Underlying spaces of noncommutative schemes.

Alexander Rosenberg

Introduction

The general philosophy adopted in noncommutative algebraic geometry is based on identifying ’spaces’ with categories regarded as categories of quasi-coherent or coherent sheaves, or derived categories of those. The purpose is to describe geometric notions in terms of related categories and functors between them and then use these descriptions for defining and studying new, noncommutative, ’spaces’. It is similar to noncommutative differential geometry started by Alain Connes, where manifolds are replaced by algebras.

In this approach only a half of the picture survives: geometric spaces become ’virtual’. Most of commutative algebraic geometry, however, is based upon studying geometric

spaces (see [Gr1], [Gr2], [Gr5], [Gr6], [Gr7], [Gr8], [GrD]). Grothendieck’s local study of schemes turns commutative algebra into a part of geometry. One of the goals of noncommutative algebraic geometry is to do the same for noncommutative algebra, in particular, to develop geometric tools and intuition for studying representations of associative algebras.

This work is a step in that direction. One of its purposes is to define underlying topological spaces of noncommutative schemes and some other noncommutative ’spaces’ of interest. A natural framework for this is provided by quasi-topological ’spaces’ represented by abelian categories (defined in 1.5). The main building block used here is the spectrum introduced in [R1] (see also [R, Ch.3]). This spectrum is defined for an arbitrary abelian category and is directly related with representation theory: isomorphism classes of simple objects are closed points of the spectrum. The spectrum of the category of modules over an arbitrary commutative ring is naturally isomorphic to the prime spectrum of this ring.

Below follows a brief outline of the content.

First three sections contain preliminaries on ’spaces’ and basic spectra used in the sequel. Section 1 starts with a short dictionary of categoric (virtual) geometry. ’Spaces’ considered here are represented by abelian categories; and morphisms of ’spaces’, X ⟶ Y, are represented by additive (inverse image) functors, C_{Y} ⟶ C_{X}, between the corresponding categories. We recall the notion of a quasi-pretopology and define the category of quasi-topological ’spaces’. Its objects are pairs (X, τ_{X}), where X is a ’space’ and τ_{X} is a set of covers. Covers considered in this work are conservative families of exact localizations (i.e. the set of their inverse image functors is a conservative family of exact localizations).

In Section 2, we define the spectrum of a ’space’ X as the spectrum (in the sense of [R1], or [R, Ch.3]) of the abelian category representing X and sketch some of its properties.

We remind the notion of a Serre subcategory, as it is introduced in [R1], and define Serre localizations as exact localizations whose kernels are Serre subcategories.

Section 3 contains preliminaries on local ’spaces’, complete spectrum, and the Sspectrum (the main references are [R, Ch.3], and [R, Ch.6], where the S-spectrum is called flat spectrum). A space X is local if the spectrum of X has a unique closed point which belongs to the support of every nonzero object of the category C_{X}. If C_{X} is the category of modules over a commutative ring R, then X is local iff the ring R is local.

The complete spectrum of X is a preordered set, Spec^{1}(X), whose elements are thick subcategories P such that the quotient category C_{X}/P is local. The S-spectrum of a ’space’ X is a preordered set, Spec^{−}(X), formed by all Serre subcategories of C_{X} which belong to the complete spectrum. The spectrum Spec(X) is naturally embedded into Spec^{−}(X), and for every exact localization U ⟶ X (an ’open immersion’ in an appropriate sense), the spectrum Spec(U) is naturally embedded into Spec^{1}(X). Note by passing that this embedding is used to formulate the local property of the spectrum: for any conservative set, {U_{i} → X | i ∈ J}, of exact localizations, Spec(X) ⊆ ⋃_{i∈J} Spec(U_{i}).

All spectra which appear in this work contain Spec(X) and are contained in Spec^{1}(X). It is argued in [R, Ch.6] that if the category C_{X} is locally noetherian, then Spec^{−}(X)

is naturally isomorphic to the Gabriel spectrum of the category C_{X} [Gab]. Recall that points of the Gabriel spectrum are isomorphism classes of indecomposable injectives. Even in the case of a locally noetherian ’space’, the (image of the) spectrum Spec(X) is, usually, a small part of Spec^{−}(X).

In Section 4 we introduce the quasi-pretopology of Serre localizations and show that its finite covers form a Grothendieck pretopology on the subcategory whose objects are arbitrary ’spaces’ X (such that C_{X} is an abelian category) and arrows are morphisms of ’spaces’ whose inverse image functors are Serre localizations. Using this fact, we establish the local property of the S-spectrum: if a cover {U_{i} → X | i ∈ J} by Serre localizations contains a finite subcover, then Spec^{−}(X) = ⋃_{i∈J} Spec^{−}(U_{i}). We deduce from these asser-

tions similar facts about complete spectrum and exact localizations: finite covers by exact localizations form a Grothendieck pretopology, and if a cover {U_{i} → X | i ∈ J} by exact localizations contains a finite subcover, then Spec^{1}(X) = ⋃_{i∈J} Spec^{1}(U_{i}).

In Section 5, we associate with every quasi-topological ’space’ (X, τ_{X}) a topological space, Spec(X, τ_{X}) which we call the upper spectrum of (X, τ_{X}). Points of Spec(X, τ_{X}) are points P of the complete spectrum of X such that P belongs to Spec(U) for some element of a τ_{X}-cover U ⟶ X. If the quasi-pretopology τ_{X} is trivial (i.e. it consists only of identical morphism), then Spec(X, τ_{X}) coincides with Spec(X). If τ_{X} is the quasi-pretopology of Serre localizations mentioned above, or even its finite version, then Spec(X, τ_{X}) is the S-spectrum, Spec^{−}(X), of the ’space’ X. If τ is (the finite version of) the quasi-pretopology of exact localizations, then Spec(X, τ_{X}) coincides with the complete spectrum of X. If a quasi-pretopology τ̃_{X} is a refinement of τ_{X}, then Spec(X, τ_{X}) ⊆ Spec(X, τ̃_{X}). In particular, Spec(X) ⊆ Spec(X, τ_{X}) ⊆ Spec^{1}(X) .

We show that, under a certain natural condition on the quasi-pretopology τ_{X}, the equality Spec(X, τ_{X}) = ⋃_{i∈J} Spec(U_{i}, τ_{U_{i}}) holds for any τ-cover {U_{i} → X | i ∈ J}. Here

τ_{U_{i}} is the quasi-pretopology on U_{i} induced by the quasi-pretopology τ_{X}.

In Section 6, we introduce the lower spectrum, otherwise called the combinatorial spectrum, of a quasi-topological ’space’ (X, τ_{X}). For a cover U = {U_{i} → X | i ∈ J}, we define the lower spectrum of U as a subset, Spec_{℘}U, of all points P of the complete spectrum of X such that, whenever P belongs to the complete spectrum of some element, U_{i}, of the cover, it belongs to its spectrum, Spec(U_{i}).

The combinatorial spectrum, Spec_{℘}(X, τ_{X}) of a quasi-topological ’space’ (X, τ_{X}) is a subspace of the complete spectrum of X formed by the points which locally belong to the spectrum. In precise terms, Spec_{℘}(X, τ_{X}) is the colimit of Spec_{℘}U, where U runs through τ_{X}-covers of X and morphisms are induced by refinements. We have the inclusions of spectra

Spec(X) ↪ Spec_{℘}(X, τ_{X}) ↪ Spec(X, τ_{X}) ↪ Spec^{1}(X).  (⋆)

Let Spec_{℘}^{−}U denote the intersection of Spec_{℘}U with the S-spectrum of X. We show that for any finite cover U of X, the natural embedding Spec(X) ↪ Spec_{℘}^{−}U is an isomorphism. This implies that the canonical map Spec(X) ⟶ Spec_{℘}^{−}(X, τ_{X}) is an isomorphism, if (X, τ_{X}) is quasi-compact, i.e. if any τ_{X}-cover of X has a finite subcover.

More generally, if X has a τ_{X}-cover U = {U_{i} → X | i ∈ J} such that all U_{i} are quasi-compact, then the canonical map Spec_{℘}^{−}U ⟶ Spec_{℘}^{−}(X, τ_{X}) is an isomorphism.

If a cover U consists of Serre localizations (i.e. their kernels are Serre subcategories), then Spec_{℘}^{−}U coincides with Spec_{℘}U. In particular, Spec_{℘}^{−}(X, τ_{X}) = Spec_{℘}(X, τ_{X}) if τ_{X} has a base formed by Serre localizations.

In Section 7, we recall the notions of closed coimmersions and immersions (introduced in slightly different terms in [R], Ch.3) and use them to define some canonical quasipretopologies, in particular, the Zariski pretopology. If C_{X} is the category of left Rmodules, then there is a bijective correspondence between isomorphism classes of closed immersions to X and two-sided ideals of the ring R. If C_{X} is the category of quasi-coherent sheaves on a scheme X = (X , O), then isomorphism classes of closed immersions to X are in one-to-one correspondence with closed subschemes of the scheme X.

We apply the results of Sections 5, 6, and 7 to commutative schemes. If C_{X} is the category of quasi-coherent sheaves on a commutative scheme X = (X , O) and τ_{X} is the Zariski pretopology, then the upper spectrum, Spec(X, τ_{X}), coincides with the combinatorial spectrum, Spec_{℘}(X, τ_{X}). If X is affine, then (by a result of Section 6) Spec_{℘}(X, τ_{X}) ≃ Spec(X), and, by [R, Ch.3], Spec(X) is isomorphic to the prime spectrum of the ring ΓO. It is easy to deduce from the latter fact that, in the case of an arbitrary scheme, Spec(X, τ_{X}) (hence Spec_{℘}(X, τ_{X})) is isomorphic to the underlying topological space X of the scheme X.

Suppose that a scheme X = (X , O) has a finite affine cover U = {U_{i} →^{u_{i}} X | i ∈ J} such that every immersion u_{i} has a direct image functor (for instance, X is quasi-compact and quasi-separated, or its underlying topological space, X, is noetherian). Then it follows from results of Section 6 that the canonical embedding Spec(X) ⟶ Spec_{℘}(X, τ_{X}) is an isomorphism. Therefore, Spec(X) ≃ Spec_{℘}(X, τ_{X}) = Spec(X, τ_{X}) ≃ X .

This straightens a confusion created in [R2], where the isomorphism Spec(X) ≃ X is claimed for an arbitrary scheme (X , O). O. Gabber showed me (by producing an example) that the claim is not correct without some finiteness conditions (which are used, implicitly, in [R2]). An obvious remedy is to replace Spec(X) by the upper spectrum Spec(X, τ_{X}^{ℨ}), where τ_{X}^{ℨ} is (a categorical incarnation of) the Zariski topology introduced in [R1]. This solution of the problem is not always satisfactory from the noncommutative point of view. The main reason is that the upper spectrum of (X, τ_{X}^{ℨ}), where X is a noncommutative affine ’space’, might be considerably larger than Spec(X).

The combinatorial spectrum seems to be a better, in a certain sense optimal, choice of the underlying topological space. If C_{X} = R−mod and (X, τ_{X}) is quasi-compact, then the combinatorial spectrum of (X, τ_{X}) is isomorphic to Spec(X). By a result of [R1], Zariski pretopology on any affine noncommutative scheme is quasi-compact.

In Section 8, we associate with a quasi-topological ’space’ (X, τ_{X}) and a choice of a spectrum, Spec^{•}(X, τ_{X}), realized as a subset of the complete spectrum and endowed with the topology induced by τ_{X}, a sheaf of commutative rings, O_{X}^{𝔠}, on Spec^{•}(X, τ_{X}). In particular, we associate with the ’space’ (X, τ_{X}) the ringed topological spaces (Spec_{℘}(X, τ_{X}), O_{X}^{𝔠}) and (Spec(X, τ_{X}), O_{X}^{𝔠}) called respectively the lower (or combinatorial) and the upper geometric center of (X, τ_{X}). Applying the facts of Section 6, we show that if C_{X} is the category of quasi-coherent sheaves on a scheme X = (X , O), and τ_{X} is the categorical analog of Zariski topology discussed in Section 7, then the upper and lower geometric centra coincide and are isomorphic to the scheme X.

In Section 9, we recall the notion of a noncommutative scheme and discuss shortly geometric realizations of noncommutative schemes based on the combinatorial spectrum and on the upper spectrum. Most of non-affine noncommutative ’spaces’ studied in literature are analogues of Cone and Proj and some related examples of noncommutative schemes. We remind the notions of the Cone of a non-unital ring and the Proj of a G-graded ring, where G is a monoid, and study the relations between their combinatorial spectra. This study involves the notion of G-spectrum. One of the applications is the description of the combinatorial spectrum of the quantum flag variety of a semisimple Lie algebra in terms of the spectrum of its base affine space.

This work was started while I was visiting Max-Planck Institut für Mathematik in Bonn during the summer of 2001 as a spectral contrepart to the papers [KR3] and [LR2]. I would like to thank the Institute for excellent working conditions.

I am grateful to O. Gabber for useful conversations and for indicating me an error.

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

1. Preliminaries on ’spaces’.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

X ⟶^{f} Y into a conservative morphism X ⟶^{f_{𝔠}} Σ_{f}^{−1}Y and a localization Σ_{f}^{−1}Y ⟶^{p_{f}} Y.

Suppose C_{Y} has finite limits (resp. finite colimits). Then f is left exact (resp. right exact) iff the class of arrows Σ_{f} is a left (resp. right) saturated multiplicative system. In this case both the localization p_{f} and the conservative morphism f_{𝔠} are left (resp. right) exact.

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

1.4. Continuous morphisms and flat morphisms. A morphism f of |Cat|^{o}, or Cat^{op}, is called continuous if its inverse image functor has a right adjoint, f_{∗}, which is called a direct image functor of f.

A morphism f is called flat if it is both exact and continuous.

One can show that a morphism f is continuous iff both the localization p_{f} and the conservative component f_{𝔠} are continuous.

1.5. Quasi-topological ’spaces’.

1.5.1. Quasi-pretopologies and quasi-presites. A quasi-pretopology on a category A is a function, τ, which assigns to each object X of A a set, τ_{X}, of covers; the latter are families of arrows {U_{i} → X | i ∈ J}. We assume that any isomorphism is a cover and the composition of covers is a cover. The pair (A, τ) is called a quasi-presite.

For any object X of A, we denote by Σ^{τ} X the set of all arrows U ⟶ X which belong to some cover of X. We denote by Σ^{τ} the union of all sets Σ^{τ} X, X ∈ ObA. It follows that Σ^{τ} is closed under composition and contains all isomorphisms of the category A, hence it defines a subcategory, A^{τ}, of A.

Let Σ_{τ} X denote the class of all morphisms Y ⟶^{f} X such that a pull-back U ×_{X} Y = {U_{i} ×_{X} Y ⟶ Y | i ∈ J} of any τ-cover U = {U_{i} → X | i ∈ J} of X along f exists and is a τ-cover of Y. One can show that the class of arrows Σ_{τ} = ⋃_{X∈ObA} Σ_{τ} X contains

all isomorphisms and is closed under the composition of morphisms, hence it defines a subcategory, A_{τ}, of the category A.

For any X ∈ ObA, the pair (X, τ_{X}) is regarded as an analog of a topological space and will be called a quasi-topological ’space’. Morphisms from (Y, τ_{Y}) to (X, τ_{X}) are given by morphisms Y ⟶ X which belong to Σ_{τ}. We denote thus defined category by 𝔗𝔬𝔭_{(A,τ)}.

A quasi-pretopology τ is a (Grothendieck) pretopology, if A_{τ} = A. If τ is a pretopology on A, the pair (A, τ) is called a presite.

Notice that if Σ^{τ} ⊆ Σ_{τ}, then the pair (A_{τ} , τ|_{A_{τ}}) is a presite.

1.5.2. The quasi-pretopology of exact localizations. Let A be the category |Cat|^{o}. We define the quasi-pretopology of exact localizations, τ^{𝔏}, as follows. For any X ∈ ObA, the set of covers τ_{X}^{𝔏} consists of all conservative families {U_{i} →^{u_{i}} X | i ∈ J}

of exact localizations; i.e. the set of inverse image functors {C_{X} →^{u_{i}^{∗}} C_{U_{i}} | i ∈ J} is a conservative family of exact localizations.

1.6. ’Spaces’ represented by abelian categories. We denote by |𝔄𝔟|^{o} the subcategory of the category |Cat|^{o} whose objects are represented by abelian categories and morphisms have additive inverse image functors.

Note that if X is an object of |𝔄𝔟|^{o} and U ⟶^{u} X is an exact localization, then the morphism u (in particular, the object U) belongs to the category |𝔄𝔟|^{o}.

In fact, an inverse image functor, C_{X} →^{u^{∗}} C_{U}, of u being an exact localization means that the category C_{U} is equivalent to a quotient category of the abelian category C_{X} by a thick subcategory Ker(u^{∗}), hence C_{U} is an abelian category, and u^{∗} is an additive functor.

1.6.1. Quasi-topological ’spaces’. In this work, we will consider quasi-presites (A, τ) such that A is a subcategory of the category |𝔄𝔟|^{o} and for any X ∈ ObA, the set of covers τ_{X} is a subset of the set τ_{X}^{𝔏} defined in 1.5.2; i.e. for every τ-cover {U_{i} →^{u_{i}} X | i ∈ J} is a conservative family of exact localizations.

Pairs (X, τ_{X}), X ∈ ObA, (objects of the category 𝔗𝔬𝔭_{(A,τ)}) are quasi-topological ’spaces’ in the sense of 1.5.1. Any cover {U_{i} →^{u_{i}} X | i ∈ J} of a ’space’ X can be represented by the set {Ker(u_{i}^{∗}) | i ∈ J} of kernels of inverse image functors. Exactness of the functor u_{i}^{∗} means that its kernel is a thick subcategory of the category C_{X}. The fact that the

family of functors {C_{X} →^{u_{i}^{∗}} C_{U_{i}} | i ∈ J} is conservative means that the intersection of their kernels is the zero subcategory.

2. The spectrum.

2.1. The preorder ≻. Let X be an object of |𝔄𝔟|^{o}, i.e. C_{X} is an abelian category. For any two objects, M and N, of C_{X}, we write M ≻ N if N is a subquotient of a finite coproduct of copies of M.

