopresheaf x has an associate sheaf iff the H_{u}^{∞}(x) exists. In this case, H_{u}^{∞}(x) is an 𝔸-sheaf associated to x.

Proof. (a) Suppose H_{u}^{∞}(x) exists. By hypothesis, the functor u^{!} preserves the colimit of functors ℕ ⟶ A, in particular u^{!} preserves the colimit of the functor H_{u}^{∼}(x). But then, since u_{!} preserves all colimits, the functor H_{u} preserves colimit of H_{u}^{∼}(x); i.e. the canonical morphism colim(H_{u} ∘ H_{u}^{∼}(x) ⟶ H_{u}(colimH_{u}^{∼}) is an isomorphism. But, obviously, the colimit of H_{u} ∘ H_{u}^{∼}(x) is naturally isomorphic to H_{u}^{∞}(x). This implies that the canonical morphism τ_{u}(H_{u}^{∞}(x)) : H_{u}^{∞}(x) ⟶ H_{u}(H_{u}^{∞}(x)) is an isomorphism. By A1.7.1, this means that H_{u}^{∞}(x) is an 𝔸-sheaf.

Let x ⟶^{f} y be an arbitrary morphism of A such that y is an 𝔸-sheaf. By A1.7.1, the canonical morphism τ_{u}(y) : y ⟶ H_{u}(y) is an isomorphism. This implies that H_{u}^{∞}(y) exists and the canonical morphism τ_{u}^{∞}(y) : y ⟶ H_{u}^{∞}(y) is an isomorphism. Thus, the morphism x ⟶^{f} y is uniquely represented as the composition of τ_{u}^{∞}(x) : x ⟶ H_{u}^{∞}(x) and a morphism H_{u}^{∞}(x) ⟶ y. Therefore H_{u}^{∞}(x) is an 𝔸-sheaf associated to x.

(b) Suppose H_{u}^{∞}(x) exists. Since x is an 𝔸-monopresheaf, the morphism τ_{u}(x) is a monomorphism (cf. A1.7.2). It follows from A1.8.2.2 that all morphisms H_{u}^{n}τ_{u}^{m−n} : H_{u}^{n}(x) ⟶ H_{u}^{m}(x) are monomorphisms. By hypothesis, the functor u^{!} preserves the colimit of monofunctors ℕ ⟶ A, in particular u^{!} preserves the colimit of the functor H_{u}^{∼}(x). Since u_{!} preserves all colimits, the functor H_{u} = u_{!}u^{!} preserves colimit of H_{u}^{∼}(x); i.e. the canonical morphism colim(H_{u} ∘ H_{u}^{∼}(x) ⟶ H_{u}(colimH_{u}^{∼}) is an isomorphism. A colimit of H_{u} ∘ H_{u}^{∼}(x) is naturally isomorphic to H_{u}^{∞}(x). By the argument (a) above, the object H_{u}^{∞}(x) is an 𝔸-sheaf associated to x.

A1.8.2.5. Proposition. Let 𝔸 = (Ā ⇄^{u} A) be a cosite. And let C be the category Sets, or the category k − mod of k-modules. Then

(a) For any functor A ⟶^{F} C, the functor H_{u}(F) is an 𝔸-monopresheaf.

(b) If F is a monopresheaf, then H_{u}(F) is a sheaf associated with F.

In particular, for any functor A ⟶^{F} C, the functor H2 ; _{u}(F) is a sheaf associated with F.

Proof. In the case 𝔸 is a cosite, the functor H_{u} is isomorphic to the Heller-Row functor, otherwise called ”+-construction” for which the assertion is a known fact.

A1.9. Subcanonical Q^{o}- and Q-categories. We call a Q^{o}-category 𝔸 = (Ā ⇄^{u} A) subcanonical if every representable presheaf on A is a sheaf on the associated quasi-site.

A Q-category is called subcanonical if its dual Q^{o}-category is subcanonical.