For any object M of the category C_{X}, we denote by [M] (resp. ⟨M⟩) the full subcategory of C_{X} whose objects are all L ∈ ObC_{X} such that M ≻ L (resp. L ⊁ M). It follows that M ≻ N ⇔ ⟨N⟩ ⊆ ⟨M⟩ ⇔ [N] ⊆ [M]. In particular, M and N are equivalent with respect to ≻ (i.e. M ≻ N ≻ M) iff ⟨M⟩ = ⟨N⟩, or [M] = [N]. Thus, the preorders ({[M] | M ∈ ObC_{X}}, ⊇ ) and ({⟨M⟩ | M ∈ ObC_{X}}, ⊇ ) are canonical realizations of the quotient of (ObC_{X}, ≻) by the equivalence relation associated with ≻.

2.2. The spectrum Spec(X). We denote by Spec(X) the family of all nonzero objects M of the category C_{X} such that L ≻ M for any nonzero subobject L of M, which is endowed with the preorder (induced by) ≻.

It follows from this definition that a nonzero object M of C_{X} belongs to Spec(X) iff it is equivalent to any of its nonzero subobjects. In particular, every simple object of the category C_{X} belongs to Spec(X).

We denote by Spec(X) the class of subcategories {[M] | M ∈ Spec(X)} endowed with the preorder ⊇. The preorder Spec(X) is called the spectrum of X. If C_{X} is equivalent to a small category, then Spec(X) is a preordered set.

Given elements [M], [P] of Spec(X), we say that [P] is a specialization of [M] if [M] ⊇ [P], or, equivalently, M ≻ P. We denote by τ^{≻} the strongest topology on Spec(X) compatible with the preorder ⊇. It can be described as follows: the closure of any subset W consists of specializations of all elements of W.

2.3. The spectrum and simple objects. Obviously, any simple object of the category C_{X} belongs to Spec(X).

2.3.1. Proposition. Let M, N be objects of the category C_{X}.

(a) If M is simple and N is nonzero, the relation M ≻ N means that N is isomorphic to a coproduct of copies of M, in particular N ≻ M, i.e. N and M are equivalent.

(b) If M and N are simple objects, then M ≻ N iff M and N are isomorphic.

Proof. Any nonzero subquotient of a semisimple object (a coproduct of simple objects) is semisimple too; hence the assertion (a). The assertion (b) is a consequence of (a).

Let τ^{≻} denote the topology on Spec(X) associated with the specialization preorder: the closure of W ⊆ Spec(X) consists of all [M] such that [M] ⊆ [M′] for some [M′] ∈ W.

2.3.2. Proposition. (a) The inclusion Simple(X) ↪ Spec(X) induces an injection of the set of isomorphism classes of simple objects of C_{X} into the set of closed points of (Spec(X), τ^{≻}).

(b) If the category C_{X} has enough objects of finite type, then this injection is a bijection, i.e. every closed point of (Spec(X), τ^{≻}) is of the form [M] for some simple object M of the category C_{X}.

Proof. (a) The assertion follows from 2.3.1.

(b) Having enough objects of finite type means that every object of C_{X} is the supremum of its subobjects of finite type. By a standard argument, this property implies that every nonzero object of C_{X} has a simple quotient. Suppose, P ∈ Spec(X) is such that [P] is a closed point. Let M be a simple quotient of P. Since P ≻ M and [P] is closed, M ≻ P, i.e. [M] = [P].

2.4. Topologizing and thick subcategories. A full subcategory, 𝕋, of C_{X} is called topologizing if it is closed under finite coproducts and subquotients taken in C_{X}.

In particular, if M is an object of 𝕋 and M ≻ L, then L belongs to 𝕋. One can show that, for any object M, the subcategory [M] is the smallest topologizing subcategory containing M.

A topologizing category is called thick if it is closed under extensions.

For a thick subcategory 𝕋 of C_{X}, we denote by X/|𝕋| the ’space’ defined by C_{X/|𝕋|} = C_{X}/𝕋. Recall that |𝕋| denote the ’space’ corresponding to the category 𝕋; i.e. it is defined by 𝕋 = C_{|𝕋|}.

2.4.1. Proposition.(a) For any topologizing subcategory 𝕋 of C_{X}, there is a natural embedding Spec(|𝕋|) ↪ Spec(X) which induces an embedding Spec(|𝕋|) ↪ Spec(X).

(b) If 𝕋 is a thick subcategory of C_{X}, then there is an embedding

Spec(X) ↪ Spec(|𝕋|) ∐ Spec(X/|𝕋|).

Sketch of proof. (a) Objects of Spec|𝕋| are exactly those objects of Spec(X) which belong to 𝕋.

(b) If 𝕋 is a thick subcategory, then the localization functor C_{X} ⟶ C_{X}/𝕋 maps objects of Spec(X) which do not belong to Spec(|𝕋|) to objects of Spec(X/|𝕋|). This map induces an injection Spec(X) − Spec|𝕋| ↪ Spec(X/|𝕋|).

For more details see [R, Ch.3].

The following fact will be refered to as the local property of the spectrum.

2.4.2. Proposition. Let {U_{i} →^{u_{i}} X | i ∈ J} be a conservative set of exact localizations. Then Spec(X) = ⋃_{i∈J} Spec(U_{i}; X), where Spec(U_{i}; X) = {[P] ∈ Spec(X) | u_{i}^{∗}(P) ≠ 0}.

The map [P] ⟼ [u_{i}^{∗}(P)] is an embedding Spec(U_{i}; X) ⟶ Spec(U_{i}).

Proof. Let [P] ∈ Spec(X). Since the family of inverse image functors {u_{i}^{∗} | i ∈ J} is conservative, u_{i}^{∗}(P) ≠ 0 for some i ∈ J. Then N ⊁ P for every N ∈ ObKer(u_{i}^{∗}), because Ker(u_{i}^{∗}), being a thick subcategory, contains with every object N all objects L of C_{X} such that N ≻ L. Therefore u_{i}^{∗}(P) ∈ Spec(U_{i}). Since the functor u_{i}^{∗} is exact, it respects the relation ≻. In particular, it maps the equivalent (with respect to ≻) objects to equivalent objects. Thus the element ⟨u_{i}^{∗}(P)⟩ does not depend on the choice of an object P representing [P], hence the assertion.

2.5. Serre subcategories. We remind the notion of a Serre subcategory of an abelian category as it is defined in [R, III.2.3.2].

Let 𝕋 be a subcategory of C_{X}. We denote by 𝕋^{−} the full subcategory of C_{X} generated by all objects L of C_{X} such that any nonzero subquotient of L has a nonzero subobject which belongs to 𝕋.

2.5.1. Proposition. Let 𝕋 be a subcategory of C_{X}. Then

(a) The subcategory 𝕋^{−} is thick.

(b) (𝕋^{−})^{−} = 𝕋^{−}.

(c) 𝕋 ⊆ 𝕋^{−} iff any subquotient of an object of 𝕋 is isomorphic to an object of 𝕋.

Proof. See [R, III.2.3.2.1].

2.5.2. Definition. A subcategory 𝕋 of C_{X} is called a Serre subcategory if 𝕋^{−} = 𝕋.

2.5.3. Proposition. Let C_{X} be an abelian category. For any P ∈ Spec(X), the subcategory ⟨P ⟩ is a Serre subcategory of the category C_{X}.

Proof. (a) By 2.5.1(c), the inclusion ⟨P ⟩ ⊆ ⟨P ⟩^{−} is equivalent to the property: any subquotient M of any object N of the subcategory ⟨P ⟩ belongs to ⟨P ⟩. This property holds. In fact, if M is a subquotient of N, then N ≻ M. If M ∉ Ob⟨P ⟩, then, by definition of ⟨P ⟩, M ≻ P. Thanks to the transitivity of ≻, this implies that N ≻ P which contradicts to the assumption that N ∈ Ob⟨P ⟩.

(b) Suppose ⟨P ⟩^{−} has an object, L, which does not belong to ⟨P ⟩. The latter means that L ≻ P, i.e. P is a subquotient of a coproduct, L^{⊕n} of n copies of L for some finite n. Since ⟨P ⟩^{−} is closed under finite coproducts, P is a nonzero subquotient of an object of ⟨P ⟩^{−}, hence it has a nonzero subobject, M, which belongs to ⟨P ⟩. But, P belongs to Spec(X), hence the object M is equivalent to P which contradicts to that M ∈ Ob⟨P ⟩. This shows that the Serre subcategory ⟨P ⟩^{−} is contained in ⟨P ⟩.

2.5.3.1. Proposition. Let C_{X} be an abelian category.

(a) For any topologizing subcategory 𝕋 of the category C_{X}, the spectrum of 𝕋 coincides with the spectrum of the minimal Serre subcategory, 𝕋^{−}, containing 𝕋.

(b) There is a natural embedding Spec(X) ↪ Spec(|𝕋|) ⋃ Spec(X/|𝕋^{−}|).

Proof. (a) Let P ∈ Spec(𝕋^{−}). Since P is a nonzero object of 𝕋^{−}, it has a nonzero subobject, P′ which belong to 𝕋. Since P is an object of the spectrum, P′ ≻ P, which implies (because 𝕋 is topologizing) that P ∈ Ob𝕋.

(b) Since 𝕋^{−} is a thick subcategory of C_{X}, by 2.4.1(b), there is a natural embedding Spec(X) ↪ Spec(|𝕋^{−}|) ⋃ Spec(X/|𝕋^{−}|), and by (a) above Spec(|𝕋^{−}|) = Spec(|𝕋|), hence the assertion.

2.5.4. The property (sup). Recall that X (or the corresponding category C_{X}) has the property (sup) if for any ascending chain, Ω, of subobjects of an object M, the supremum of Ω exists, and for any subobject L of M, the natural morphism

sup(N ∩ L | N ∈ Ω) ⟶ (supΩ) ∩ L

is an isomorphism.

Recall that a subcategory 𝕊 of C_{X} is called coreflective if the inclusion functor 𝕊 ↪ C_{X} has a right adjoint. In other words, the subcategory 𝕊 is coreflective iff every object of C_{X} has a biggest subobject which belongs to 𝕊.

2.5.5. Lemma. (a) Any coreflective thick subcategory of an abelian category C_{X} is a Serre subcategory.

(b) If X has the property (sup), then any Serre subcategory of C_{X} is coreflective.

Proof. See [R, III.2.4.4].

2.5.5.1. Note. If C_{X} is a category with small coproducts, then a thick subcategory of C_{X} is coreflective iff it is closed under small coproducts (taken in C_{X}).

2.5.6. Proposition. Let 𝕋 be a full subcategory of an abelian category C_{X} which is closed under taking subquotients.

(a) The smallest thick subcategory containing 𝕋 is generated by objects M which have a finite filtration, 0 = M_{0} ↪ M_{1} ↪ … ↪ M_{n} = M such that M_{i+1}/M_{i} belongs to 𝕋 for 0 ≤ i ≥ n − 1.

(b) If X has the property (sup), then an object M of C_{X} belongs to 𝕋^{−} iff it has an increasing filtration {M_{i} | i ≥ 0} such that M = sup(M_{i}|i ∈ J) and M_{i} = sup(M_{j}|j < i) for every limit ordinal i, and M_{i+1}/M_{i} belongs to 𝕋 if i is not a limit ordinal.

Proof. (a) Let 𝕋_{n} denote the full subcategory of C_{X} whose objects, M, have a finite filtration, 0 = M_{0} ↪ M_{1} ↪ … ↪ M_{n} = M such that M_{i+1}/M_{i} belongs to 𝕋 for 0 ≤ i ≥ n − 1. Thus, 𝕋_{1} = 𝕋. It is easy to see that all 𝕋_{n} are subcategories of the smallest thick subcategory, 𝕋_{∞}, containing 𝕋. On the other hand, if 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence such that M′ ∈ Ob𝕋_{n} and M′′ ∈ Ob𝕋_{m}, then M ∈ Ob𝕋_{n+m}. Besides, every subcategory 𝕋_{n} is closed under taking subquotients. Therefore 𝕋_{∞} = ⋃_{n≥0} 𝕋_{n}.

(b) Let 𝕋̃ denote (temporarily) the full subcategory of C_{X} whose objects are supremums of their subobjects which belong to 𝕋_{∞}. The claim is that 𝕋̃ coincides with 𝕋^{−}.

Since 𝕋 is closed under taking subquotients, it is contained in 𝕋^{−}, hence 𝕋_{∞} is a subcategory of 𝕋^{−}, Thanks to the property (sup), 𝕋^{−} is closed under supremums, which shows that 𝕋̃ ⊆ 𝕋^{−}.

Notice that the map assigning to every object M of the category C_{X} the supremum of all its subobjects which belong to 𝕋_{∞} defines a functor which is a right adjoint to the inclusion functor 𝕋̃ ↪ C_{X}. This shows that 𝕋̃ is coreflective. It is easy to show that it is also thick. By 2.5.5, 𝕋̃ = 𝕋̃^{−}. Therefore, the inclusions 𝕋 ⊆ 𝕋̃ ⊆ 𝕋^{−} imply that 𝕋̃ = 𝕋^{−}.

Thus, every object of 𝕋^{−} has an increasing filtration by objects of 𝕋_{∞}. The assertion now follows from (a).

2.5.6.1. Remark. Suppose 𝕋 is a strictly full coreflective subcategory of C_{X}, i.e. the inclusion functor has a right adjoint, or, equivalently, every object, M, of C_{X} has the 𝕋-torsion which is, by definition, the biggest subobject, 𝔱_{𝕋}(M), of M which belongs to 𝕋. Then the object M has a canonical filtration defined as follows: M_{0} = 0; if i is not a limit ordinal, then M_{i+1}/M_{i} = 𝔱_{𝕋}(M/M_{i}) (in particular, M_{1} = 𝔱_{𝕋}(M)); if i is a limit ordinal, then M_{i} = sup(M_{j}|j < i). If X has the property (sup) and the subcategory 𝕋 is closed under taking subquotients, than sup(M_{i}) is the 𝕋^{−}-torsion of M. In particular, 𝕋^{−} consists of all objects M such that M = sup(M_{i}).

3. Local ’spaces’, the S-spectrum, and the complete spectrum.

3.1. Local ’spaces’ and categories. Let X be an object of |𝔄𝔟|^{o}, i.e. C_{X} is an abelian category. A nonzero object P of the category C_{X} is called quasi-final if N ≻ P for every nonzero object N of C_{X}, or, equivalently, ⟨P ⟩ = 0.

It follows that any quasi-final object of C_{X} belongs to Spec(X) and all quasi-final objects are equivalent to each other.

We call X (and the category C_{X}) local if C_{X} has a quasi-final object.

If X is local, then Spec(X) has a (unique) final object which coincides with the smallest topologizing subcategory of C_{X}. This final object is a unique closed point of the spectrum (in the topology associated with the preorder ⊇).

3.1.1. Proposition. Let X be local, and let the category C_{X} have simple objects. Then all simple objects of C_{X} are isomorphic to each other, and any quasi-final object is a direct sum of copies of a simple object.

Proof. See [R, III.3.1.2].

3.1.2. Proposition. For any P ∈ Spec(X), the quotient ’space’ X/|⟨P ⟩| (defined by C_{X/|⟨P ⟩|} = C_{X}/⟨P ⟩) is local.

Proof. The localization functor q^{∗} : C_{X} ⟶ C_{⟨P ⟩} is exact and its kernel coincides with ⟨P ⟩. The latter means that every nonzero object of C_{X/|⟨P ⟩|} is isomorphic to an object q^{∗}(N) such that N ≻ P. Since q^{∗} is exact and preserves colimits of small diagrams, it respects the preorder ≻. In particular, q^{∗}(N) ≻ q^{∗}(P). Since P does not belong to the subcategory ⟨P ⟩, the object q^{∗}(P) is nonzero. All together shows that q^{∗}(P) is a quasi-final object.

3.2. Residue ’space’ and residue (skew) field of a local category. Let X be local, and let P be a quasi-final object of the category C_{X}. Since N ≻ P for every nonzero object of C_{X}, the object P belongs to all nonzero topologizing subcategories of C_{X}, i.e. [P] is the smallest nonzero topologizing subcategory of C_{X}. We denote the local ’space’ |[P]| by X_{⋆} and call it the residue ’space’ of X. The inclusion functor [P] ↪ C_{X} is regarded as an inverse image functor of a morphism X → X_{⋆}.

3.2.1. The residue skew field of a local ’space’. Suppose that the category C_{X} has a simple object, M. We denote by k_{X} the ring C_{X}(M, M)^{o} opposite to the ring of endomorphisms of the object M. Since M is simple, k_{X} is a skew field which we call the residue skew field of the local ’space’ X. It follows from 3.1.1 that the residue skew field of X (if any) is defined uniquely up to isomorphism.

It follows that x̄ is naturally isomorphic to the (categoric) spectrum of k_{X}, i.e. the category Cx̄ is equivalent to the category k_{X} − mod of modules (vector spaces) over k_{X}.

3.3. The S-spectrum. We define the S-spectrum of X as the preorder (with respect to ⊇) of Serre subcategories, P, of the category C_{X} such that the quotient ’space’ X/|P| (or the quotient category C_{X}/P) is local. We denote the S-spectrum of X by Spec^{−}(X).

The S-spectrum of X can be viewed as the preordered set of equivalence classes of exact localizations X_{P} →^{q_{P}} X such that the ’space’ X_{P} is local and Ker(q_{P}^{∗}) is a Serre subcategory

of C_{X}. The preorder is given by ’specialization’: a point x′ →^{q′} X of the spectrum is a specialization of x →^{q} X if q′ factors through q. One might regard local ’spaces’ as ’fat points’, and interpret exact localizations from ’fat points’ to X as geometric points of X.

3.3.1. The spectrum and the S-spectrum. It follows from 2.5.3 and 3.1.2 that for any X ∈ Ob|𝔄𝔟|^{o}, the map

Spec(X) ⟶ Spec^{−}(X), [P] ⟼ ⟨P ⟩,

is a monomorphism of preorders which identifies Spec(X) with a subpreorder of Spec^{−}(X).

3.3.2. Locally noetherian ’spaces’. We call X ∈ Ob|𝔄𝔟|^{o} locally noetherian if the category C_{X} is a locally noetherian abelian category. It is argued in [R, Ch. VI] that if X is locally noetherian, then elements of Spec^{−}(X) are in one-to-one correspondence with the set of isomorphism classes of indecomposable injectives of the category C_{X}. In other words, Spec^{−}(X) is isomorphic to the Gabriel spectrum of the category C_{X}.

3.3.3. Remark. If X is the spectrum of a commutative noetherian ring R (i.e. C_{X} = R − mod), then the Gabriel spectrum of C_{X} (hence Spec^{−}(X)) is isomorphic to the prime spectrum of the ring R [Gab]. If R is a non-noetherian commutative ring, Spec^{−}(X) might be much bigger than the prime spectrum of R, while Spec(X) is naturally isomorphic to the prime spectrum of R (cf. [R], Ch.3).

This suggests that Spec(X) is a better candidate for the role of the underlying space of a commutative (and a noncommutative) scheme X. Here C_{X} is the category of quasicoherent sheaves on X. We shall see that this is the case if C_{X} is the category of quasicoherent sheaves of a quasi-compact quasi-separated scheme. In the case of a general scheme, Spec(X) can be used as a building block for a (re)construction of the underlying space.

3.4. Functorial properties of the S-spectrum. Let |L_{s}𝔄𝔟|^{o} denote a diagram scheme (in terminology of [GZ, Dictinary]) whose objects are X ∈ Ob|𝔄𝔟|^{o} with the property (sup). Morphisms of |L_{s}𝔄𝔟|^{o} are exact localizations X ⟶^{u} Y such that the kernel, Ker(u^{∗}), of an inverse image functor of u is a Serre subcategory of the category C_{Y}.

3.4.1. Lemma. The composition of morphisms of |L_{s}𝔄𝔟|^{o} is a morphism of |L_{s}𝔄𝔟|^{o}; i.e. |L_{s}𝔄𝔟|^{o} is a subcategory of |𝔄𝔟|^{o}.

Proof. Let X ⟶^{f} Y and Y ⟶^{g} Z be morphisms of |L_{s}𝔄𝔟|^{o}. Set 𝕊 = Ker(f^{∗}) and 𝕋 = Ker(g^{∗}). Let M be an object of C_{Z} which does not belong to the kernel of

C_{Z} ⟶^{f^{∗}g^{∗}} C_{X}, i.e. g^{∗}(M) does not belong to the Serre subcategory 𝕊 of C_{Y}. Since Y has the property (sup), the category 𝕊 is coreflective (cf. 2.5.5). Denote by M_{𝕊} the quotient of the object M by its 𝕊-torsion and by ϕ the canonical epimorphism g^{∗}(M) ⟶ M_{𝕊}.

Since g^{∗} is a localization, there exists a diagram M ⟵^{s} K′ ⟶^{ϕ′} M′ such that g^{∗}(s) is an isomorphism, ϕ′ is an epimorphism, and there exists an isomorphism g^{∗}(M′) ⟶^{t} M_{𝕊} making the diagram

g^{∗}(M) ⟶^{ϕ}  M_{𝕊}
g^{∗}(s) ↑  ↑ t
g^{∗}(K′) ⟶^{ϕ′} g^{∗}(M′)

commute. Replacing K′ by the image, K, of s and M′ by the coproduct M′′ = K ∐_{K′} M′, we can assume that s is a monomorphism, i.e. M′ is a subquotient of the object M. Finally, replacing M′ by the quotient of M′ by its 𝕋-torsion (which is possible due to the coreflectivity of 𝕋), we obtain a nonzero subquotient of M which is both 𝕊- and 𝕋torsion free. Therefore it cannot have a nonzero subobject which is Ker(f^{∗}g^{∗})-free. This shows that M ∉ Ker(f^{∗}g^{∗})^{−}. Therefore Ker(f^{∗}g^{∗})^{−} ⊆ Ker(f^{∗}g^{∗}), i.e. Ker(f^{∗}g^{∗}) = Ker(f^{∗}g^{∗})^{−}.

3.4.2. Proposition. Any morphism Y ⟶^{ϕ} X of the category |L_{s}𝔄𝔟|^{o} induces an

injective map of spectra Spec^{−}(Y) ⟶^{ϕ^{a}} Spec^{−}(X). The correspondence

X ⟼ Spec^{−}(X), ϕ ↦ ϕ^{a},  (1)

is functorial.

Proof. The map Spec^{−}(Y) ⟶^{ϕ^{a}} Spec^{−}(X) is given by q ⟼ ϕ ∘ q. The functoriality of the map (1) is immediate.

3.4.3. Remark. Let U ⟶^{u} X be a morphism of |L_{s}𝔄𝔟|^{o}; i.e. u^{∗} is a localization whose kernel is a Serre subcategory of C_{X}. By 3.4.2 and 3.3.1, we have canonical injective maps Spec(U) ⟶ Spec^{−}(U) ⟶ Spec^{−}(X). This allows to identify Spec(U) with a subset of Spec^{−}(X).

3.5. The complete spectrum. The complete spectrum of X is a preorder (with resp. to ⊇) of all thick subcategories P such that the quotient ’space’ X/|P| is local. We denote it by Spec^{1}(X). Obviously, Spec^{−}(X) ⊆ Spec^{1}(X). It follows that the map X ⟼ Spec^{1}(X) is functorial with respect to exact localizations. Thus, for any exact localization U ⟶^{u} X, we have canonical embeddings

Spec(U) ↪ Spec^{−}(U) ↪ Spec^{1}(U) ↪ Spec^{1}(X)

which allow to identify Spec(U) with a subset of Spec^{1}(X). In terms of this identification, the local property of the spectrum (2.4.2) can be reformulated as follows:

For any set {U_{i} ⟶^{u_{i}} X | i ∈ J} of exact localizations, Spec(X) ↪ ⋃_{i∈J} Spec(U_{i}).

3.5.1. Note. The complete spectrum was first introduced in [R, Ch.VI], together with the S-spectrum. The S-spectrum of a ’space’ X can be viewed as the smallest extension of the spectrum Spec(X) containing Spec(U) for all Serre localizations U ↪ X. Similarly, the complete spectrum of a ’space’ X is the smallest extension of Spec(X) containing Spec(U) for all exact localizations U ↪ X. If C_{X} is a category with colimits (say, a Grothendieck category), then Serre localizations are exact localizations with a direct image functor. Even if X corresponds to a commutative scheme (i.e. C_{X} is the category of quasi-coherent sheaves on a scheme) and U ⟶^{u} X corresponds to an open immersion, the existence of a direct image functor is a certain finiteness condition.

4. The pretopology of Serre localizations and the S-spectrum.

4.1. Lemma. Let C_{X} be an abelian category. For any finite set {T_{i} | i ∈ J} of topologizing subcategories of C_{X}, we have the equality ( ⋂_{i∈J} T_{i})^{−} = ⋂_{i∈J} T_{i}^{−}.

Proof. Clearly ( ⋂_{i∈J} T_{i})^{−} ⊆ ⋂_{i∈J} T_{i}^{−}. We need to prove the inverse inclusion.

Let J = {1, 2, ..., n}. Let M be a nonzero object of ⋂_{i∈J} T_{i}^{−}. And let L be any nonzero

subquotient of the object M. Since M is a nonzero object of T_{1}^{−}, the object L has a nonzero subobject, L_{1}, which belongs to T_{1}. Since M is a nonzero object of T_{2}^{−} and L_{1} is a nonzero subquotient of M, the object L_{1} has a nonzero subobject, L_{2}, which belongs to T_{2}. Since T_{1} contains all subobjects of its objects, L_{2} ∈ T_{1} ⋂ T_{2}. Continuing this way, we obtain a descending chain of nonzero subobjects of L, L_{n} → L_{n−1} → ... → L_{1} → L, such

that L_{i} ∈ Ob ⋂_{1≤j≤i} T_{j}. This shows that M ∈ ( ⋂_{i∈J} T_{i})^{−}.

4.2. The Gabriel multiplication. The Gabriel product of two subcategories 𝕋 and 𝕊 of an abelian category C_{X} is the full subcategory 𝕋•𝕊 of C_{X} generated by all M ∈ ObC_{X} for which there exists an exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

with M′ ∈ Ob𝕊 and M′′ ∈ Ob𝕋. If 𝕋 and 𝕊 are topologizing subcategories, then such is 𝕋 • 𝕊. This multiplication is associative and has an identity element – the subcategory 0. Note that a topologizing subcategory 𝕋 of C_{X} is thick iff 𝕋 • 𝕋 = 𝕋.

By Lemma III.6.2.1 in [R], if 𝕋, 𝕊 are coreflective subcategories of C_{X}, their Gabriel product 𝕋 • 𝕊 is coreflective too.

4.2.1. Lemma. Let C_{X} be an abelian category. For any finite set {S, T_{i} | i ∈ J} of topologizing subcategories of the category C_{X}, the following equalities hold:

( ⋂_{i∈J} T_{i}) • S = ⋂_{i∈J} T_{i} • S and S • ( ⋂_{i∈J} T_{i}) = ⋂_{i∈J} S • T_{i}.

Proof. (a) The inclusions ( ⋂_{i∈J} T_{i}) • S ⊆ ⋂_{i∈J} T_{i} • S and S • ( ⋂_{i∈J} T_{i}) ⊆ ⋂_{i∈J} S • T_{i} are

evident. We need to prove the inverse inclusions.

(b) Let M ∈ Ob ⋂_{i∈J} T_{i} • S; i.e. for any i ∈ J, there is a monomorphism f_{i} : M_{i} → M

such that M_{i} ∈ ObT_{i} and Cok(f_{i}) ∈ ObS. This gives an exact sequence

0 ⟶ ⋂_{i∈J} M_{i} ⟶ M ⟶ ∏_{i∈J} Cok(f_{i})

Clearly ⋂_{i∈J} M_{i} ∈ Ob ⋂_{i∈J} T_{i}. Since J is finite, ∏_{i∈J} Cok(f_{i}) ∈ ObS, hence M ∈ Ob( ⋂_{i∈J} T_{i}) • S.

(c) Suppose S • M ∈ Ob ⋂_{i∈J} T_{i}; i.e. for any i ∈ J, there is a monomorphism M_{i} ⟶^{f_{i}} M

such that M_{i} ∈ ObS and Cok(f_{i}) ∈ ObT_{i}. Since J is finite, sup(M_{i}|i ∈ J) ∈ ObS, and

M/sup(M_{i}|i ∈ J) ∈ Ob ⋂_{i∈J} T_{i}, hence M ∈ ⋂_{i∈J} S • T_{i}.

For any pair S and T of Serre subcategories of the category C_{X}, the symbol S ∨ T denotes the minimal Serre subcategory of C_{X} containing S and T. It follows that S ∨ T = (S • T)^{−}.

4.3. Proposition. Let C_{X} be an abelian category. For any finite set {T_{i}|i ∈ J} of Serre subcategories of the category C_{X}, the equality ( ⋂_{i∈J} T_{i}) ∨ S = ⋂_{i∈J}(T_{i} ∨ S) holds.

Proof. By Lemma 4.1, ⋂_{i∈J}(T_{i} ∨ S) = ⋂_{i∈J}(T_{i} • S)^{−} = ( ⋂_{i∈J}(T_{i} • S))^{−}. By Lemma 4.2.1,

( ⋂_{i∈J}(T_{i} • S))^{−} = ((⋂_{i∈J} T_{i}) • S)^{−} = ( ⋂_{i∈J} T_{i}) ∨ S.

4.4. The pretopology of Serre localizations. We define the quasi-pretopology of Serre localizations, τ_{𝔏_{𝔰}}, on the category |𝔏_{𝔰}𝔄𝔟|^{o} by taking as covers all families of morphisms {U_{i} →^{u_{i}} X | i ∈ J} such that the corresponding family of inverse image functors is conservative. We define by τ_{𝔏_{𝔰}}^{𝔣} the quasi-pretopology on |𝔏_{𝔰}𝔄𝔟|^{o} obtained by taking all covers of τ_{𝔏_{s}} containing a finite subcover.

It follows from Proposition 4.3 that τ_{𝔏_{s}}^{𝔣} is a Grothendieck pretopology. We call it the pretopology of Serre localizations.

4.5. The local property of the S-spectrum.

4.5.1. Proposition. Let {U_{i} →^{u_{i}} X | i ∈ J} be a cover in the pretopology τ_{𝔏_{s}}^{𝔣}. Then Spec^{−}(X) = ⋃_{i∈J} Spec^{−}(U_{i}).

Proof. The inclusion ⋃_{i∈J} Spec^{−}(U_{i}) ⊆ Spec^{−}(X) follows from the functoriality of the

S-spectrum (see 3.4.1). We need to check the inverse inclusion.

Let x →^{q} X be any point of Spec^{−}(X). Consider cartesian squares

\begin{matrix}U_{i}^{x} & ⟶^{q_{i}} & U_{i} \\ u_{i}^{x} ↓ & & ↓ u_{i} \\ x & ⟶^{q} & X & i ∈ J \end{matrix}  (1)

Since τ_{𝔏_{s}}^{𝔣} is a pretopology, the pull-back {U_{i}^{x} →^{u_{i}^{x}} x | i ∈ J} of the cover {U_{i} →^{u_{i}} X | i ∈ J} is a cover. Let P be a quasi-final object of the local category C_{x}. Since (by the definition of

a cover) the set of inverse image functors {C_{x} ⟶^{u_{i}^{x∗}} C_{U_{i}^{x}} | i ∈ J} is conservative, u_{j}^{x∗}(P) ≠ 0 for some j ∈ J. Since P is the point of the spectrum of x, the object u_{j}^{x∗}(P) belongs to the spectrum, hence to the flat spectrum, of U_{j}^{x}. By functoriality of the S-spectrum (cf.

3.4.1), the morphism U_{j}^{x} ⟶^{q_{j}} U_{j} induces a map Spec^{−}(U_{j})^{x} ⟶ Spec^{−}(U_{j}) which sends the point ⟨u_{j}^{x∗}(P)⟩ to a point P_{j} of Spec^{−}(U_{j}). It follows from the commutativity of (1)

that the image of P_{j} by U_{j} ⟶^{u_{j}} X coincides with x →^{q} X.

4.5.2. Corollary. Let {U_{i} ⟶^{u_{i}} X | i ∈ J} be a finite conservative set of exact localizations. Then Spec^{−}(X) ⊆ ⋃_{i∈J} Spec^{−}(U_{i}). Here Spec^{−}(X) and Spec^{−}(U_{i}), i ∈ J,

are realized as subsets of the complete spectrum Spec^{1}(X).

Proof. Let T_{i} denote the kernel of the localization functor C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}}, i ∈ J. The condition that {U_{i} ⟶^{u_{i}} X | i ∈ J} is conservative means that ⋃_{i∈J} T_{i} = 0. By

4.1, ⋂_{i∈J} T_{i}^{−} = ( ⋂_{i∈J} T_{i})^{−} = 0, or, equivalently, \{Ũ_{i} \underrightarrow{_{u_{i}}} X | i ∈ J\} is a finite cover of X by Serre localizations. Here Ũ_{i} denote the ’space’ X/|T_{i}^{−}| (i.e. C_{U_{i}} = C_{X}/T_{i}^{−}).

Since T_{i} ⊆ T_{i}^{−}, the localization Ũ_{i} ⟶ X factors through a localization Ũ_{i} ⟶ U_{i} with the kernel T_{i}^{−}/T_{i}. The latter is a Serre subcategory of the quotient category C_{X}/T_{i}. Therefore, Spec^{−}(Ũ_{i}) ⊆ Spec^{−}(U_{i}). By 4.5.1, Spec^{−}(X) = ⋃_{i∈J} Spec^{−}(Ũ_{i}) ⊆ ⋃_{i∈J} Spec^{−}(U_{i}).

4.6. The pretopology of exact localizations and the complete spectrum. There are similar facts for exact localizations and complete spectrum.

For any pair S and T of Serre subcategories of the category C_{X}, the symbol S ⊔ T denotes the minimal thick subcategory of C_{X} containing S and T.

4.6.1. Proposition. Let C_{X} be an abelian category. Let {T_{i}|i ∈ J} be a finite family of thick subcategories of the category C_{X}. Then ( ⋂_{i∈J} T_{i}) ⊔ S = ⋂_{i∈J}(T_{i} ⊔ S) for any thick

subcategory S.

Proof. The inclusion ( ⋂_{i∈J} T_{i}) ⊔ S ⊆ ⋂_{i∈J}(T_{i} ⊔ S) is evident. We need to prove the

inverse inclusion. Let T denote the intersection ⋃_{i∈J} T_{i}. Replacing S by S ⊔ T and C_{X} by the quotient category C_{X}/T, we can assume that T = 0. Then by 4.3, we have the inclusions

S = ( ⋂_{i∈J} T_{i}) ⊔ S ⊆ ⋂_{i∈J}(T_{i} ⊔ S) ⊆ ⋂_{i∈J}(T_{i} ∨ S) = ( ⋂_{i∈J} T_{i}) ∨ S = S,

hence the assertion.

4.6.2. The pretopology of exact localizations. Let |𝔏_{𝔢}𝔄𝔟|^{o} be a category whose objects are ’spaces’ X with abelian category C_{X} and morphisms are exact localizations. We define the quasi-pretopology of exact localizations, τ_{𝔏_{𝔢}}, on the category |𝔏_{𝔢}𝔄𝔟|^{o} by taking as covers all families of morphisms {U_{i} →^{u_{i}} X | i ∈ J} such that the corresponding family of inverse image functors is conservative. We define by τ_{𝔏_{𝔢}}^{𝔣} the quasi-pretopology on |𝔏_{𝔢}𝔄𝔟|^{o} obtained by taking all covers of τ_{𝔏_{e}} containing a finite subcover.

It follows from Proposition 4.6.1 that τ_{𝔏_{e}}^{𝔣} is a Grothendieck pretopology. We call it the pretopology of exact localizations, or simply the pretopology of localizations.

The following assertion is refered to as the local property of the complete spectrum.

4.6.3. Proposition. Let {U_{i} →^{u_{i}} X | i ∈ J} be a cover in the pretopology τ_{𝔏_{e}}^{𝔣}. Then Spec^{1}(X) = ⋃_{i∈J} Spec^{1}(U_{i}).

Proof. The argument is similar to that of 4.5.1. Details are left to the reader.

5. The upper spectrum of a quasi-topological ’space’.

We fix a quasi-presite (A, τ), where A is a subcategory of |𝔄𝔟|^{o} and τ-covers are sets

of morphisms {U_{i} →^{u_{i}} Y | i ∈ J} such that {C_{Y} →^{u_{i}^{∗}} C_{U_{i}} | i ∈ J} is a conservative family of exact localizations.