A1.9.1. Classical examples of subcanonical pretopologies. One of the most important examples is the category of (commutative) affine schemes with the Zariski, or étale, or fpqc pretopology. Recall that fpqc covers are families of affine scheme morphisms {ϕ_{i} : Spec(R_{i}) ⟶ Spec(R)| i ∈ J} such that all inverse image functors ϕ_{i}^{∗} : R − mod ⟶ R_{i} −mod are exact, and the family {ϕ_{i}^{∗}| i ∈ J} contains a finite conservative (i.e. reflecting isomorphisms) subfamily. In the case of Zariski topology, rings R_{i} are localizations at finitely generated multiplicative sets and ϕ_{i} universal morphisms. In the case of étale pretopology, all morphisms ϕ_{i} are étale.

It is well known that the fpqc topology is subcanonical. In particular, Zariski and étale topologies are subcanonical.

A1.9.2. A standard noncommutative example. Fix an associative ring k and take as A the category of rings over k, i.e. (unital) ring morphisms k ⟶ R. Objects of the category Ā are faithfully flat morphisms, i.e. morphisms ϕ : R ⟶ T of rings over k such that the inverse image functor ϕ^{∗} : R − mod ⟶ T − mod is exact and faithful. The functor u_{∗} : Ā ⟶ A is defined by (R ⟶^{ϕ} T) ⟼ R.

For a Q-category 𝔸, let 𝔗𝔸 denote the quasi-cosite associated with 𝔸 (cf. 2.4 and 3.9).

A1.9.2.1. Proposition. (a) The Q-category 𝔸 defined above is subcanonical.

(b) For any ȳ ∈ ObĀ, the canonical morphism 𝔯_{u} : u_{∗}(ȳ) ⟶ u_{!}(ȳ) induces a strict epimorphism of the corresponding 𝔸-spaces.

Proof. The assertions follow from [R4], Proposition 4.3.2.

A1.9.3. Subcanonical Q^{o}-categories in terms of covers. Consider a Q^{o}-category 𝔸_{τ}. We say that τ is subcaninical if 𝔸_{τ} is subcanonical.

Let τ be a subcanonical quasi-pretopology. Then (4) is an exact diagram of sheaves of sets. Taking in (5) F = A(−, x), we obtain the diagram

A(y, x) ⟶ ∏_{i∈I} A(y_{i}, x) \overrightarrow{⟶} ∏_{i,j∈I} A(y_{i} ×_{y} y_{j}, x).  (6)

Since τ is subcanonical, the presheaf A(−, x) is a sheaf for any x ∈ ObA, i.e. the diagram (6) is exact for any x ∈ ObA. But this means that the diagram (4) is an exact diagram of sheaves for any ȳ ∈ ObA_{τ}.

A1.9.3.2. Note. Suppose the object ȳ = (y_{i} → y| i ∈ I) of A_{τ} is such that there exist coproducts ∐_{i∈I} y_{i} and ∐_{i,j∈I} y_{i} ×_{y} y_{j}. Then the diagram (6) is isomorphic to the diagram

A(y, x) ⟶ A(∐_{i∈I} y_{i}, x) \overrightarrow{⟶} A( ∐_{i,j∈I} y_{i} ×_{y} y_{j}, x).  (7)

In this case, the condition “τ is subcanonical” implies the that the diagram

∐_{i,j∈I} y_{i} ×_{y} y_{j} \overrightarrow{⟶} ∐_{i∈I} y_{i} ⟶ y  (8)

is exact. Conversely, if for any ȳ = (y_{i} → y| i ∈ I) in 𝔗, the coproducts ∐_{i∈I} y_{i} and ∐_{i,j∈I} y_{i} ×_{y} y_{j} exist and the diagram (8) is exact, then τ is subcanonical.

A1.10. Q-categories and sites.