For any X ∈ ObA, we define the upper spectrum, Spec(X, τ_{X}), of the quasi-topological ’space’ (X, τ_{X}) as the (preordered) subset of the complete spectrum Spec^{1}(X) formed by all elements P such that the embedding of the quotient local ’space’ X/|P| → X factors through an embedding Spec(U) ↪ Spec^{1}(X) for some element of a cover U ⟶^{u} X. In other words, Ker(u^{∗}) ⊆ P and P/Ker(u^{∗}) = ⟨M⟩ for some object M of Spec(U). It follows from this definition that

Spec(X, τ_{X}) = ⋃_{(U→X)∈Σ^{τ} X} Spec(U).

Here Spec(U) is identified with its image in Spec^{1}(X). Since Σ^{τ} is closed under composition, the map X ⟼ Spec(X, τ_{X}) is functorial with respect to Σ^{τ}.

5.1. Observations. (a) If A coincides with the category |𝔏_{𝔢}𝔄𝔟|^{o} and τ is a discrete pretopology (i.e. all covers are isomorphisms), then Spec(X, τ_{X}) = Spec(X).

(b) If A is the category |𝔏_{𝔰}𝔄𝔟|^{o} and τ is the quasi-pretopology of Serre localizations, τ_{𝔏_{s}}, or its finite version, the pretopology τ_{𝔏_{s}}^{𝔣} (cf. 4.3, 4.4), then Spec(X, τ_{X}) coincides with the S-spectrum: Spec(X, τ_{X}) = Spec^{−}(X).

(c) If A is the category |𝔏_{𝔰}𝔄𝔟|^{o} and τ is the quasi-pretopology of exact localizations, τ_{𝔏_{𝔢}}, or its finite version, the pretopology τ_{𝔏_{𝔢}}^{𝔣} (cf. 4.3, 4.4), then Spec(X, τ_{X}) coincides with the complete spectrum: Spec(X, τ_{X}) = Spec^{1}(X).

(d) In general, there are inclusions

Spec(X) ⊆ Spec(X, τ_{X}) ⊆ Spec^{1}(X).

If all elements of τ_{X}-covers are Serre localizations, than

Spec(X) ⊆ Spec(X, τ_{X}) ⊆ Spec^{−}(X).

Due to (b) above, the following fact might be regarded as an extension to the general case of the ’local property’ of the S-spectrum (see 4.5.1).

5.2. Proposition. Suppose Σ^{τ} X ⊆ Σ_{τ} X. Then Spec(X, τ_{X}) = ⋃_{i∈J} Spec(U_{i}, τ_{U_{i}})

for any τ-cover {U_{i} → X | i ∈ J}.

Proof. The inclusion ⋃_{i∈J} Spec(U_{i}, τ_{U_{i}}) ⊆ Spec(X, τ_{X}) follows from the definition of

Spec(X, τ_{X}) and does not require any additional assumptions.

Let P be an arbitrary element of Spec(X, τ_{X}) and V → X a morphism of Σ^{τ} X such that P ∈ Spec(V, τ_{V}). Let {Ũ_{i} → V | i ∈ J} be a pull-back of the cover {U_{i} → X | i ∈ J} along the morphism V → X. The condition Σ^{τ} ⊆ Σ_{τ} means that {Ũ_{i} → V | i ∈ J} is a cover. In particular, the set of inverse image functors of morphisms of the cover is conservative. Therefore, there exists i ∈ J such that P ∈ Spec(Ũ_{i}) (see 2.4.4). Since the projection Ũ_{i} → U_{i} is an element of a cover (belongs to Σ^{τ}, this implies that P ∈ Spec(U_{i}, τ_{U_{i}}).

5.3. A natural topology on the upper spectrum. We define a topology, τ^{X}, on Spec(X, τ_{X}) by taking 𝔅_{τ} X = {Spec(U, τ_{U}) ⊆ Spec(X, τ_{X}) | (U → X) ∈ Σ^{τ} X} as a base of its open sets.

5.3.1. Lemma. Suppose that Σ^{τ} X ⊆ Σ_{τ} . Then the intersection of any pair of sets of the base 𝔅_{τ} X belongs to 𝔅_{τ} X.

Proof. The condition Σ^{τ} X ⊆ Σ_{τ} X implies that for any pair U ⟶ X ⟵ V of morphisms of Σ^{τ} X, the other two morphisms (projections) of the cartesian square

U ×_{X} V ⟶ U
↓ ↓
V  ⟶ X

and the composition U ×_{X} V ⟶ X belong to Σ^{τ}. The fact follows now from the equality

Spec(U ×_{X} V, τ_{U×_{X}V}) = Spec(U, τ_{U}) ⋂ Spec(V, τ_{V}).

The inclusion Spec(U ×_{X} V, τ_{U×_{X}V}) ⊆ Spec(U, τ_{U}) ⋂ Spec(V, τ_{V}) is evident. The inverse inclusion follows from the definition of the spectrum and the universal property of fibered products. Details are left to the reader.

5.4. The upper spectrum of covers. Let U = {U_{i} ⟶^{u_{i}} X | i ∈ J} be a set of exact localizations. We call Spec^{†}U = ⋃_{i∈J} Spec(U_{i}) the upper spectrum of the family U. If follows from 2.4.2 that if a cover V of X is a refinement of the cover U, then there is a natural injective map Spec^{†}U ⟶ Spec^{†}V. It follows that Spec(X, τ_{X}) is isomorphic to the colimit of Spec^{†}U, where U runs through the set of all τ-covers of X.

6. The combinatorial spectrum.

6.1. The lower spectrum of a cover and of a quasi-topological ’space’. Fix a quasi-presite (A, τ), where A is a subcategory of the category |𝔄𝔟|^{o}. We assume that all τ-covers are conservative families {U_{j} ⟶^{u_{j}} X | j ∈ J} of exact localizations; i.e. all inverse

image functors C_{X} ⟶^{u_{j}^{∗}} C_{U_{j}} are exact localizations and the family of functors {u_{j}^{∗} | j ∈ J} is conservative.

For any cover U = {U_{i} ⟶^{u_{i}} X | i ∈ J}, we define the lower spectrum of U as the set Spec_{℘}U of all P ∈ Spec^{1}(X) such that if P ∈ Spec^{1}(U_{j}) (i.e. Ker(u_{j}^{∗}) ⊆ P) for some j ∈ J, then P ∈ Spec(U_{j}). The latter means that there exists an object L of C_{X} such that P/Ker(u_{j}^{∗}) = ⟨u_{j}^{∗}(L)⟩.

If follows from 2.4.2 that if a cover V of X is a refinement of the cover U, then there is a natural injective map Spec_{℘}U ⟶ Spec_{℘}V. We denote by Spec_{℘}(X, τ_{X}) the colimit of Spec_{℘}U, where U runs through the set Cov_{τ} (X) of all τ-covers of X.

For any cover U of X, we have Spec(X) ⊆ Spec_{℘}U ⊆ Spec(X, τ_{X}), which imply the inclusions

Spec(X) ⊆ Spec_{℘}(X, τ_{X}) ⊆ Spec(X, τ_{X}).

Let τ^{X} denote the topology on Spec_{℘}(X, τ_{X}) induced by the natural topology on Spec(X, τ_{X}) (cf. 5.3). We call the topological space (Spec_{℘}(X, τ_{X}), τ^{X}) the combinatorial τ-spectrum of X.

6.1.1. A remark on spectra of a commutative scheme. For an arbitrary ringed topological space X = (X , O), we denote by SpX the categoric spectrum of X defined by taking as C_{SpX} the category Qcoh_{X} of quasi-coherent sheaves on X. Let A be the category whose objects are categoric spectra of commutative schemes and morphisms are morphisms of spectra corresponding to open immersions. Let τ be the pretopology on A corresponding to the Zariski pretopology on the category of schemes. Then for any object X = Sp(X , O) of the category A, the combinatorial spectrum Spec_{℘}(X, τ_{X}) and the spectrum Spec(X, τ_{X}) of (X, τ_{X}) both coincide with the underlying space X of the scheme X. But, according to an observation by O. Gabber, the spectrum Spec(X) might differ from the space X.

6.1.2. The combinatorial S-spectrum. For any cover U = {U_{i} ⟶^{u_{i}} X | i ∈ J} of (X, τ_{X}), we define the lower S-spectrum, Spec_{℘}^{−}U, of the cover U as the intersection of Spec_{℘}U and the S-spectrum of X. In other words, Spec_{℘}^{−}U is the set of all P ∈ Spec^{−}(X) such that if P ∈ Spec^{1}(U_{j}) for some j ∈ J, then P ∈ Spec(U_{j}).

Let Spec_{℘}^{−}(X, τ_{X}) denote the intersection of the combinatorial spectrum of (X, τ_{X}) with the S-spectrum of X and call it the combinatorial S-spectrum of (X, τ_{X}).

We show below, among other facts, that the natural map Spec(X) ⟶ Spec_{℘}^{−}(X, τ_{X}) is bijective, if (X, τ_{X}) is quasi-compact (i.e. every τ-cover of X contains a finite subcover).

One of the consequences of this fact is that if C_{X} is the category of quasi-coherent sheaves on a scheme X, then Spec(X) is isomorphic to the set of points of the underlying topological space of X, provided that the scheme X is quasi-compact and quasi-separated, or its underlying space is noetherian.

We need some preparation.

6.2. Proposition. Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a finite set of exact localizations such that the family of inverse image functors {u_{i}^{∗} | i ∈ J} is conservative. Let M, N be objects of C_{X} such that N is Ker(u_{i}^{∗})-torsion free. Then M ≻ N iff u_{i}^{∗}(M) ≻ u_{i}^{∗}(N) for all i ∈ J.

Proof. The implication [M ≻ N] ⇒ [u_{i}^{∗}(M) ≻ u_{i}^{∗}(N) for all i ∈ J] holds without any additional conditions on N, because all functors u_{i}^{∗} are exact, hence they respect the relation ≻. The rest of the argument shows that the inverse implication holds provided the object N is Ker(u_{i}^{∗})-torsion free for all i ∈ J.

(a) The relation u_{i}^{∗}(M) ≻ u_{i}^{∗}(N) means that there exists a diagram

u_{i}^{∗}(M)^{⊕J_{i}} ⟵^{𝔧_{i}} K̃_{i} ⟶^{𝔢_{i}} u_{i}^{∗}(N)  (1)

in which 𝔧_{i} is a monomorphism, 𝔢_{i} is an epimorphism, and J_{i} is finite. Since u_{i}^{∗} is a localization functor, we can assume that K̃_{i} = u_{i}^{∗}(K_{i}) for some K_{i} ∈ ObC_{X}. To the diagram (1) there corresponds a diagram

M^{⊕J_{i}} ⟶^{s} M′_{i} ⟵^{𝔧′_{i}} K_{i} ⟶^{𝔢′_{i}} N_{i} ⟵^{t} N  (2)

such that the morphisms u_{i}^{∗}(s) and u_{i}^{∗}(t) are invertible, and the arrows of (1) are described by 𝔧_{i} = u_{i}^{∗}(s)^{−1}u_{i}^{∗}(𝔧′_{i}) and 𝔢_{i} = u_{i}^{∗}(t)^{−1}u_{i}^{∗}(𝔢′_{i}). Here we identify u_{i}^{∗}(M)^{⊕J_{i}} with u_{i}^{∗}(M^{⊕J_{i}}).

Since 𝔧_{i} is a monomorphism, the Ker(𝔧′_{i}) belongs to the kernel of the functor u_{i}^{∗}. Replacing K_{i} by the image of the morphism 𝔧′_{i} and N_{i} by the cokernel of the composition

Ker(𝔧′_{i}) ⟶ K_{i} ⟶^{𝔢′_{i}} N_{i}, we can (and will) assume that M′_{i} ⟵^{𝔧′_{i}} K_{i} is a monomorphism. This implies that the projection K′_{i} = K_{i} ×_{M′_{i}} (M^{⊕J_{i}}) ⟶ M^{⊕J_{i}} is a monomorphism.

(b) Consider now the cartesian square

K′′_{i}  ⟶^{𝔢′′_{i}} N
𝔧′′_{i} ↓ ↓ t
K′_{i} ⟶ N_{i}

in which the lower horizontal arrow is the composition of the projection K′_{i} ⟶ K_{i} and

the morphism K_{i} ⟶^{𝔢′_{i}} N_{i}. Since u_{i}^{∗}(t) is invertible, the kernel of the morphism t belongs to Ker(u_{i}^{∗}). By hypothesis, N is Ker(u_{i}^{∗})-torsion free. Therefore t is a monomorphism.

This implies that K′′_{i} ⟶^{𝔧′′_{i}} K′_{i} is a monomorphism, hence the composition of K′_{i} ⟶^{𝔧′′_{i}} K′_{i} and K′_{i} ⟶ M^{⊕J_{i}} is a monomorphism. Thus, we have obtained a diagram

M^{⊕J_{i}} \underleftarrow{_{𝔧_{i}}} K′′_{i} \underrightarrow{_{𝔢_{i}}} N  (3)

whose left arrow is a monomorphism and such that its image by the localization functor u_{i}^{∗} is isomorphic to the diagram (1).

(c) Let I denote the disjoint union of J_{i}, i ∈ J, and let K = ⊕_{i∈J}K′′_{i}. Monomorphisms {𝔧̃_{i} | i ∈ J} determine a monomorphism K ⟶^{𝔧} M^{⊕I}. The morphisms 𝔢̃_{i}, i ∈ J, determine a morphism K ⟶^{𝔢} N. We claim that 𝔢 is an epimorphism.

Since the set of functors {u_{i}^{∗} | i ∈ J} is conservative, in particular it reflects epimor-

phisms, it suffices to show that u_{i}^{∗}(K̃) ⟶^{u_{i}^{∗}(𝔢)} u_{i}^{∗}(N) is an epimorphism for every i ∈ J. This is, indeed, the case, because the composition of u_{i}^{∗}(𝔢) with the natural morphism u_{i}^{∗}(K′′_{i}) ⟶ u_{i}^{∗}(K) coincides with u_{i}^{∗}(𝔢̃_{i}), and the latter is an epimorphism, as it is argued above.

6.2.1. Corollary. Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a finite set of exact localizations such that the family of inverse image functors {u_{i}^{∗} | i ∈ J} is conservative. Let P be an object of C_{X} such that P is Ker(u_{i}^{∗})-torsion free for all i ∈ J. Then P ∈ Spec(X) iff u_{i}^{∗}(P) ∈ Spec(U_{i}) for every i ∈ J.

Proof. (a) Suppose P ∈ Spec(X). Since P is Ker(u_{i}^{∗})-torsion free for all i ∈ J, the object u_{i}^{∗}(P) is nonzero for every i ∈ J. Thus u_{i}^{∗}(P) ∈ Spec(U_{i}) for every i ∈ J.

(b) Conversely, suppose that u_{i}^{∗}(P) ∈ Spec(U_{i}) for all i ∈ J. Let M ⟶ P be a nonzero monomorphism. Since P is Ker(u_{i}^{∗})-torsion free for all i ∈ J, the object M has the same property. In particular, u_{i}^{∗}(M) ⟶ u_{i}^{∗}(P) is a nonzero monomorphism for every i ∈ J. Since u_{i}^{∗}(P) ∈ Spec(U_{i}), u_{i}^{∗}(M) ≻ u_{i}^{∗}(P) for all i ∈ J. By 6.2, this implies that M ≻ P. Therefore P ∈ Spec(X).

6.3. Proposition. Suppose C_{X} is an abelian category with the property (sup). Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a finite set of exact localizations such that the family of inverse image functors {u_{i}^{∗} | i ∈ J} is conservative. Then the canonical map

Spec(X) ⟶ Spec_{℘}^{−}U = Spec_{℘}U ⋂ Spec^{−}(X), [P] ⟼ ⟨P ⟩,

is an isomorphism.

Proof. (a) Let P be an element of Spec_{℘}U, and let q^{∗} = q_{P}^{∗} be the canonical localization functor C_{X} ⟶ C_{X}/P. We denote by J_{P} the set of all i ∈ J such that Ker(u_{i}^{∗}) ⊆ P. Set J^{P} = J − J_{P}. Fix an object P of C_{X} such that q^{∗}(P) is a quasi-final object of the category C_{X}/P.

There exists an object L of the subcategory ⋂_{i∈J^{P}} Ker(u_{i}^{∗}) such that q^{∗}(L) is isomorphic

to the quasi-final object q^{∗}(P).

By definition, j ∈ J^{P} iff Ker(u_{j}^{∗}) is not contained in Ker(q^{∗}). Let M be an object of Ker(u_{j}^{∗}) which does not belong to Ker(q^{∗}). Then q^{∗}(M) ≻ q^{∗}(P), i.e. there exists a diagram

q^{∗}(M^{⊕I}) ⟵^{𝔧} q^{∗}(K) ⟶^{𝔢} q^{∗}(P)  (4)

whose left arrow is a monomorphism and the right arrow is an epimorphism. To the diagram (4), there corresponds the diagram

M^{⊕I} ⟶^{s} M′ ⟵^{𝔧′} K ⟶^{𝔢′} P′ ⟵^{t} P  (5)

such that q^{∗}(s) and q^{∗}(t) are invertible and 𝔧 = q^{∗}(s)^{−1}q^{∗}(𝔧′), 𝔢 = q^{∗}(t)^{−1}q^{∗}(𝔢′).

Moreover, we can assume that the arrow 𝔧′ in (5) is a monomorphism (see the part (a) of the argument of 6.2). This implies that the projection K′′ = M^{⊕I} ×_{M′} K ⟶ M^{⊕I} is a monomorphism too. Thus, we obtain the diagram

M^{⊕I} ⟵^{𝔧′′} K′′ ⟶^{𝔢′′} P′_{j} ⟶^{s′} P′ ⟵^{t} P  (6)

in which the arrow 𝔧′′ is a monomorphism, the object P′_{j} is the image of the composition K′′ ⟶ K ⟶ P′ and 𝔢′′ and s′ are resp. an epimorphism and a monomorphism. It follows that q^{∗}(s′) is invertible. Let P_{j} denote the intersection (pull-back) of P′_{j} ⟶^{s′} P′ and the

image of P ⟶^{t} P′. If follows that P_{j} is a subquotient of M^{⊕I} (see (6)). Since M^{⊕I} ∈ ObKer(u_{j}^{∗}), and Ker(u_{j}^{∗}), being a thick subcategory, is closed under taking subquotients, the object P_{j} belongs to Ker(u_{j}^{∗}). It follows from the construction that q^{∗}(P_{j}) is isomorphic to the quasi-final object q^{∗}(P).

Observe that if the initially chosen object P belongs to some thick subcategory 𝕋 (or any subcategory 𝕋 closed under subquotients taken in C_{X}), then the object P_{j}, being a subquotient of P, belongs to the intersection 𝕋 ⋂ Ker(u_{j}^{∗}).

We replace P by P_{j} and apply the same construction to another i ∈ J^{P}. As a result, we obtain an object, P_{ij} which belongs to the intersection Ker(u_{j}^{∗}) ⋂ Ker(u_{i}^{∗}) and such that q^{∗}(P_{ij}) ≃ q^{∗}(P_{j}). Continuing this procedure until all elements of J^{P} are exhausted, we obtain a Ker(q^{∗})-torsion free object P̃ of the subcategory ⋂_{i∈J^{P}} Ker(u_{i}^{∗}) such that q^{∗}(P̃)

is isomorphic to q^{∗}(P); so that q^{∗}(P̃) is a quasi-final object of the category C_{X}/P.

(b) This shows, in particular, that a point P of Spec^{1}(X) can be regarded as a point of Spec^{1}(V_{P}), where V_{P} is defined by C_{V_{P}} = ⋂_{i∈J^{P}} Ker(u_{i}^{∗}). The ’space’ V_{P} is interpreted

as the complement to the ’open subspace’ ⋃_{i∈J^{P}} U_{i} in X.

The set of inverse image functors {C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}} | i ∈ J_{P}} unduces a cover (i.e. a conservative set of localizations) of V_{P}.

In fact, to any morphism u_{i}, i ∈ J_{P}, there corresponds a localization of C_{V_{P}} with the kernel Ker(u_{i}^{∗}) ⋂ C_{V_{P}} (which is a thick subcategory of the category C_{V_{P}}). The intersection of all these kernels coincides with

( ⋂_{i∈J_{P}} Ker(u_{i}^{∗})) ⋂ C_{V_{P}} = ( ⋂_{i∈J_{P}} Ker(u_{i}^{∗})) ⋂ ( ⋂_{j∈J^{P}} Ker(u_{j}^{∗})) = ⋂_{i∈J} Ker(u_{i}^{∗})

and ⋂_{i∈J} Ker(u_{i}^{∗}) = 0, because the set of functors {u_{i}^{∗} | i ∈ J} is conservative by hypothesis.

(c) Replacing X by V_{P}, we shall assume for the rest of the argument that J_{P} = J, i.e. Ker(u_{i}^{∗}) ⊆ P for any i ∈ J.

Let L be an object of C_{X} such that q^{∗}(L) is a quasi-final object of the local category C_{X}/P. There exists an object P of the category C_{X} such that for every i ∈ J, the object u_{i}^{∗}(P) belongs to the spectrum, Spec(U_{i}), of U_{i} and such that q^{∗}(P) is isomorphic to q^{∗}(L). In particular, q^{∗}(P) is a quasi-final object of the local category C_{X}/P.

(c1) Fix an i ∈ J. By hypothesis, Ker(u_{i}^{∗}) ⊆ P and P/Ker(u_{i}^{∗}) belongs to Spec(U_{i}), i.e. there exists an object P_{i} of the category C_{X} such that P/Ker(u_{i}^{∗}) = ⟨u_{i}^{∗}(P_{i})⟩. The object q^{∗}(P_{i}) is quasi-final, hence equivalent to the object q^{∗}(L). In particular, q^{∗}(P_{i}) ≻ q^{∗}(L). The latter relation is expressed by the following diagram:

P_{i}^{⊕n} ⟵^{𝔧} K ⟶^{𝔢} P′_{i} ⟶^{𝔧′} P̃′ ⟵^{𝔢′} L  (7)

in which 𝔧, 𝔧′ are monomorhisms, 𝔢, 𝔢′ are epimorphisms, and the arrows 𝔧′, 𝔢′ belong to Σ_{q^{∗}} = {s ∈ HomC_{X} | q^{∗}(s) is invertible}.

We claim that u_{i}^{∗}(P_{i}) is equivalent (with respect to ≻) to u_{i}^{∗}(P′_{i}).

Let q_{i}^{∗} denote the localization functor C_{U_{i}} ⟶ C_{X}/P = C_{U_{i}}/(P/Ker(u_{i}^{∗})); so that q^{∗} ≃ q_{i}^{∗}u_{i}^{∗}. If u_{i}^{∗}(P′_{i}) ⊁ u_{i}^{∗}(P_{i}), then q^{∗}(P′_{i}) ≃ q_{i}^{∗}(u_{i}^{∗}(P′_{i})) = 0 which contradicts to the isomorphism (following from (7)) q^{∗}(P′_{i}) ≃ q^{∗}(L) and the fact that, by definition, q^{∗}(L) is a nonzero object. Thus, u_{i}^{∗}(P′_{i}) ≻ u_{i}^{∗}(P_{i}). The inverse relation, u_{i}^{∗}(P_{i}) ≻ u_{i}^{∗}(P′_{i}), follows from the relation P_{i} ≻ P′_{i} (see (7)) and the fact that any exact functor, in particular u_{i}^{∗}, preserves the relation ≻.

Notice that since the arrows 𝔧′, 𝔢′ of the diagram (7) belong to Σ_{q^{∗}}, the object q^{∗}(P′_{i}) is isomorphic to q^{∗}(L).

(c2) Now fix another element, j, of J. Repeating the argument of (c1) with the object L replaced by the object P′_{i}, we obtain an object P′_{j} of C_{X} such that P_{j} ≻ P′_{j} and there exists a diagram

P′_{i} ⟶^{𝔢} M ⟵^{𝔧} P′_{j}  (8)

in which 𝔢 is an epimorphism, 𝔧 is a monomorphism, and both arrows belong to Σ_{q^{∗}} (i.e. q^{∗} maps them to isomorphisms). By the argument (c1), the object u_{j}^{∗}(P′_{j}) is equivalent to u_{j}^{∗}(P_{j}), and q^{∗}(P′_{j} is isomorphic to q^{∗}(P′_{j}). Since P′_{i} ≻ P′_{j} (see (8) above) and u_{i}^{∗} is an exact functor, u_{i}^{∗}(P′_{i}) ≻ u_{i}^{∗}(P′_{j}).

On the other hand, u_{i}^{∗}(P′_{j}) ≻ u_{i}^{∗}(P′_{i}), because if not, u_{i}^{∗}(P′_{j}) would belong to Ker(q_{i}^{∗}) = ⟨u_{i}^{∗}(P′_{i})⟩. But, q_{i}^{∗}u_{i}^{∗}(P′_{j}) ≃ q^{∗}(P′_{j}) ≃ q^{∗}(P′_{i}) ≃ q^{∗}(L) ≠ 0..

(c3) Replacing P′_{i} by P′_{j}, we repeat the procedure, and continue doing so until J will be exhausted. As a result, we obtain an object P of C_{X} such that q^{∗}(P) ≃ q^{∗}(L), and u_{i}^{∗}(P) belongs to Spec(U_{i}) for every i ∈ J.

(d) Since X has the property (sup), every Serre subcategory of C_{X}, in particular Ker(q^{∗}), is coreflective (cf. 2.5.5). Replacing the object P constructed in (c) by its quotient by Ker(q^{∗})-torsion of P, we assume that P is Ker(q^{∗})-torsion free. Since Ker(u_{i}^{∗}) is contained in Ker(q^{∗}) for all i ∈ J (cf (c) above), the object P is Ker(u_{i}^{∗})-torsion free for all i ∈ J. It follows from 6.2.1 that P belongs to Spec(X).

6.3.1. Note. The hypothesis that X has the property (sup) was used only in the last step of the proof of 6.3 in order to ensure the existence of a Ker(q^{∗})-torsion free object P such that q^{∗}(P) is a quasi-final object. Thus, the assertion 6.3 is valid without any hypothesis on X (except for C_{X} being abelian), if we will consider only P ∈ Spec_{℘}U and [M] ∈ Spec(X) such that the subcategories P and ⟨M⟩ are coreflective.

6.4. The combinatorial spectrum of a locally quasi-compact ’space’. Fix a quasi-presite (A, τ), where A is a subcategory of |𝔄𝔟|^{o} and τ is a quasi-pretopology on A such that for any cover {U_{j} ⟶^{u_{j}} X | j ∈ J}, the family of inverse image functors

{C_{X} ⟶^{u_{j}^{∗}} C_{U_{j}} | j ∈ J} is a conservative set of exact localizations.

Recall that for any object X of A, we denote by Σ^{τ} X the set of all arrows U ⟶ X which belong to some τ-cover of X. And Σ_{τ} X denotes the class of all morphisms Y ⟶^{f} X such that the pull-back of a τ-cover along f is a τ-cover.

6.4.1. Proposition. Let X be an object of A with the property (sup).

(a) If X is τ-quasi-compact, then the canonical map Spec(X) ⟶ Spec_{℘}^{−}(X, τ_{X}) is an isomorphism.

(b) Suppose X has the property Σ^{τ} X ⊆ Σ_{τ} X. Then Spec_{℘}^{−}(X, τ_{X}) = Spec_{℘}^{−}U for any τ-cover U = {U_{i} →^{u_{i}} X | i ∈ J} such that all U_{i} are quasi-compact.

Proof. Let U and Ũ be τ-covers of X. Suppose U is a subcover of Ũ. Then there are inclusions

Spec(X) ↪ Spec_{℘}^{−}Ũ ↪ Spec_{℘}^{−}U.  (1)

(a) Suppose X is τ-quasi-compact, that is any τ-cover Ũ of X has a finite subcover, say U. By 6.3, the embedding Spec(X) ↪ Spec_{℘}^{−}U (which is the composition of the maps of (1)) is an isomorphism. Therefore Spec(X) ↪ Spec_{℘}^{−}Ũ is an isomorphism. This implies that the map Spec(X) ↪ Spec_{℘}^{−}(X, τ_{X}) := colim (Spec_{℘}^{−}Ũ | Ũ ∈ Cov_{τ} X) is an isomorphism.

(b) Fix a τ-cover U = {U_{i} →^{u_{i}} X | i ∈ J} such that all U_{i} are quasi-compact. Let V = {V_{ν} →^{𝔳_{i}} X | ν ∈ 𝔍} be any other τ-cover of X. The inclusion Σ^{τ} X ⊆ Σ_{τ} X garantees that the covers V and U have a common refinement, V ×_{X} U = {U_{iν}→X | (i, ν) ∈ J × 𝔍}. Here U_{iν} denotes U_{i} ×_{X} V_{ν}. By a general standard argument, the canonical morphism

colim (Spec_{℘}(V ×_{X} U) | V ∈ Cov_{τ} X) ⟶ colim (Spec_{℘}(V) | V ∈ Cov_{τ} X)

is an isomorphism, and, by definition, colim (Spec_{℘}(V) | V ∈ Cov_{τ} X) = Spec_{℘}(X, τ_{X}).

On the other hand, for any τ-cover V = {V_{ν} →^{𝔳_{i}} X | ν ∈ 𝔍} of X, the τ-cover V ×_{X} U ⟶ X is a refinement of the τ-cover U. Thus, we have a canonical morphism Spec_{℘}^{−}U ⟶ Spec_{℘}^{−}(V ×_{X} U). This morphism is an isomorphism.

In fact, for any i ∈ J, the set of morphisms {U_{iν} → U_{i} | ν ∈ 𝔍} is a cover of U_{i}. Since U_{i} is, by hypothesis, quasi-compact, this cover has a finite subcover, Ũ_{i}. There are obvious inclusions (see (1) above):

Spec(U_{i}) ↪ Spec_{℘}^{−}(V ×_{X} U) ↪ Spec_{℘}^{−}(Ũ_{i}).  (2)

Since the τ-cover Ũ_{i} is finite, the canonical map Spec(U_{i}) ↪ Spec_{℘}^{−}(Ũ_{i}), the composition of the arrows (2), is an isomorphism. Therefore the map

Spec(U_{i}) ↪ Spec_{℘}^{−}(V ×_{X} U)  (3)

in (2) is an isomorphism. Since all arrows here are functorial, the isomorphisms (3), i ∈ J, induce an isomorphism Spec_{℘}^{−}U ⟶ Spec_{℘}^{−}(V ×_{X} U), Therefore, the canonical morphism

Spec_{℘}^{−}U ⟶ colim (Spec_{℘}^{−}(V ×_{X} U) | V ∈ Cov_{τ} X)  (4)

is an isomorphism. This shows that Spec_{℘}^{−}U ⟶ Spec_{℘}^{−}(X, τ_{X}) is an isomorphism.

6.5. Extended spectra.

6.5.1. The extended complete spectrum and the extended S-spectrum. For any X ∈ Ob|𝔄𝔟|^{o}, we denote by Spec_{⋆}^{1}X the ordered set Spec^{1}(X) ∪ {⋆_{X}}, where Spec^{1}(X) is a complete spectrum of the ’space’ X and ⋆_{X} is the added point.

If Spec^{1}(X) is realized as the set of ’geometric points’, i.e. morphisms x →^{q} X from local ’spaces’ such that q^{∗} is an exact localization, then ⋆_{X} is becomes the (unique) morphism from the zero ’space’ represented by the zero category. Notice that the zero ’space’ is both final and initial object of the category |𝔄𝔟|^{o}.

If Spec^{1}(X) is realized as the set thick subcategories, P, of C_{X} such that C_{X}/P is a local category, then the zero ’space’ corresponds to the unproper thick subcategory C_{X} ⟶^{Id} C_{X}; i.e. ⋆_{X} is realized as the category C_{X} itself.

We define the extended S-spectrum, by Spec_{⋆}^{−}(X) = Spec^{−}(X) ∪ {⋆_{X}}.

6.5.2. The extended spectrum. If 𝔜 is a (preordered) subset of Spec^{−}(X), the symbol 𝔜_{⋆} will denote 𝔜 ∪ {⋆_{X}}. In particular, we have the extended spectrum, Spec_{⋆}(X) = Spec(X) ∪ {⋆_{X}}.

The extended spectrum Spec_{⋆}(X) can be naturally obtained via an extended version of the spectrum Spec(X) (cf. 2.2). Namely, we define Spec_{⋆}(X) by adjoining to Spec(X) the zero object and imposing its minimality. Notice that ⟨0⟩ is the empty category, ∅. So that passing from Spec_{⋆}(X) to Spec_{⋆}(X) via the map P ⟼ ⟨P ⟩ gives another realization of the initial element, dual to the one above: instead of the trivial subcategory C_{X} ⟶^{Id} C_{X}, we use the trivial subcategory ∅ ↪ C_{X}.

6.5.3. Functorial properties of the extended spectra. Let |𝔏_{𝔢}𝔄𝔟|^{o} denote the subcategory of the category |𝔄𝔟|^{o} formed by all morphisms X ⟶ Y of |𝔄𝔟|^{o} whose inverse image functors are exact localizations (in particular, Ob|𝔏_{𝔢}𝔄𝔟|^{o} = Ob|𝔄𝔟|^{o}). On the other hand, let 𝔒𝔯𝔡_{⋆} denote the category whose objects are preordered sets with the initial (i.e. the smallest) element and arrows are morphisms of ordered sets which map the initial element to the initial element. Finally, let 𝔒𝔯𝔡_{⋆} denote the category objects of which are preorders (i.e. categories with no more than one arrow between any two objects) with an initial object and morphisms are functors which map initial objects to initial objects.

6.5.3.1. Proposition. (a) The map X ⟼ Spec_{⋆}^{−}(X) extends to a contravariant functor, Spec_{⋆}^{−}, from the category |𝔏_{𝔢}𝔄𝔟|^{o} to the category 𝔒𝔯𝔡_{⋆} of preordered sets with the smallest element.

(b) The map X ⟼ Spec_{⋆}(X) extends to a contravariant pseudo-functor from the category |𝔏_{𝔢}𝔄𝔟|^{o} to the category 𝔒_{⋆}Cat of preorders with an initial object. This functor gives a rise to a subfunctor, Spec_{⋆}, or the functor Spec_{⋆}^{−}.

Proof. (a) Let Y ⟶^{f} X be a morphism of |𝔏_{𝔢}𝔄𝔟|^{o}; i.e. C_{X} ⟶^{f^{∗}} C_{Y} is an exact localization. We define a map f^{−} : Spec_{⋆}^{−}X ⟶ Spec_{⋆}^{−}(Y) as follows. Let P ∈ Spec^{−}(X). If Ker(f^{∗}) ⊆ P, then f^{−}(P) = (f^{∗}(P)^{−} – the minimal Serre subcategory spanned by the image of P. If C_{Y} is identified with the quotient category C_{X}/Ker(f^{∗}), then f^{−}(P) is identified with the subcategory P/Ker(f^{∗}). By transitivity of localizations, the quotient category (C_{X}/Ker(f^{∗}))/(P/Ker(f^{∗})) is equivalent to C_{X}/P, hence it is local. This shows that, f^{−}(P) belongs, indeed, to the subset Spec^{−}(Y) of Spec_{⋆}^{−}(Y). If Ker(f^{∗}) is not contained in P, then f^{−} maps the point P to the initial object ⋆_{Y}. It is required that f^{−}(⋆_{X}) = ⋆_{Y}. It is easy to check that thus defined map f ⟼ f^{−} is a functor (|𝔏_{𝔢}𝔄𝔟|^{o})^{op} ⟶ 𝔒𝔯𝔡_{⋆}.

(b) For any morphism Y ⟶^{f} X of the category |𝔏_{𝔢}𝔄𝔟|^{o} we choose an inverse image

functor, f^{∗}. The functor f^{∗} induces a morphism (a functor) Spec_{⋆}(X) \underrightarrow{_{f^{a}}} Spec_{⋆}(Y) between preorders (see the argument of 2.4.2) which, obviously, maps initial objects to initial objects: f^{∗}(0) = 0. Being exact, the functor f^{∗} respects the preorder ≻, hence

defines, via P ⟼ ⟨P ⟩, a map Spec(X) ∪ ∅ ⟶^{f^{a}} Spec(Y) ∪ ∅ uniquely determined by the commutativity of the diagram

Spec_{⋆}(X) \underrightarrow{_{f^{a}}} Spec_{⋆}(Y)
↓  ↓
Spec(X) ∪ ∅ ⟶^{f^{a}} Spec(Y) ∪ ∅

It remains to identify the ∅ with ⋆_{X}.