A1.10.1. Topologies on a given category. Let A be a category. Consider the set A_{ex}^{∧} of strictly full subcategories B of A^{∧} (strictly means that any object of A^{∧} isomorphic to an object of B belongs to B) such that the inclusion functor B ↪ A^{∧} has a left adjoint, i_{B}^{∗}, which is left exact, i.e. it preserves finite limits. Since i_{B}^{∗} preserves all colimits, it is exact. Recall the latter means that i_{B}^{∗} is a localization functor at a class of morphisms which admit left and right fractions [GZ, I.3.4].

On the other hand, denote by 𝔗𝔬𝔭/A the set of topologies on A. There is a canonical map

Ψ : A_{ex}^{∧} ⟶ 𝔗𝔬𝔭/A  (1)

defined as follows. To each subcategory B of A^{∧} such that the inclusion functor i_{∗} : B ↪ A^{∧} has an exact left adjoint, i^{∗} : A^{∧} ⟶ B, Ψ assigns a topology 𝔗_{B} such that for any object X of A, 𝔗_{B}(X) consists of all subobjects R of A(−X) such that i^{∗}(R ↪ A(−, X)) is an isomorphism.

A1.10.2. Theorem (Giraud) The map (1) is a bijection. The inverse map assigns to each topology 𝔗 on A the corresponding subcategory of sheaves of sets on the site (A, 𝔗).

Proof. See [SGA4, Exp.II], Theorem 5.5.

A1.10.3. The case of arbitrary Q-categories. Denote by A_{rex}^{∧} the set of all strictly full subcategories of A^{∧} such that the inclusion functor B ↪ A^{∧} has a (not necessarily exact) left adjoint, i_{B}^{∗}. Let B be a subcategory from A_{rex}^{∧}. To any X ∈ ObA, we assign the set Ā_{B}(X) of all subfunctors R of A(−, X) such that i_{B}^{∗}(R ↪ A(−X)) is an isomorphism. Denote by Ā_{B} the category whose objects are pairs (X, R), where R ∈ Ā_{B}(X). Morphisms from (X, R) to (X′, R′) are given by morphisms f : X → X′ such that there is a commutative diagram

R ↪ X
g ↓ ↓ f
R′ ↪ X′

(with uniquely defined arrow g). The composition is defined in an obvious way. There is a natural fully faithful functor A ⟶ Ā_{B}, X ⟼ (X, X), which is right adjoint to the functor Ā_{B} ⟶ A, (X, R) ⟼ X. This defines a Q-category 𝔸_{B} = (Ā_{B}^{op} ⇄ A^{op}).

The map B ⟼ 𝔸_{B} defines a functor

Ψ_{A} : A_{rex}^{∧} ⟶ QCat/A^{op}  (1)

Here QCat/A^{op} is the category whose objects are Q-categories of the form (Ā ⇄ A^{op}) (with fixed A) and morphisms are morphisms of Q-categories identical on A^{op}.

Let 𝔸 = (Ā ⇄ A^{op}) be a Q-category. The map which assigns to the Q-category 𝔸 the subcategory Sp_{𝔸} of A^{∧} formed by 𝔸-spaces defines a functor

Φ_{A} : QCat/A^{op} ⟶ A_{rex}^{∧}  (2)

A1.10.4. Proposition. The functor Φ_{A} is left adjoint to Ψ_{A}, and Φ_{A}∘Ψ_{A} = Id_{A_{rex}^{∧}}.

Proof. The functor Ψ_{A} ∘Φ_{A} assigns to any Q-category 𝔸 = (Ā ⇄ A^{op}) the Q-category associated with the category of 𝔸-spaces. The adjunction morphism,

Id_{QCat/A^{op}} ⟶ Ψ_{A} ∘ Φ_{A}

assigns to each Q-category 𝔸 = (Ā ⇄ A^{op}) the canonical morphism 𝔸 ⟶ Ψ_{A}(Sp_{𝔸}) which is identical on A and sends each ȳ ∈ Ā to the image of the canonical morphism Ā(ȳ, u^{∗}(−)) ⟶ A(u_{∗}(ȳ), −).

Appendix 2: Finiteness conditions.

In (commutative) algebraic geometry, there are essentially two notions of ”global finiteness”: the notion of a quasi-compact morphism and the notion of a quasi-separated morphism. These are derivatives of the notion of a quasi-compact object. In noncommutative geometry, there is more than one meaningful interpretation of quasi-compactness (like there is more than one meaningful interpretation of smoothness). By this reason, we discuss here a relative version of a quasi-compact and a quasi-separated morphisms. ”Relative” means that they depend on a subcategory, P, thought as a category of ”quasicompact objects”. Notice that this relative version is mentioned in [SGA4, Expose VI, Remarque 1.9.1].

A2.1. Setting. Fix a category 𝔅 and its subcategory P such that any object of 𝔅 isomorphic to an object of P belongs to P. We shall sometimes assume that the following condition holds:

(#) If Y ⟶^{f} X is a split monomorphism in 𝔅 (i.e. g ∘ f = id_{Y} for some X ⟶^{g} Y) and X ∈ P, then f ∈ P.

A2.2. Examples.

A2.2.1. Objects of finite type and finitely presentable objects. Suppose 𝔅 has colimits of filtered inductive systems. An object X of a category 𝔅 is said to be of finite type (resp. finitely presentable) if for any filtered inductive systeme D ⟶^{D} 𝔅, the canonical map

colim𝔅(X, D) ⟶ 𝔅(X, colimD)  (1)

is injective (resp. bijective).

We denote by 𝔅_{ft} the full subcategory of 𝔅 formed by objects of finite type and by 𝔅_{fp} its full subcategory formed by finitely presented objects. The following assertion shows that the subcategories 𝔅_{ft} and 𝔅_{fp} satisfy the condition A2.1(#).

A2.2.1.1. Lemma. Any retract of an object of finite type (resp. of a finitely presentable object) is of finite type (resp. finitely presentable).

Proof. Let X be of finite type, and let Y be a retract of X; i.e. there exist morphisms Y ⟶^{ϕ} X and X ⟶^{ψ} Y such that ψ ∘ϕ = id_{X}. Let D ⟶^{D} 𝔅 be a filtered inductive system. Consider the commutative diagram

colim𝔅(Y, D) ⟶ 𝔅(Y, colimD)
ψ_{•} ↓ ↓ ψ_{•}
colim𝔅(X, D) ⟶ 𝔅(X, colimD)

(2)

in which vertical arrows are given by functor morphism

𝔅(Y, −) ⟶^{ψ_{•}} 𝔅(X, −), g ⟼ g ∘ ψ.

Since ψ ∘ ϕ = id_{Y}, ϕ_{•} ∘ ψ_{•} = id_{𝔅(Y,−)}. In particular, vertical arrows in (2) are injective.

(a) If X is of finite type, the lower horizontal arrow is injective. Therefore, the canonical morphism colim𝔅(Y, D) ⟶ 𝔅(Y, colimD) is injective, i.e. Y is of finite type.

(b) Suppose X is finitely presentable. Note that Y is a cokernel of the pair of morphisms id_{X}, ϕ ∘ ψ : X ⇉ X. Thus we have a commutative diagram

colim𝔅(Y, D) ⟶ 𝔅(Y, colimD)
ψ_{•} ↓  ↓ ψ_{•}
colim𝔅(X, D) ⟶^{∼} 𝔅(X, colimD)
↓↓  ↓↓
colim𝔅(X, D) ⟶^{∼} 𝔅(X, colimD)

such that the vertical diagrams · → · ⇉ · are exact and two (lower) horizontal arrows are isomorphisms. Therefore, the canonical morphism colim𝔅(Y, D) ⟶ 𝔅(Y, colimD) is bijective, i.e. Y is finitely presentable.