6.5.4. The extended upper spectrum and combinatorial spectrum of a quasi-topological ’space’. If X is an object of quasi-presite (A, τ), where A is a subcategory of |𝔄𝔟|^{o} and τ-covers are conservative families of localizations, then we have the extended upper spectrum, Spec_{⋆}(X, τ_{X}) = Spec(X, τ_{X}) ∪ {⋆_{X}}, and the extended combinatorial spectrum, Spec_{℘⋆}(X, τ_{X}) = Spec_{℘}(X, τ_{X})∪{⋆_{X}} of the quasi-topological ’space’ (X, τ_{X}). We define the extended versions of the upper S-spectrum and the lower (combinatorial) S-spectrum as intersections of the respective spectra with the family of all Serre subcategories of C_{X}. The inclusions

Spec_{⋆}(X) ⟶ Spec_{℘⋆}^{−}(X, τ_{X}) ⟶ Spec_{⋆}^{−}(X, τ_{X}) ⟶ Spec_{⋆}^{−}(X)
↓  ↓  ↓
Spec_{℘⋆}(X, τ_{X}) ⟶ Spec_{⋆}(X, τ_{X}) ⟶ Spec_{⋆}^{1}(X)

(1)

follow from the definitions of these spectra.

Let 𝔏_{𝔢}A denote the subcategory of the category A formed by all morphisms X ⟶^{f} Y whose inverse image is an exact localization. Since τ-covers consist of exact localizations, quasi-topological ’spaces’ on (𝔏_{𝔢}A, τ) and on (A, τ) are same.

6.5.4.1. Proposition. The maps

(X, τ_{X}) ⟼ Spec_{⋆}(X, τ_{X}) and (X, τ_{X}) ⟼ Spec_{℘⋆}(X, τ_{X})

are extended to contravariant functors from the category 𝔗𝔬𝔭_{(𝔏_{𝔢}A,τ)} of quasi-topological ’spaces’ on (𝔏_{𝔢}A, τ) to the category 𝔒𝔯𝔡_{⋆} of preordered sets with the smallest element.

Proof. The fact can be deduced from 6.5.3.1. Details are left to the reader.

6.5.5. Towards computations of the (extended) combinatorial spectrum. Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a cover of a ’space’ X. For every subset I of J, let C_{U_{I}} be the quotient of the category C_{X} by the thick subcategory 𝕋_{I} = ⋂_{i∈I} Ker(u_{i}^{∗}). The

localization functor C_{X} ⟶^{u_{I}^{∗}} C_{U_{I}} is regarded as an inverse image functor of a morphism U_{I} ⟶^{u_{I}} X. If ℑ is another subset of J, then we have a diagram

U_{𝔍} ; ↓ ; U_{I} ⟶ U_{I∩𝔍} ⟶ X  (1)

Let J_{𝔣𝔱} denote the preordered (with respect to the inclusion) set of finite subsets of J. The map ℑ ⟼ U_{ℑ} is a functor from J_{𝔣𝔱} to the category of ’subspaces’ of the ’space’ X. We denote by U^{𝔣𝔱} the set of morphisms {U_{ℑ} →^{u_{ℑ}} X | ℑ ∈ J_{𝔣𝔱}}.

6.5.5.1. Lemma. There is a natural isomorphism Spec_{℘}^{−}(U^{𝔣𝔱}) ≃ Spec_{℘}^{−}U.

Proof. For any ℑ ∈ J_{𝔣𝔱}, the set of morphisms {U_{i} ⟶^{u_{i}} X | i ∈ ℑ} induces a finite conservative set of localizations U^{ℑ} = {U_{i} → U_{ℑ} | i ∈ ℑ}. By 6.3, the natural map Spec(U_{ℑ}) ⟶ Spec_{℘}^{−}(U^{ℑ}) is an isomorphism. This implies that a canonical map Spec_{℘}^{−}(U^{𝔣𝔱}) ⟶ Spec_{℘}^{−}U (which is due to the fact that the cover U is a refinement of the cover U^{𝔣𝔱}) is an isomorphism. Details are left to the reader.

Let Spec_{℘⋆}U denote the extended spectrum of the cover U defined by Spec_{℘⋆}U = Spec_{℘}U ∪ {⋆_{X}}. In the notations above, we have the following

6.5.5.2. Lemma. Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a τ-cover of a ’space’ X, and let U^{𝔣𝔱} = {U_{𝔍} →^{u_{𝔍}} X | 𝔍 ∈ J_{𝔣𝔱}} be the associated with U filtered cover. There is a natural isomorphism

lim_{𝔍∈J_{𝔣𝔱}} Spec_{℘⋆}^{−}U_{ℑ} ⟶^{∼} Spec_{℘⋆}^{−}U.  (2)

Proof. Thanks to 6.5.5.1, it suffices to show that the canonical morphism

lim_{𝔍∈J_{𝔣𝔱}} Spec_{℘⋆}^{−}U_{ℑ} ⟶^{ϕ_{U}} Spec_{℘⋆}^{−}U^{𝔱𝔱}  (3)

is an isomorphism. But, this follows from the definition of the morphism (3) which assigns to every element (P_{ℑ} | ℑ ∈ J_{𝔣𝔱}) of lim_{𝔍∈J_{𝔣𝔱}} Spec_{℘⋆}^{−}U_{ℑ} the element P of Spec_{⋆}^{−}(X) uniquely

defined by the condition that the image of P in Spec_{⋆}^{−}U_{ℑ} coincides with P_{ℑ} ∈ Spec_{℘⋆}^{−}U_{ℑ}. The map which assigns to each element P of Spec_{℘⋆}^{−}U^{𝔱𝔱} the element (P_{ℑ} | ℑ ∈ J_{𝔣𝔱}) of ∏_{ℑ∈J_{𝔣𝔱}} Spec_{⋆}^{−}U_{ℑ}, where P_{ℑ} is the image of P in Spec_{⋆}^{−}U_{ℑ}, takes values in lim_{𝔍∈J_{𝔣𝔱}} Spec_{℘⋆}^{−}U_{ℑ}

and (its corestriction to lim_{𝔍∈J_{𝔣𝔱}} Spec_{⋆}^{−}U_{ℑ}) is the inverse map to (3).

6.5.5.3. Proposition. Let (A, τ) be a quasi-presite such that A is a subcategory of |𝔄𝔟|^{o} and τ-covers are conservative sets of exact localizations. Let U = {U_{i} →^{u_{i}} X | i ∈ J} be a τ-cover of a ’space’ X, and let U^{𝔱} = {U_{𝔍} →^{u_{𝔍}} X | 𝔍 ∈ J_{𝔣𝔱}} be the associated with U filtered cover. Suppose X has the property (sup) and Σ_{τ} X ⊆ Σ^{τ} X. If U_{i} is quasi-compact for all i ∈ J, then there is a natural isomorphism

lim_{𝔍∈J_{𝔣𝔱}} Spec_{℘⋆}^{−}U_{ℑ} ⟶^{∼} Spec_{℘⋆}^{−}(X, τ_{X}).  (4)

Proof. The assertion follows from 6.5.5.2 and 6.4.1.

7. Closed coimmersions and immersions. Zariski pretopology.

7.1. Closed coimmersions. We call a morphism X ⟶^{𝔳} V_{𝔳} a closed coimmersion if its inverse image functor (is fully faithful and) induces an equivalence between C_{V_{𝔳}} and a coreflective topologizing subcategory of the category C_{X}. In particular, 𝔳 has a direct image functor (a right adjoint to 𝔳^{∗}).

Notice that the spectrum is (contravariantly) functorial with respect to closed coimmersions: to any closed coimmersion X ⟶ Y, there corresponds a natural embedding Spec(Y) ↪ Spec(X) and this correspondence commutes with the composition.

This functoriality with respect to closed coimmersions extends to the S-spectrum.

7.2. Quasi-pretopologies associated with families of closed coimmersions. Let C_{V_{𝔳}^{−}} be the Serre subcategory of C_{X} generated by the image of C_{V_{𝔳}}. Let C_{U_{𝔳}} be the quotient C_{X}/C_{V_{𝔳}^{−}} and u_{𝔳}^{∗} the localization functor. We have an exact sequence of categories and functors C_{V_{𝔳}^{−}} ⟶ C_{X} ⟶ C_{U_{𝔳}} which corresponds to the first two arrows of the diagram of ’spaces’

U_{𝔳} ⟶^{u_{𝔳}} X ⟶^{𝔳^{−}} V_{𝔳}^{−} ⟶ V_{𝔳}.  (1)

Let A be a subcategory of |𝔄𝔟|^{o}. For any X ∈ ObA, we choose a family ℭ_{X} of closed coimmersions. For any element X ⟶^{𝔳} V_{𝔳} of ℭ_{X}, we declare the morphism U_{𝔳} ⟶^{u_{𝔳}} X being an element of a cover, an ’open subspace’. Covers are sets {U_{𝔳} ⟶^{u_{𝔳}} X | 𝔳 ∈ 𝔙} of ’open subspaces’ such that the family of inverse image functors {u_{𝔳}^{∗} | 𝔳 ∈ 𝔙} is conservative.

We assume that ℭ_{X} contains the zero coimmersion X ⟶ •. Here C_{•} = 0. The corresponding open ’subspace’ is X ⟶^{id_{X}} X.

7.2.1. A special case. If A = |𝔄𝔟|^{o} and for any ’space’ X the family ℭ_{X} consists of morphisms corresponding to inclusions of all topologizing subcategories of C_{X}, then the resulting quasi-pretopology is (the restriction to A of) the quasi-pretopology τ^{𝔏} of exact localizations defined in 1.5.2.

Notice that ℭ_{X} is closed with respect to all intersections (which are colimits of ’small’ families {X ⟶^{𝔳} V_{𝔳} | 𝔳 ∈ 𝔙} of arrows of ℭ_{X}) and with respect to the Gabriel multiplication (cf. 4.2). It follows from 4.1 that the ’open subspace’ corresponding to the intersection of a finite family {X ⟶^{𝔳} V_{𝔳} | 𝔳 ∈ 𝔙} of arrows of ℭ_{X} is the union (i.e. colimit) of the corresponding family {U_{𝔳} ⟶^{u_{𝔳}} X | 𝔳 ∈ 𝔙} of ’open subspaces’.

The ’open subspace’ corresponding to the Gabriel product of two closed coimmersions, V_{𝔳} ⟵^{𝔳} X ⟶^{𝔴} V_{𝔴}, is the intersection of the associated ’open subspaces’, i.e. the fibered product of U_{𝔳} ⟶^{u_{𝔳}} X ⟵^{u_{𝔴}} U_{𝔴} (see the paragraph preceeding to 4.3).

7.2.2. The quasi-pretopology τ_{𝔷}. Let the family of closed coimmersions ℭ_{X} consists of morphisms X ⟶ |[M]|, where M is an object of C_{X} of finite type, [M] the minimal topologizing coreflective subcategory of C_{X} containing M, and C_{|[M]|} = [M].

If C_{X} = R−mod for a commutative ring R, then Spec(X) = SpecR, and the topology on SpecR corresponding to τ_{𝔷} coincides with the Zariski topology.

7.2.3. The quasi-pretopology τ_{𝔯}. This time, ℭ_{X} consists of the zero morphism and all morphisms X ⟶ |[P]|, with P ∈ Spec(X).

If the category C_{X} has enough objects of finite type (that is every nonzero object of C_{X} has a nonzero subobject of finite type), then the quasi-pretopology τ_{𝔯} is coarser than the quasi-pretopology τ_{𝔷}.

7.3. Closed immersions. Zariski quasi-presite. We call a morphism V ⟶^{𝔣} X a closed immersion if it is continuous and its direct image functor 𝔣_{∗} : C_{V} ⟶ C_{X} identifies the category C_{V} with a topologizing subcategory of the category C_{X}. In other words, 𝔣_{∗} is an inverse image functor of a closed coimmersion.

7.3.1. Proposition. Let C_{X} be the category R − mod of R-modules over an associative unital ring R. Then there is a bijective correspondence between isomorphism classes of closed immersions and two-sided ideals of the ring R.

Proof. To any two-sided ideal α in the ring R, we assign the category R/α − mod of R/α-modules which we identify with the full subcategory of the category R − mod whose objects are modules annihilated by α. It is shown in [R, Ch.3, 6.4.1] that this map establishes the desired bijective correspondence.

7.3.2. Zariski quasi-presite. Let τ_{X}^{ℨ} be the family of all isomorphism classes of closed immersions to X. We denote the corresponding class of covers of X by τ_{X}^{ℨ}. This defines the Zariski quasi-pretopology on |𝔄𝔟|^{o}.

7.4. Proposition. Let C_{X} be the category of quasi-coherent sheaves on a scheme X = (X , O). Let τ_{X} be the Zariski pretopology τ^{ℨ}, and τ^{X} topology induced by τ_{X}^{ℨ} on the spectrum Spec(X, τ_{X}^{ℨ}). Then the underlying space X of the scheme X is isomorphic to the upper spectrum (Spec(X, τ_{X}^{ℨ}), τ^{X}) of the quasi-topological ’space’ (X, τ_{X}^{ℨ}).

The upper spectrum Spec(X, τ_{X}^{ℨ}) coincides with the lower spectrum Spec_{℘}(X, τ_{X}^{ℨ}).

Proof. Let Y →^{𝔧} X be a closed subscheme of the scheme X with the defining ideal J_{Y}. Then a direct image functor, 𝔧_{∗}, of the embedding 𝔧 induces an equivalence of the category Qcoh_{Y} of quasi-coherent sheaves on Y to the full subcategory 𝕋_{Y} of the category C_{X} = Qcoh_{X} formed by quasi-coherent O-modules annihilated by the ideal J_{Y}. The functor M ⟼ O/J_{Y} ⊗_{O} M is a left adjoint to the inclusion functor 𝕋_{Y} ↪ C_{X}; i.e. 𝔧_{∗} is a direct image of a closed immersion in the sense of 7.3.

Conversely, let 𝕋 be a subcategory of C_{X} such that the inclusion functor, 𝕋 ↪^{𝔧_{∗}} C_{X}, is a direct image functor of a closed immersion (in the sense of 7.3). Then an adjunction morphism Id_{C_{X}} ⟶^{η_{𝔧}} 𝔧_{∗}𝔧^{∗} is an epimorphism. In particular, we have an epimorphism O ⟶ 𝔧_{∗}𝔧^{∗}(O) of quasi-coherent sheaves with the kernel J_{𝕋}. Objects of the subcategory 𝕋 are precisely those quasi-coherent O-modules which are annihilated by the ideal J_{𝕋}. Thus, the pretopology on X defined by τ is the same as the pretopology defined by Zariski topology. Therefore the map of sets X ⟶ Spec(X, τ_{X}) is a bijection, which implies that the map X ⟶ (Spec(X, τ_{X}), τ^{X}) of topological spaces is an isomorphism.

The equality Spec_{℘}(X, τ_{X}^{ℨ}) = Spec(X, τ_{X}^{ℨ}) is left to the reader.

7.5. Locally Zariski quasi-pretopology. Let (X, τ_{X}) be a quasi-topological space. We call the quasi-pretopology τ_{X} locally Zariski, if there exists a conservative family of morphisms {U_{i} ⟶^{u_{i}} X∥ i ∈ J} such that all u_{i} belong to Σ^{τ} (cf. 1.5.1) and for every i ∈ J, the restriction τ_{U_{i}} of τ_{X} to U_{i} is the Zariski pretopology.

8. The center and the geometric center of a quasi-topological ’space’.

8.1. The center. For any ’space’ X, we denote by O^{𝔠}(X) the center of the category C_{X}, i.e. the (commutative) monoid of all endomorphisms of the identical functor Id_{C_{X}}. If C_{X} is an additive category, then O^{𝔠}(X) is a commutative unital ring.

8.1.1. Example. If R is an associative unital ring and X = Sp(R), i.e. C_{X} = R − mod, then O^{𝔠}(X) is naturally isomorphic to the center of the ring R.

8.1.2. Lemma. The map X ⟼ O^{𝔠}(X) is functorial with respect to localizations.

Proof. Let C_{Y} ⟶^{u^{∗}} C_{X} be a localization, and let ξ ∈ O^{𝔠}(Y). Then u^{∗}(ξ) ∈ O^{𝔠}(X). In fact, let Σ_{u} denote the family of all s ∈ HomC_{Y} such that u^{∗}(s) is invertible.

Any morphism M_{1} ⟶^{φ} M_{n} of the category C_{X} can be represented as the composition

u^{∗}(f_{n})u^{∗}(s_{n})^{−1} … u^{∗}(f_{1})u^{∗}(s_{1})^{−1}, where M_{i} ⟵^{s_{i}} N_{i} ⟶^{f_{i}} M_{i+1} and s_{i} ∈ Σ_{u} for all i. Since

the diagram

M_{i} ⟵^{s_{i}}  N_{i}  ⟶^{f_{i}} M_{i+1}
ξ(M_{i}) ↓  ξ(N_{i}) ↓  ↓ ξ(M_{i+1})
M_{i} ⟵^{s_{i}}  N_{i}  ⟶^{f_{i}} M_{i+1}

commutes, u^{∗}(ξ(M_{i+1})) ∘ (u^{∗}(f_{i})u^{∗}(s_{i})^{−1}) = (u^{∗}(f_{i})u^{∗}(s_{i})^{−1}) ∘ u^{∗}(ξ(M_{i})) for all i. Therefore,

u^{∗}(ξ(M_{n})) ∘ φ = u^{∗}(ξ(M_{n})) ∘ (u^{∗}(f_{n})u^{∗}(s_{n})^{−1} … u^{∗}(f_{1})u^{∗}(s_{1})^{−1})
= (u^{∗}(f_{n})u^{∗}(s_{n})^{−1} … u^{∗}(f_{1})u^{∗}(s_{1})^{−1}) ∘ u^{∗}(ξ(M_{1})) = φ ∘ u^{∗}(ξ(M_{1})).

Hence the assertion.