A2.2.2. Example: quasi-compact objects of a site and quasi-compact objects of a topos. Let 𝔅 be a site. An object X of 𝔅 is called quasi-compact if for any cover {X_{i} → X| i ∈ J}, there exists a finite subset I of J such that {X_{i} → X| i ∈ I} is still a cover.

Let 𝔅^{∼} denote the topos of sheaves of sets on 𝔅, and let h^{+} : 𝔅 ⟶ 𝔅^{∼} be the composition of the Ioneda embedding 𝔅 ⟶ 𝔅^{∧}, X ⟼ 𝔅(−, X), and the sheafification functor 𝔅^{∧} ⟶ 𝔅^{∼}.

A2.2.2.1. Proposition. An object X of 𝔅 is quasi-compact iff its image, h^{+}(X), in the topos 𝔅^{∼} is of finite type (in the sense of A2.2.1).

Proof. The fact follows from [SGA4], Exp.VI, Proposition 1.2 and Theorem 1.23.

Notice that if 𝔅 itself is a topos with the canonical topology, then the functor h^{+} is a category equivalence. Therefore quasi-compact objects of a topos are exactly objects of finite type.

It follows from A2.2.2.1 and A2.2.1.1 that for any site 𝔅, the class 𝔅_{qc} of quasicompact objects of 𝔅 satisfies the condition A2.1(#).

A2.3. Weakly P-representable, P-quasi-separated, and P-coherent morphisms. We call a morphism X ⟶^{f} Y of 𝔅 weakly P-representable if for any commutative diagram

Z ⟶^{s} X
t ↓  ↓ f
T ⟶^{g} Y

with T ∈ P, there exist a commutative diagram

Z′  ⟶^{s′} X
t′ ↓  ↓ f
T  ⟶^{g} Y

and a morphism γ : Z ⟶ Z′ such that t′ ∈ P, s = s′ ∘ γ and t = t′ ∘ γ.

We denote the class of all weakly P-representable morphisms of 𝔅 by Σ^{P}.

A2.3.1. Proposition. Suppose fiber products exist in 𝔅 and the subcategory P satisfies A2.1(#). Then a morphism X ⟶^{f} Y belongs to Σ^{P} iff for any morphism T ⟶ Y such that T ∈ P, the projection X ×_{Y} T ⟶ T belongs to P. In other words, Σ^{P} is the class of P-representable morphism.

Proof. Suppose (X ⟶^{f} Y) ∈ P, and let g : T ⟶ Y be a morphism with T ∈ P. By definition, there exists a commutative diagram

Z′  ⟶^{s′} X
t′ ↓  ↓ f
T  ⟶^{g} Y

and a morphism γ : X ×_{Y} T ⟶ Z′ such that t′ ∈ P, s = s′ ∘ γ and t = t′ ∘ γ. Here s, t denote the canonical projections T ⟵ X ×_{Y} T ⟶ X . By the universal property of fiber products, there exists a unique morphism ϕ : Z′ ⟶ X ×_{Y} T such that s′ = s ∘ ϕ and t′ = t ∘ ϕ. Thus s = s ∘ ϕ ∘ γ and t = t ∘ ϕ ∘ γ which implies that ϕ ∘ γ = id. By (#), X ×_{Y} T ⟶ T is a composition of two morphisms of P, hence it belongs to P.

A2.3.2. P-quasi-separated morphisms. We call a morphism X ⟶^{f} Y P-quasi-separated if for any pair of morphisms t_{1}, t_{2} : T ⟶ X such that T ∈ P and f ∘ t_{1} = f ∘ t_{2}, any morphism g : Z ⟶ T which equalizes t_{1}, t_{2} factorizes through a morphism (g′ : Z′ ⟶ T) ∈ P such that t_{1} ∘ g′ = t_{2} ∘ g′.

Denote by Σ_{qs}^{P} the class of all P-quasi-separated morphisms.

A2.3.3. Propostion. (a) Any monomorphism of 𝔅 is P-quasi-separated.

(b) Suppose the P has the property A2.1(#) and X ×_{Y} X exists. Then f ∈ Σ_{qs}^{P} iff the diagonal morphism, Δ_{X/Y} : X ⟶ X ×_{Y} X belongs to Σ^{P}.

Proof. (a) If X ⟶ Y is a monomorphism, then the morphism Δ_{X/Y} : X ⟶ X ×_{Y} X is an isomorphism, hence the assertion.

(b) The argument is similar to that of A2.3.1.

A2.3.4. P-coherent morphisms. We call a morphism P-coherent if it belongs to the class Σ_{coh}^{P} = Σ^{P} ∩ Σ_{qs}^{P}.

A2.3.5. Proposition. (a) Any isomorphism is P-coherent.

(b) The classes Σ^{P} and Σ_{qs}^{P} are closed under composition. In particular Σ_{coh}^{P} is closed under composition.

(c) Let X ⟶^{f} Y and g : T ⟶ Y be morphisms such that X ×_{Y} T exists. If f ∈ Σ^{P} (resp. f ∈ Σ_{qs}^{P}, resp. f ∈ Σ_{coh}^{P}), then the canonical projection X ×_{Y} T ⟶ T belongs to the same class.

(d) Let X ⟶^{f} Y, Y ⟶^{g} Z be morphisms.

(i) If g ∘ f ∈ Σ_{qs}^{P}, then f ∈ Σ_{qs}^{P}.

(ii) Suppose g ∈ Σ_{qs}^{P}. If g∘f ∈ Σ^{P} (resp. g∘f ∈ Σ_{coh}^{P}), then f ∈ Σ^{P} (resp. f ∈ Σ_{coh}^{P}).

Proof. (a) Obvious.

(b) (i) Let X ⟶^{f} Y ⟶^{g} Z ⟵^{s} S be a diagram such that f, g ∈ Σ^{P} and S ∈ ObP. Let

W ⟶^{ϕ} X
ψ ↓  ↓ gf
S ⟶^{s} Z

be a commutative diagram. Since g ∈ Σ^{P}, there exists a commutative diagram

W ⟶^{ϕ} X
ξ ↓  ↓ f
T ⟶^{t} Y
ψ′ ↓  ↓ g
S ⟶^{s} Z

such that ψ′ ∘ ξ = ψ and T ∈ P. Since f ∈ Σ^{P}, there exists a commutative diagram

W′ ⟶^{ϕ′} X
ξ′ ↓  ↓ f
T ⟶^{t} Y

and a morphism γ : W ⟶ W′ such that ξ = ξ′ ∘ γ, ϕ = ϕ′ ∘ γ and W′ ∈ P, hence the assertion.

(ii) Suppose morphisms X ⟶^{f} Y and Y ⟶^{g} Z are P-quasi-separated. Let

\begin{matrix}W & ⟶^{h} & T & ⟶^{t_{1}} & X \\ h ↓ & & t_{2} ↓ & & ↓ gf \\ T & ⟶^{t_{1}} & X & ⟶^{gf} & Z \end{matrix}  (1)

be a commutative diagram such that T ∈ P. Consider instead of (1) the commutative diagram

\begin{matrix}W & ⟶^{h} & T & ⟶^{ft_{1}} & Y \\ h ↓ & & ft_{2} ↓ & & ↓ g \\ T & ⟶^{ft_{1}} & Y & ⟶^{g} & Z \end{matrix}  (1)

Since g ∈ Σ_{qs}^{P}, the morphism h : W ⟶ T is a composition of a morphism (h′ : S ⟶ T) ∈ P such that f ∘ t_{1} ∘ h′ = f ∘ t_{2} ∘ h′ and a morphism ξ : W ⟶ S. Since h equalizes t_{1}, t_{2}, the morphism ξ equalizes t′_{1} = t_{1} ∘ h′ and t′_{2} = t_{2} ∘ h′. Since f is P-quasi-separated, ξ factors through a morphism (ξ′ : U ⟶ S) ∈ P such that t′_{1} ∘ ξ′ = t′_{2} ∘ ξ′. Thus t_{1} ∘ (h′ ∘ ξ′) = t_{2} ∘ (h′ ∘ ξ′), hence the assertion.