8.1.3. The center of a quasi-topological ’space’. Let (X, τ) be a quasi-topological ’space’ such that C_{X} is an additive category. By 8.1.2, the map (U →^{u} X) ⟼ O^{𝔠}(U) defines a presheaf, O_{X}^{𝔠}, of commutative rings on (X, τ). We call this presheaf (or, ruther, the ringed ’space’ (X, τ; O_{X}^{𝔠})) the center of (X, τ).

8.2. Spectra of a ’space’ and the prime spectrum of its center.

8.2.1. Proposition. Suppose C_{X} is an abelian category.

(a) If X is local, then O^{𝔠}(X) is a local ring.

(b) Let P be an element of Spec^{1}(X), i.e. P is a thick subcategory of C_{X} such that X/|P| is local; and let X/|P| ⟶^{u_{P}} X be the localization at P. Suppose M is a quasi-final object of C_{X}/P. Then 𝔭_{P} = {ξ ∈ O^{𝔠}(X) | u_{P}^{∗}(ξ)(M) = 0} is a prime ideal in the ring O^{𝔠}(X). The ideal 𝔭_{P} does not depend on the choice of a quasi-final object M.

(c) For every object M of Spec(X) and every ξ ∈ O^{𝔠}(X), either ξ(M) = 0, or ξ(M) is a monomorphism. In particular, the set {ξ ∈ O^{𝔠}(X) | ξ(M) = 0} is a prime ideal in the ring O^{𝔠}(X). This ideal depends only on the equivalence class of M.

Proof. (a) Let M be a quasi-final object of the category C_{X}. By the argument of [R, 3.2.1], the set 𝔪_{X} = {ξ ∈ O^{𝔠}(X) | ξ(M) = 0} is the unique maximal ideal in the ring O^{𝔠}(X). In other words, for any ξ ∈ O^{𝔠}(X), either ξ(M) = 0, or ξ is an invertible element.

(b) By (the argument of) 8.1, the map ξ ⟼ u_{P}^{∗}(ξ) is a ring morphism O^{𝔠}(X) ⟶ O^{𝔠}(X/|P|). By (a), the set 𝔭_{P} = {ξ ∈ O^{𝔠}(X) | u_{P}^{∗}(ξ)(M) = 0} is the preimage of the unique maximal ideal of the local ring O^{𝔠}(X/|P|), hence it is a prime ideal.

(c) See [R, 7.1.2].

8.2.2. Corollary. Let C_{X} be an abelian category. There exist natural maps from the complete spectrum Spec^{1}(X) of the ’space’ X to the prime spectrum, Spec(O_{X}^{𝔠}), of the center of X and from the spectrum Spec(X) to Spec(O_{X}^{𝔠}). These maps are related by the commutative diagram

Spec(X) ⟶ Spec^{1}(X)
↑ ↓
Spec(X) ⟶ Spec(O_{X}^{𝔠})

(1)

The diagram (1) is functorial with respect to localizations.

Proof. The right vertical arrow and the lower horizontal arrow of (1) are provided by the assertions resp. 8.2.1(b) and 8.2.1(c). The left vertical arrow in (1) is the canonical map M ⟼ [M], and the upper horizontal arrow assigns to each element [M] of Spec(X) the corresponding Serre subcategory ⟨M⟩. Verifying details is left to the reader.

8.3. The geometric centers of a quasi-topological ’space’. Let (X, τ_{X}) be a quasi-topological ’space’. Let Spec^{•}(X, τ_{X}) be any spectrum of this work realized as a subset of the complete spectrum Spec^{1}(X) and endowed with the topology induced by τ_{X}. With any subset V of Spec^{1}(X), we associate the quotient category C_{X}/𝕊_{V}, where 𝕊_{V} = ⋂_{P∈V} P. The map Õ_{X}^{𝔠} which assigns to every open set U of Spec^{•}(X, τ_{X}) the center of the category C_{X}/𝕊_{U} is a presheaf of commutative rings on Spec^{•}(X, τ_{X}). We denote by O_{X}^{𝔠} the associated sheaf and call the ringed topological space (Spec^{•}(X, τ_{X}), O_{X}^{𝔠}) the geometric center of the ’space’ (X, τ_{X}) associated with the spectrum Spec^{•}(X, τ_{X}).

In particular, we associate with the ’space’ (X, τ_{X}) the ringed topological spaces (Spec_{℘}(X, τ_{X}), O_{X}^{𝔠}) and (Spec(X, τ_{X}), O_{X}^{𝔠}) called respectively the lower (or combinatorial) and the upper geometric center of (X, τ_{X}).

8.4. Zariski geometric centers. The reconstruction of commutative schemes. Taking the Zariski pretopology, we assign to every X ∈ Ob|𝔄𝔟|^{o} the lower and the upper Zariski geometric center of X, resp. (Spec_{℘}(X, τ_{X}^{ℨ}), O_{X}^{𝔠}) and (Spec(X, τ_{X}^{ℨ}), O_{X}^{𝔠}). If C_{X} is the category of quasi-coherent sheaves on a commutative scheme, then, by 7.4, these two spectra coincide.

8.4.1. Proposition. Let C_{X} be the category of quasi-coherent sheaves on a scheme X = (X , O). Then the Zariski geometric center of X is isomorphic to the scheme X.

Proof. (a) Suppose first that the space X is quasi-compact. By 7.4, the topological space (Spec(X, τ_{X}^{ℨ}), τ^{ℨ}) is isomorphic to the underlying space X of the scheme X. Let U = (U, O|_{U}) be an affine open subscheme of the scheme X corresponding to an element U ⟶ X of an affine cover. Then the category Qcoh_{U} of quasi-coherent sheaves on U is equivalent to the category C_{U}. On the other hand, it is equivalent to the category of O(U)-modules. The center of the category of O(U)-modules is naturally isomorphic to the ring O(U). This shows that the presheaf which assigns to any affine open set U corresponding to an element U ⟶ X of τ^{ℨ}-cover of X the center of U (that is the center of the category C_{U}) is isomorphic to a presheaf which assigns to an affine open set U the ring O(U) of global sections of the structure sheaf O over U. Therefore associated sheaves of these presheaves are isomorphic, hence the assertion.

(b) Consider now the general case. Let X be represented as a union (colimit) of a filtered system, {U_{ν} | ν ∈ 𝔍}, of its quasi-compact open subschemes, U_{ν} = (U_{ν}, O|_{U_{ν}}). Let C_{U_{ν}} be the category of quasi-coherent sheaves on the scheme U_{ν}. By 6.5.5.3,

lim_{ν∈𝔍} Spec_{⋆}(U_{ν}) ⟶^{∼} Spec_{℘⋆}(X_{𝔠}, τ_{X_{𝔠}}^{ℨ}).

This isomorphism extends to an isomorphism of ringed spaces

lim_{ν∈𝔍}(Spec_{⋆}(U_{ν}), O_{U_{ν}}^{𝔠}) ⟶^{∼} (Spec_{℘⋆}(X_{𝔠}, τ_{X_{𝔠}}^{ℨ}), O_{X}^{𝔠}).  (1)

By (a), for any ν ∈ 𝔍, there is a natural isomorphism

(Spec(U_{ν}), O_{U_{ν}}^{𝔠}) ⟶^{∼} (U_{ν}, O|_{U_{ν}}), and lim_{ν∈𝔍}(U_{ν}, O|_{U_{ν}})_{⋆} ⟶^{∼} (X , O)_{⋆} = X_{⋆}.

Here (X , O)_{⋆} is a ringed space obtained by adjoining to (X , O) one point with zero ring.

9. Noncommutative schemes and their spectra.

9.1. Affine morphisms. A continuous morphism X ⟶^{f} Y of ’spaces’ is called affine if its direct image functor is conservative and has a right adjoint. It follows that the composition of affine morphisms is affine, and any isomorphism is affine.

9.1.1. Example: affine morphisms to Sp(R). For any associative ring R, the ’space’ Sp(R) is defined by C_{Sp(R)} = R − mod. The ’space’ Sp(R) is called the categoric spectrum of the ring R. Let R ⟶^{φ} A be an associative ring morphism. We denote by φ̃ the morphism Sp(A) ⟶ Sp(R) with the inverse image functor

R − mod \underrightarrow{φ^{∗}} A − mod, M ⟼ A ⊗_{R} M.

The morphism φ̃ is affine. Its direct image functor, φ̃_{∗}, is the pull-back along φ; and the functor

R − mod \underrightarrow{φ^{!}} A − mod, M ⟼ Hom_{R}(φ̃_{∗}(A), M),

is right adjoint to φ̃_{∗}. By [KR3, 6.6.1], any affine morphism X ⟶ Sp(R) is a composition of an isomorphism X ⟶^{∼} Sp(A) for an associative ring A (hence the category C_{X} is equiv-

alent to the category of left A-modules) and a morphism Sp(A) \underrightarrow{_{φ}} Sp(R) corresponding to a unital ring morphism R ⟶^{φ} A.

The ring morphism R ⟶^{φ} A can be chosen uniquely up to isomorphism. In fact, let X ⟶^{f} Sp(R) be an affine morphism with an inverse image functor f^{∗}. Set O = f^{∗}(R) and denote by φ the composition of the natural ring isomorphism R^{o} ⟶^{∼} Hom_{R}(R, R) and the morphism Hom_{R}(R, R) ⟶ C_{X}(O, O), u ↦ f^{∗}(u). Here R^{o} is the ring opposite to R. Since an inverse image functor is determined uniquely up to isomorphism, the object O and the ring morphism R ⟶^{φ^{o}} A = C_{X}(O, O)^{o} are determined uniquely up to isomorphism.

9.2. Locally affine ’spaces’ over a ’space’. Let (X, τ) be a quasi-topological ’space’, and let X ⟶^{f} Y be a morphism of ’spaces’. We call (X, τ; f) a quasi-topological ’space’ over Y.

A morphism U ⟶^{u} X is called f-affine if the composition U ⟶^{fu} Y is affine. The morphism U ⟶^{u} X is called an affine open ’supspace’ of (X, τ; f) if fu is affine and u belongs to Σ^{τ} (i.e. u belongs to some τ-cover).

A quasi-topological ’space’ (X, τ; f) over Y is said to be locally affine if the family of morphisms {U ⟶^{u} X | u ∈ Σ^{τ} , fu is affine} is conservative.

Any conservative set, 𝔘 = {U_{i} ⟶^{u_{i}} X | i ∈ J}, of affine open ’subspaces’ is called an affine cover of the scheme (X, τ; f).

9.2.1. Locally affine ’spaces’ over Sp(R). Let (X, τ) be a quasi-topological ’space’ and R a unital associative ring. Let X ⟶^{f} Sp(R) be a locally affine morphism, and let 𝔘 = {U_{i} ⟶^{u_{i}} X | i ∈ J} be an affine cover of (X, f). Set O = f^{∗}(R) and O_{U_{i}} = u_{i}^{∗}(O), i ∈ J. Since morphisms fu_{i} are affine and O_{U_{i}} = u_{i}^{∗}f^{∗}(R) ≃ (fu_{i})^{∗}(R), the object O_{U_{i}} is a projective cogenerator of finite type with a structure of a right R-module. The latter is a ring morphism R^{o} ⟶^{ϕ_{i}} C_{U_{i}}(O_{U_{i}}, O_{U_{i}}).

9.3. Quasi-topological ℤ-’spaces’ and ℤ-schemes. A ℤ-’space’ is a pair (X, f), where f is a continuous morphism X ⟶ Sp(ℤ). We denote by 𝔈𝔰𝔭_{ℤ} the full subcategory of |𝔄𝔟|^{o}/Sp(ℤ) whose objects are ℤ-’spaces’.

Let Esp_{ℤ} denote the category whose objects are pairs (X, O_{X}), where X is a ’space’ such that C_{X} is an abelian category and O_{X} is an object of C_{X} such that there exist coproducts of copies of O. Morphisms (X, O_{X}) ⟶ (Y, O_{Y}) are morphisms X ⟶^{f} Y such that f^{∗}(O_{Y}) ≃ O_{X}.

For every object (X, O_{X}) of the category Esp_{ℤ}, let ΓO_{X} denote the ring C_{X}(O_{X}, O_{X})^{o} opposite to the ring of endomorphisms of the object O_{X}. By [BD, 6.6.23], the functor

C_{X} ⟶^{Γ_{∗}} ΓO_{X} − mod, M ⟼ C_{X}(O_{X}, M),

has a left adjoint; i.e. it is a direct image functor of a continuous morphism.

9.3.1. Proposition. The category 𝔈𝔰𝔭_{ℤ} of ℤ-’spaces’ is naturally equivalent to the category Esp_{ℤ}.

Proof. Let Φ_{∗} denote the map which assigns to each ℤ-’space’ (X, f) the pair (X, O_{X}), where O_{X} = f^{∗}(ℤ) for some inverse image functor f^{∗} of the morphism X ⟶^{f} Sp(ℤ), and to every morphism (X, f) ⟶^{h} (Y, g) the morphism (X, O_{X}) ⟶^{h} (Y, O_{Y}). Since the functor f^{∗} has a right adjoint, it preserves ’small’ coproducts; in particular it maps free ℤ-modules to coproducts of copies of O_{X} = f^{∗}(ℤ). Thus Φ_{∗} is a functor from 𝔈𝔰𝔭_{ℤ} to Esp_{ℤ}. The quasi-inverse functor, Φ^{∗}, assigns to each object (X, O_{X}) of the category Esp_{ℤ} the pair (X, f), where f is the composition of the ”global sections” morphism X ⟶ Sp(ΓO_{X}) and the natural morphism Sp(ΓO_{X}) ⟶ Sp(ℤ).

9.3.2. Definitions. (a) A quasi-topological ℤ-’space’ is a triple (X, τ; f), where (X, τ) is a quasi-topological ’space’ and (X, f) is a ℤ-’space’. In other words, (X, τ; f) is a quasi-topological ’space’ over Sp(ℤ) such that f is a continuouos morphism.

(b) A quasi-topological ℤ-’space’ (X, τ; f) is an affine (Zariski) scheme if the morphism f is affine and τ is the Zariski pretopology.

(c) A quasi-topological ℤ-’space’ (X, τ; f) is called a (Zariski) ℤ-scheme if there exists a conservative cover {U_{i} ⟶^{u_{i}} X | i ∈ J} such that all u_{i} belong to Σ^{τ}, fu_{i} is an affine morphism, and the induced quasi-pretopology τ_{U_{i}} on U_{i} is the Zariski pretopology.

9.3.3. Proposition. (a) For every quasi-topological ℤ-’space’ (X, τ; O_{X}), the presheaf Γ̃O_{X} defined by Γ̃O_{X}(U → X) = ΓO_{U} = C_{U}(O_{U}, O_{U})^{o} is a presheaf of algebras over the center of (X, τ).

(b) A quasi-topological ℤ-’space’ (X, τ; f) is affine iff O_{X} = f^{∗}(ℤ) is a projective generator of finite type and τ is the Zariski pretopology.

(c) A quasi-topological ℤ-’space’ (X, τ; f) is a ℤ-scheme iff there exists a conservative family of morphisms {U_{i} ⟶^{u_{i}} X | i ∈ J} such that each u_{i} belongs to Σ^{τ} and O_{U_{i}} = u_{i}^{∗}(O_{X}) is a projective generator of finite type and the induced quasi-pretopology τ_{U_{i}} on U_{i} is the Zariski pretopology.

Proof. (a) In fact, for every arrow U ⟶^{u} X of Σ^{τ}, we have a natural ring morphism O_{U}^{𝔠} ⟶ ΓO_{U} = C_{U}(O_{U}, O_{U})^{o} taking values in the center of the ring ΓO_{U}.

(b) A morphism X ⟶^{f} Sp(ℤ) is affine iff the associated with f canonical morphism X ⟶ Sp(ΓO_{X}) is an isomorphism, or, equivalently, the functor

C_{X} ⟶^{Γ_{∗}} ΓO_{X} − mod, M ⟼ C_{X}(O_{X}, M),

is an equivalence of categories. By a well-known result of Gabriel-Mitchell, this happens iff O_{X} is a projective generator of finite type.

(c) The assertion follows from (a) and the definition of a ℤ-scheme.

9.4. Spectra of noncommutative schemes.

9.4.1. Spectra of affine schemes. Let (X, τ) be a quasi-topological ’space’ such that X = Sp(R) for some associative unital ring R and τ is the Zariski topology. Then (X, τ) is a quasi-compact ’space’ [R1]. Therefore, by 6.4.1(a), the combinatorial spectrum, Spec_{℘}(X, τ), of the ’space’ (X, τ) coincides with Spec(X).

If the ring R is noncommutative, then the upper spectrum of (X, τ) might be larger than Spec(X).

9.4.2. The general case. Let (X, τ; X →^{f} Y) be a scheme over Y with an affine cover 𝔘 = {U_{i} →^{u_{i}} X | i ∈ J}. Suppose that X has the property (sup), Σ^{τ} ⊆ Σ_{τ} (cf. 1.5.1), and that every U_{i} → X, i ∈ J, is a Serre localization. Then, by 6.4.1, the combinatorial spectrum of (X, τ) coincides with the spectrum Spec_{℘}𝔘 of the cover 𝔘. If the cover 𝔘 is finite, then Spec_{℘}(X, τ) = Spec(X).

By 5.2, the upper spectrum of (X, τ) equals to the union, ⋃_{i∈J} Spec(U_{i}, τ_{U_{i}}), of the upper spectra of affine ’spaces’ (U_{i}, τ_{U_{i}}). Thus, we have the following inclusions

Spec(X) ⊆ Spec_{℘}(X, τ) ⊆ ⋃_{i∈J} Spec(U_{i}) ⊆ ⋃_{i∈J} Spec(U_{i}, τ_{U_{i}}) = Spec(X, τ) ⊆ Spec^{−}(X).

9.4.3. Example: quasi-coherent sheaves of algebras, D-Schemes. Let X = (X , O_{X}) be a ringed topological space, and let (A, φ) be a quasi-coherent sheaf of associative rings on X; i.e. A is a sheaf of associative rings on the topological space X and φ is a morphism O_{X} ⟶ A of sheaves of rings which turns A into a sheaf of quasi-coherent left O_{X} -modules on X. Let C_{Sp(A)} denote the category of quasi-coherent A-modules on X. This defines the ’space’ Sp(A). The morphism O_{X} ⟶^{φ} A induces two functors,

C_{Sp(A)} \underrightarrow{_{φ_{∗}}} C_{Sp(O_{X})} = Qcoh(X) \underrightarrow{φ^{∗}} C_{Sp(A)}

which are interpreted as resp. direct and inverse image functors of a morphism of ’spaces’

Sp(A) \underrightarrow{_{φ}} Sp(O_{X}). If X is a scheme, then every affine cover of X induces an affine cover of Sp(A). Therefore, Sp(A) is a scheme.