(c) Let X ⟶^{f} Y, g : T ⟶ Y be morphisms such that X ×_{Y} T exists. Suppose f ∈ Σ^{P}. Consider a commutative diagram

\begin{matrix}W & ⟶^{ϕ} & X ×_{Y} T & ⟶^{t_{1}} & X \\ ψ ↓ & & f_{1} ↓ & & ↓ f \\ S & ⟶^{s} & T & ⟶^{t} & Y \end{matrix}  (3)

such that the right square is cartesian and S ∈ P. Since f ∈ Σ^{P}, there exists a commutative diagram

\begin{matrix}W′ & ⟶^{ϕ′} & X ×_{Y} T & ⟶^{t_{1}} & X \\ ψ′ ↓ & & & & ↓ f \\ S & ⟶^{s} & T & ⟶^{t} & Y \end{matrix}  (4)

with ψ′ ∈ P and a morphism γ : W ⟶ W′ such that ϕ = ϕ′ ∘ γ, ψ = ψ′ ∘ γ. If follows from the universal property of fiber products that the diagram

W′ ⟶^{ϕ′} X ×_{Y} T ⟶^{t_{1}} X
ψ′ ↓  f_{1} ↓  ↓ f
S ⟶^{s}  T  ⟶^{t} Y

commutes.

The remaining assertions are proved in a similar way. Details are left to the reader.

A2.4. P-quasi-separated and P-coherent objects.

A2.4.1. Definition. We call an object X of 𝔅 P-quasi-separated if any morphism T ⟶ X with T ∈ P, belongs to Σ^{P}.

A2.4.2. Definition. We call an object X P-coherent if it is P-quasi-separated and belongs to P. We denote by P_{coh} the full subcategory of P formed by P-coherent objects.

A2.4.3. Proposition. Let X ⟶^{f} Y be a morphism of 𝔅.

(a) If Y ∈ P and f ∈ Σ^{P}, then X ∈ P.

(b) If Y is P-quasi-separated and X ∈ P, then f ∈ Σ^{P}.

(c) If Y is P-coherent, then f ∈ Σ^{P} iff X ∈ P.

(d) If Y ∈ P (resp. Y is P-quasi-separated, resp. Y is P-coherent) and if f ∈ Σ^{P} (resp. f ∈ Σ_{qs}^{P}, resp. f ∈ Σ_{coh}^{P}), then X ∈ P (resp. X is P-quasi-separated, resp. X is P-coherent).

Proof. (a) Since Y ∈ P, X ≃ X ×_{Y} Y ∈ P (see A2.3.1).

(b) Follows from definition.

(c) Follows from (a) and (b).

(d) (i) Let Y and X ⟶^{f} Y be P-quasi-separated. Let S ⟶^{g} X be a morphism with S ∈ P. Since Y is P-quasi-separated and S ∈ P, the composition f ∘ g belongs to Σ^{P}. By A2.3.5(d)(ii), g ∈ Σ^{P}, hence X is P-quasi-separated.

Suppose Y and X ⟶^{f} Y are P-coherent. Then by (i), X is P-quasi-separated and by (c), X ∈ P, hence X is P-coherent.

A2.4.4. Proposition. Subobjects of P-quasi-separated objects are P-quasi-separated.

Proof. Let Y ⟶^{f} X be a monomorphism, and let X be a P-quasi-separated object. Let

Z ⟶^{ϕ} S
ψ ↓  ↓ s
T ⟶^{t} Y

be a commutative diagram such that S, T ∈ P. Since X is P-quasi-separated, there is a commutative diagram

Z′ ⟶^{ϕ′} S
ψ′ ↓  ↓ f ∘ s
T ⟶^{f∘t} X

with ψ′ ∈ Σ^{P} and a morphism Z ⟶^{h} Z′ such that ϕ = ϕ′ ∘ h, ψ = ψ′ ∘ h. Since f is a monomorphism, the diagram

Z′ ⟶^{ϕ′} S
ψ′ ↓  ↓ s
T ⟶^{t} Y

is commutative.

A2.5. P-constructible objects. Suppose the category 𝔅 has a final object. An object X of 𝔅 is called constructible if it is coherent over a final object. We denote by 𝔅_{P−cons} the full subcategory of 𝔅 formed by P-constructible objects.

A2.6. Finiteness conditions in a Q-category. Fix a Q-category 𝔸 = (Ā ⇄^{u} A).

A2.6.1. Lemma. Let ȳ ∈ ObĀ be such an object that u_{!}(ȳ) exists. If ȳ is of finite type (resp. finitely presentable), then u_{!}(ȳ) has the same property.

Proof. Let D : D ⟶ A be a filtered inductive system. We have a commutative diagram

colimA(u_{!}(ȳ), D) ⟶ A(u_{!}(ȳ), colimD))
↓  ↑
colimĀ(ȳ, u^{∗} ∘ D) ⟶ Ā(ȳ, colim(u^{∗} ∘ D))

(1)

Here the right vertical arrow is the composition of the canonical morphism

Ā(ȳ, colim(u^{∗} ∘ D)) ⟶ Ā(ȳ, u^{∗}(colimD))  (2)

and the isomorphism Ā(ȳ, u^{∗}(colimD)) ⟶ A(u_{!}(ȳ), colimD). Since u^{∗} has a right adjoint, it preserves colimits. In particular, the morphism (2) is an isomorphism. If ȳ is of finite type (resp. finitely presentable), the lower horizontal arrow is injective (resp. bijective), hence the upper one is injective (resp. bijective).

A2.7. Quasi-compact spaces and quasi-separated morphisms. Let 𝔸 = (Ā ⇄^{u} A) be a Q-category, and let τ be a quasi-topology on the category Esp_{𝔸} of 𝔸-spaces (cf. 8.1 and 8.2). We call an 𝔸-space X τ-quasi-compact if any τ-cover of X has a finite subcover. Denote by 𝔎_{τ} the full subcategory of the category Esp_{𝔸} whose objects are τquasi-compact spaces. Applying to the subcategory 𝔎_{τ} the formalism presented above, we obtain the notions of weakly 𝔎_{τ} -representable, τ-quasi-separated, and τ-coherent morphisms and spaces (see A2.3, A.2.3.2, A2.3.4, A2.4).

A2.7.1. Proposition. (a) Any isomorphism is τ-coherent.

(b) The class of weakly 𝔎_{τ} -representable morphisms and the class of τ-quasi-separated morphisms are closed under composition and base change. In particular, the class of τcoherent morphisms (which is the intersection of other two) is closed under composition and base change.

(c) Suppose the composition g∘f of morphisms is a weakly 𝔎_{τ} -representable morphism (resp. a τ-quasi-separated morphism). Then f is weakly 𝔎_{τ} -representable (resp. τ-quasi-separated). In particular, if g ∘ f is τ-coherent, then f is quasi-coherent.

Proof. The assertion is a special case of A2.3.5.

A2.7.2. Proposition. (i) Let X ⟶^{f} Y be an 𝔸-space morphism. If Y is τquasi-compact (resp. Y is τ-quasi-separated, resp. Y is τ-coherent) and if f is weakly 𝔎_{τ} -representable (resp. f is τ-quasi-separated, resp. f is τ-coherent), then X is τ-quasi-compact (resp. X is τ-quasi-separated, resp. X is τ-coherent).

(ii) Subspaces of τ-quasi-separated 𝔸-spaces are τ-quasi-separated.

Proof. The assertion (i) is a special case of the assertion (d) in A2.4.3. The assertion (ii) is a specialization of A2.4.4.

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