Let τ be the Zariski pretopology on Sp(A). Let 𝔘 = {U_{i} → X | i ∈ J} be an affine cover of the scheme (X , O_{X}). Then the upper spectrum of (Sp(A), τ) is the union ⋃_{i∈J} Spec(Sp(A(U_{i})), τ_{U_{i}}) of the upper spectra of the schemes (Sp(A(U_{i})), τ_{U_{i}}).

Suppose, for example, that X = (X , O_{X}) is a smooth scheme over Spec(k), where k is a field of zero characteristic; and let A = D_{X} be the sheaf of differential operators on X. Since the scheme X is smooth, one can choose an affine open cover formed by affine spaces. Let U = {U_{i} → X | i ∈ J} be such a cover. Then all algebras A(U_{i}) are isomorphic to the Weyl algebra of the rank equal to the dimension of U_{i}. In particular, they are simple, hence the Zariski pretopology on Sp(A(U_{i}) is trivial. Therefore, the upper spectrum of the D-scheme (X, D_{X}) is the union of the spectra of Weyl algebras.

Other examples of this work are noncommutative versions of quasi-affine and projective schemes.

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

We define the cone of R_{+} by taking as C_{Cone(R_{+})} the quotient category R−mod/T_{R_{+}}^{−}.

The localization functor R − mod ⟶^{u^{∗}} R − mod/T_{R_{+}}^{−} is an inverse image functor of a morphism of ’spaces’ Cone(R_{+}) ⟶^{u} Sp(R). The functor u^{∗} has a (necessarily fully faithful) right adjoint, i.e. the morphism u is continuous. If R_{+} is a unital ring, then u is an isomorphism (see [KR3, C3.2.1]). The composition of the morphism u with the canonical affine morphism Sp(R) ⟶ Sp(R_{0}) is a continuous morphism Cone(R_{+}) ⟶ Sp(R_{0}). Its direct image functor is (regarded as) the global sections functor.

It follows from [R, 3.6.5.2] that the Zariski pretopology on Cone(R_{+}) is quasi-compact if R_{+} is an R_{0}-module of finite type. Therefore, by 6.4.1(a),

Spec_{℘}^{−}(Cone(R_{+}), τ^{ℨ}) = Spec(Cone(R_{+})).

In the general case, R_{+} is the union of a filtered family, {α_{i} | i ∈ J}, of finitely generated two-sided ideals in R. Then Cone(R_{+}) = sup_{i∈J}Cone(α_{i}), hence, by 6.5.5.3 and 6.4.1,

Spec_{℘⋆}^{−}(Cone(R_{+}), τ^{ℨ}) ≃ lim_{i∈J} Spec_{⋆}(Cone(α_{i})).

Note that, in general, Cone(R_{+}) is not a scheme.

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

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

Here gr_{G}T_{R_{+}}^{−} is the Serre subcategory of the category gr_{G}R − mod spanned by gr_{G}T_{R_{+}}.

We show below that gr_{G}T_{R_{+}}^{−} = gr_{G}R − mod ∩ T_{R_{+}}^{−} (see 9.8, 9.8.1). Therefore, there is a canonical projection

Cone(R_{+}) ⟶^{𝔭} Proj_{G}(R).

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

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

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

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

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

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

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

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

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

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

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

9.7. Actions. Let G be a monoid. An action of G on a ’space’ X is a monoidal functor G ⟶^{𝔏} Ẽnd(C_{X}). Here G is viewed as a discrete monoidal category and Ẽnd(C_{X}) denote the (strict) monoidal category of endofunctors C_{X} ⟶ C_{X}; i.e. Ẽnd(C_{X}) = (End(C_{X}), ∘).

9.7.1. Examples. (a) Let R_{+} be a (non-unital in general) G-graded R_{0}-ring (cf. 9.5); and let C_{Y} = gr_{G}R_{+} − mod_{1}. For any G-graded R_{+}-module N = ⊕_{ν∈G}N_{ν} and any γ ∈ G, we denote by N[γ] the G-graded R_{+}-module defined by N[γ]_{σ} = N_{σγ}. This defines a strict action of G on the ’space’ Y. Here strict means that the monoidal functor G ⟶^{𝔏} Ẽnd(C_{X}) is strict, that is N[γ_{1}γ_{2}] = (N[γ_{2}])[γ_{1}] for all N.

(b) The action of G on the ’space’ Y in (a) (i.e. on the category gr_{G}R_{+} − mod_{1}), induces an action of G on Proj_{G}(R_{+}).

9.8. Proposition. Let 𝕋 be a G-stable, topologizing subcategory of gr_{G}R_{+} − mod_{1}, and let 𝕋̃ denote the image of 𝕋 in R_{+} − mod_{1}. Then π^{∗^{−1}}(𝕋̃^{−}) = 𝕋^{−}.

Proof. (a) Since the functor (1) is exact, the preimage, π^{∗^{−1}}(𝕋̃^{−}), of the Serre subcategory 𝕋̃^{−} is a Serre subcategory of the category gr_{G}R_{+} − mod_{1}. The inclusion π^{∗^{−1}}(𝕋̃^{−}) ⊆ 𝕋^{−} is equivalent to that every nonzero object of π^{∗^{−1}}(𝕋̃^{−}) has a nonzero subobject which belongs to 𝕋; or, what is the same, for any nonzero object, N, of π^{∗^{−1}}(𝕋̃^{−}), there exists a nonzero morphism L ⟶^{g} N, with L ∈ Ob𝕋. We can and will assume that L is generated by one of its homogeneous components. Then R_{+} − mod_{1}(L, N) is a Ggraded ℤ-module, and some of homogeneous components of the morphism g are nonzero. Replacing the module L by the module L[γ] for an appropriate γ ∈ G, we can assume that the homogeneous component of g of zero degree is nonzero. Thus, there exists a nonzero morphism L[γ] ⟶ N of graded R_{+}-modules. Since the subcategory 𝕋 is stable under the action of G, the object L[γ] belongs to 𝕋.

(b) Every object, M, of 𝕋^{−} has a filtration, {M_{i} | i ≥ 0} such that M_{i} = sup(M_{j}|j < i), if i is a limit ordinal, and M_{i+1}/M_{i} belongs to 𝕋. But, this implies that M is an object of 𝕋̃^{−}; i.e. we have the inverse inclusion, 𝕋^{−} ⊆ π^{∗^{−1}}(𝕋̃^{−}).

9.8.1. Corollary. gr_{G}T_{R_{+}}^{−} = π^{∗^{−1}}(T_{R_{+}}^{−}).

Proof. Set 𝕋 = gr_{G}T_{R_{+}}. Then 𝕋̃^{−} coincides with T_{R_{+}}^{−}, hence the assertion.

9.8.2. Corollary. The forgetful functor (1) induces a faithful exact functor

C_{Proj_{G}(R_{+})} ⟶^{𝔭^{∗}} C_{Cone(R_{+})}  (2)

Proof. By 9.8.1(b), gr_{G}T_{R_{+}}^{−} = π^{∗^{−1}}(T_{R_{+}}^{−}), where gr_{G}R_{+}−mod_{1} ⟶^{π^{∗}} R_{+}−mod_{1} is the

forgetful functor. The functor π^{∗^{−1}} induces a faithful functor between quotient categories

gr_{G}R_{+} − mod_{1}/gr_{G}T_{R_{+}}^{−} ⟶ R_{+} − mod_{1}/T_{R_{+}}^{−}.

This functor is exact because the inclusion functor gr_{G}R_{+} −mod_{1} ⟶ R_{+} −mod_{1} is exact. Hence the assertion.

The functor (2) is regarded as an inverse image functor of a morphism (’projection’) Cone(R_{+}) ⟶^{𝔭} Proj_{G}(R_{+}).

9.9. G-Spectrum. Fix an action of a monoid G on a ’space’ X. For every M ∈ ObC_{X}, we denote by [M]_{G} the smallest G-stable topologizing subcategory of C_{X} containing M. Let ⟨M⟩_{G} be the full subcategory of C_{X} whose objects are all N ∈ ObC_{X} such that M does not belong to [N]_{G}. It follows from this definition that the subcategory ⟨M⟩_{G} is G-stable.

We write N ≻^{G} M if M is contained in [N]_{G}. It is easy to see that N ≻^{G} M ⇔ [M]_{G} ⊆ [N]_{G} ⇔ ⟨M⟩_{G} ⊆ ⟨N⟩_{G}.

We denote by Spec_{G}(X) the preorder defined as follows. Objects of Spec_{G}(X) are nonzero objects P of C_{X} such that if there exists a non-trivial morphism N ⟶ M, then

N ≻^{G} M. The preorder is given by ≻^{G}.

We denote by Spec_{G}(X) the preorder whose objects are G-stable topologizing subcategories [M]_{G}, M ∈ Spec_{G}(X), and the preorder is the inverse inclusion, ⊇. The map M ⟼ [M]_{G} induces an epimorphism of preorders Spec_{G}(X) ⟶ Spec_{G}(X).

It follows that the relation N ≻ M implies that N ≻^{G} M. In particular, there is a natural embedding Spec(X) ⟶ Spec_{G}(X). This embedding induces a morphism Spec(X) ⟶ Spec_{G}(X) such that the diagram

Spec(X) ⟶ Spec_{G}(X)
↓ ↓
Spec(X) ⟶ Spec_{G}(X)

commutes.

9.9.1. Proposition. Let a monoid G act on a ’space’ X. Suppose that there exists a family {𝕋_{i} | i ∈ J} of G-stable Serre subcategories of the category C_{X} such that

(a) ⋂_{i∈J} 𝕋_{i} = 0;

(b) every γ ∈ G induces a trivial (that is isomorphic to the identical functor) autoequivalence of C_{X}/𝕋_{i} for all i ∈ J.

Then the canonical map Spec(X) ⟶ Spec_{G}(X) is an isomorphism.

Proof. Let U_{i} denote the ’space’ defined by C_{U_{i}} = C_{X}/𝕋_{i}. The condition (a) means that every localization functor C_{X} ⟶ C_{U_{i}} factors through a localization func-

tor C_{X} ⟶^{q_{i}^{∗}} C_{U_{i}}, and the family of localization functors {q_{i}^{∗} | i ∈ J} is conservative. The condition (b) implies that G acts trivially on each category C_{U_{i}}, hence it acts trivially on Spec(U_{i}) for every i ∈ J. This means that Spec(U_{i}) = Spec_{G}(U_{i}). By 2.4.2, there is a natural embedding Spec(X) ↪ ⋃_{i∈J} Spec(U_{i}), where each Spec(U_{i}) is identified with the

corresponding subset of the complete spectrum Spec^{1}(X). Therefore, G acts trivially on Spec(X), which implies that Spec(X) = Spec_{G}(X).

9.9.1.2. Note. It is useful to see what is going on with representatives of the points of the spectrum, i.e. objects of Spec(X). Let M be an object of Spec(X) and let γ be an arbitrary element of G. Since the family of localizations is conservative, there exist i ∈ J such that q_{i}^{∗}(M) ≠ 0. Since M belongs to Spec(X) this means that M is 𝕋_{i}-torsion free and q_{i}^{∗}(M) is an object of Spec(U_{i}) (see the argument of 2.4.2). Since q_{i}^{∗}(M[γ]) = q_{i}^{∗}(M)[γ], the condition (b) implies that q_{i}^{∗}(M) ≃ q_{i}^{∗}(M[γ]). In particular, [q_{i}^{∗}(M)] = [q_{i}^{∗}(M[γ])].

9.10. Proposition. An object M of C_{Proj_{G}(R_{+})} belongs to Spec_{G}(Proj_{G}(R_{+})) iff its image in C_{Cone(R_{+})} belongs to Spec(Cone(R_{+})).

Proof. (a) Let M be a graded grT_{R_{+}}-torsion free R_{+}-module such that its image in C_{Cone(R_{+})} belongs to Spec(Cone(R_{+})). Let 𝕋 be any G-stable Serre subcategory of grR_{+} − mod_{1} (i.e. 𝕋^{−} = 𝕋) which contains the subcategory grT_{R_{+}} and does not contain the object M. Let 𝕋̃ denote the image of 𝕋 in R_{+} − mod_{1}. It follows that 𝕋̃^{−} contains the Serre subcategory T_{R_{+}}^{−}. By 9.8(b), 𝕋 coincides with the preimage of 𝕋̃^{−} in grR_{+} − mod_{1}.

Therefore, the Serre subcategory 𝕋̃^{−} does not contain the image M̃ of M in R_{+} −mod_{1} (M̃ coincides with the R_{+}-module M with forgotten grading). This means that 𝕋̃ ⊆ ⟨M̃⟩_{T_{R_{+}}} ,

hence 𝕋 ⊆ π^{∗^{−1}}(⟨M̃⟩_{T_{R_{+}}} ). Here ⟨M̃⟩_{T_{R_{+}}} denotes the union of all Serre subcategories of

R_{+} − mod_{1} which contain the subcategory T_{R_{+}}, but do not contain the object M̃. Notice that the subcategory ⟨M̃⟩_{T_{R_{+}}} is the preimage in R_{+} − mod_{1} of the subcategory ⟨M̃′⟩,

where M̃′ denote the image in C_{Cone(R_{+})} of the object M̃. By hypothesis, the object M̃′ belongs to Spec(Cone(R_{+})) which means precisely that ⟨M̃′⟩ is a Serre subcategory of C_{Cone(R_{+})}. Therefore ⟨M̃⟩_{T_{R_{+}}} is a Serre subcategory of R_{+} − mod_{1}. The latter implies

that the preimage, π^{∗^{−1}}(M̃⟩_{T_{R_{+}}} ), of ⟨M̃⟩_{T_{R_{+}}} is the largest G-stable Serre subcategory of grR_{+}−mod_{1} containing grT_{R_{+}} which does not contain the object M. Therefore, the image of M in C_{Proj_{G}(R_{+})} belongs to Spec_{G}(Proj_{G}(R_{+})).

(b) Let M be a graded grT_{R_{+}}-torsion free R_{+}-module such that its image in C_{Proj_{G}(R_{+})}

belongs to Spec_{G}(Proj_{G}(R_{+})). Since by 9.8.2, the functor C_{Proj_{G}(R_{+})} ⟶^{𝔭^{∗}} C_{Cone(R_{+})} is faithful, the image M̃ of M in R_{+} − mod_{1} is T_{R_{+}}-torsion free. Let N ⟶ M̃ be a nonzero monomorphism. Because N is a nonzero R_{+}-module, there exists a nonzero monomorphism L̃ ⟶ N for some graded R_{+}-module L generated by one of its homogeneous components. Since the composition L̃ ⟶ M is a nonzero (mono)morphism, one of its graded components is nonzero; i.e. there exists a nonzero morphism L[γ] ⟶^{g} M of graded R_{+}-modules. Since M belongs to Spec_{G}(Proj_{G}(R_{+})), there exists a graded submodule M′ of M such that L[γ] ≻ M′ (i.e. M′ ∈ Ob[L[γ]]) and M/M′ belongs to grT_{R_{+}}^{−}. Since the forgetful func-

tor gr_{G}R_{+}−mod_{1} ⟶ R_{+}−mod_{1} is exact, L̃[γ] ≻ M̃′, and the quotient M̃/M̃′ = { \atop M/M′} is

an object of T_{R_{+}}^{−}. Let q^{∗} denote the localization functor R_{+} − mod_{1} ⟶ C_{Cone(R_{+})}. Since the functor q^{∗} is exact, q^{∗}(L) ⟶ N is a monomorphism; in particular, q^{∗}(N) ≻ q^{∗}(L̃). Notice that L̃ ≃ L̃[γ] and q^{∗}(M̃′) ≃ q^{∗}(M̃). Thus, q^{∗}(N) ≻ q^{∗}(L̃[γ]) ≻ q^{∗}(M̃′) ≃ q^{∗}(M̃). This proves that the image, q^{∗}(M̃), of M in C_{Cone(R_{+})} belongs to Spec(Cone(R_{+})).

9.10.1. Corollary. The canonical morphism Cone(R_{+}) ⟶^{𝔭} Proj_{G}(R_{+}) induces a map

Spec_{G}(Proj_{G}(R_{+})) ⟶ Spec(Cone(R_{+})).  (1)

Proof. By 9.10, the inverse image functor of the morphism 𝔭 induces a functor

Spec_{G}(Proj_{G}(R_{+})) ⟶ Spec(Cone(R_{+}))

which gives a rise to the map (1).

9.10.2. Proposition. Suppose that there exists a family {𝕋_{i} | i ∈ J} of G-stable Serre subcategories of the category gr_{G}R_{+} − mod_{1} such that

(a) ⋂_{i∈J} 𝕋_{i} = gr_{G}T_{R_{+}}^{−};

(b) every γ ∈ G induces a trivial (that is isomorphic to the identical functor) autoequivalence of gr_{G}R_{+} − mod_{1}/𝕋_{i} for all i ∈ J. Then Spec_{G}(Proj_{G}(R_{+}) = Spec(Proj_{G}(R_{+}), and the canonical map (1) is injective and identifies Spec(Proj_{G}(R_{+})) with the subset of all [M] such that M is a graded R_{+}-module which belongs to Spec(Cone(R_{+})).

Proof. The assertion is a corollary of 9.9.1.

9.10.3. Proposition. Let 𝔤 be a semisimple Lie algebra over a field of zero characteristic, and let R = ⊕_{λ∈G_{+}}R_{λ} is the algebra of functions on the base affine ’space’ of 𝔤 (cf. 9.6.3). Then

Spec(Proj_{G}(R)) ≃ Spec_{G}(Proj_{G}(R)) ≃ {[M] | M ∈ Spec(Cone(R)) ⋂ gr_{G}R − mod}.

Proof. The canonical cover {Sp_{G}(S_{w}^{−1}R) ⟶ Proj_{G}(R) | w ∈ W } (cf. 9.6.4) satisfies the conditions of 9.9.1, hence the assertion.

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