Spectra of ’spaces’ represented by abelian categories.

Alexander Rosenberg

Introduction

Spectral theory of abelian categories was started by P. Gabriel in early sixties [Gab]. Elements of the Gabriel’s spectrum are isomorphism classes of indecomposable injectives. If R is a commutative noetherian ring, the Gabriel’s spectrum of the category of R-modules is naturally isomorphic to the prime spectrum of the ring R. More generally, the Gabriel’s spectrum of the category of quasi-coherent sheaves on a noetherian scheme is isomorphic to the underlying space of the scheme [Gab, Ch. VI, Theorem 1].

The Gabriel’s spectrum was the only spectrum of an abelian category for more than a quarter of century (except for it was rediscovered a couple of times). Then three other spectra were assigned to an abelian category C_{X}: the spectrum Spec(X) (see [R1] and [R, Ch. 3]), the S-spectrum Spec^{−}(X), and the complete spectrum Spec^{1}(X) [R, Ch.6]. These spectra make sense without noetherian hypothesis. If C_{X} is the category of modules over an arbitrary commutative unital ring R, then Spec(X) is naturally isomorphic to the prime spectrum of R. More generally, if C_{X} is the category of quasi-coherent sheaves on a quasi-compact quasi-separated scheme, then Spec(X) is isomorphic to the underlying space of the scheme. This fact is obtained in [R4] as a consequence of a theorem on local properties of the spectrum Spec(X). In Section 9 of this work, we prove a refined version of this theorem. For an arbitrary abelian category C_{X}, isomorphism classes of simple objects of C_{X} correspond to closed points of Spec(X).

The S-spectrum is larger than Spec(X) – there is a natural embedding of Spec(X) into Spec^{−}(X). If C_{X} is a locally noetherian Grothendieck category (or, more generally, a Grothendieck category with a Gabriel-Krull dimension), then the S-spectrum is naturally isomorphic to the Gabriel spectrum. An advantage of the S-spectrum is that it makes sense for abelian categories without injective objects, like, for example, the category of coherent sheaves on a scheme, or the category of finitely generated modules over a ring.

One of practical merits of the complete spectrum is that it contains the S-spectrum and is functorial with respect to arbitrary open immersions, while the S-spectrum is functorial only with respect to open immersion having direct image functor. As a result, the underlying space of a scheme is naturally embedded into the complete spectrum of the category of quasi-coherent sheaves on the scheme.

In [R5], several new spectra were defined which make sense for ’spaces’ represented by (not necessarily abelian) categories, in particular, the spectra of ’spaces’ represented by triangulated categories. Finally, a natural, very general construction of spectra was proposed in [R6]. This construction is tested here on the spectra Spec(X), Spec^{−}(X), and Spec^{1}(X); that is we use the construction to study these spectra, as well as some other spectra which appeared in [R5] or suggested by the construction of [R6].

The main thrust of the paper is establishing local properties of the spectra which are needed to study the spectra of non-affine noncommutative schemes, and are crucial for reconstruction problems. These local properties are also used in computations of spectra and applications of noncommutative local algebra and algebraic geometry to representation theory [R7].

Section 1 contains the necessary preliminaries on ’spaces’ and morphisms of ’spaces’, topologizing, thick, and Serre subcategories, and the spectra Spec(X), Spec^{−}(X), and Spec^{1}(X) mentioned above. Section 2 presents, for the reader’s convenience a short exposition of the main construction of [R6] whose applications are studied in this work. Sections 3 and 4 are dedicated to the spectrum Spec_{𝔱}^{1}(X) introduced in [R6, 2.3.5] and its counterpart Spec_{𝔱}^{0}(X) suggested by the general construction of Section 2. The natural decompositions of these spectra allow to recover the spectrum Spec(X) and see some of its important properties unnoticed earlier. In Section 5, we look at the complete spectrum and the S-spectrum. The general pattern invoke another spectrum, Spec_{⋊}(X), which is a natural extension of Spec(X) and plays a similar role. In Section 7 we study functorial properties of these spectra. In Section 8, we apply general pattern developed in the previous sections to the spectra related with exact localizations. We start with the thick spectrum introduced in [R5] and recover other spectra via its canonical decomposition.

In Section 9, we study local properties of the spectra with respect to finite covers. The results we obtain here are stronger than the results in this direction obtained in [R4]. The refinements are due to different arguments making use of a simple algebra of the monoid of topologizing subcategories which is sketched in Appendix 1.

Section 10 is dedicated to the spectra related to reflective topologizing categories and their local properties. The material of this section is entirely new. The general pattern of Section 2 produce a pair spectra – Spec_{𝔠}^{0}(X) and Spec_{𝔠}^{1}(X), together with a canonical morphism between them, which, for these particular spectra, turns out to be an isomorphism. For an arbitrary abelian category C_{X}, there is a natural embedding Spec(X) ↪ Spec_{𝔠}^{0}(X). If C_{X} is the category of quasi-coherent sheaves on a quasi-compact (noncommutative) scheme, then Spec_{𝔠}^{0}(X) and Spec(X) coincide.

Some time ago, O. Gabber constructed an example showing that the spectrum Spec(X) is not sufficient to recover the underlying topological space of an arbitrary non-quasi-compact (commutative) scheme from the category C_{X} of quasi-coherent sheaves on the scheme. He observed that the spectrum Spec_{𝔠}^{0}(X), on the other hand, suffices to reconstruct the underlying space in his example which gave a ground to the conjecture that it might suffice in the general case, under some mild restrictions. We show that this conjecture is, indeed, true. It follows from the local property of the spectrum Spec_{𝔠}^{0}(X) (or, ruther, of its counterpart Spec_{𝔠}^{1}(X)) with respect to infinite covers, which we formulate and prove in 10.6. If C_{X} is the category of quasi-coherent sheave on a scheme X, then Spec_{𝔠}^{0}(X) endowed with the Zariski topology (which is defined in terms of topologizing subcategories) is naturally isomorphic to the underlying topological space of the scheme X, under the condition that X admits an affine cover {U_{i} ↪ X | i ∈ J} such that each immersion U_{i} ↪ X has a direct image functor. The latter condition holds if the scheme is quasi-separated.

In Appendix 1, we relate topologies on spectra with some natural ’topological’ structures on the monoid of topologizing subcategories. Besides, this appendix contains facts (mostly borrowed from [R4]) which are used in the main body of the paper, expecially in Sections 9 and 10. Appendix 2 contains some complements on (differences between) spectra Spec_{𝔱}^{0}(X) and Spec(X). It is a complement to Sections 3 and 4. We consider Spec_{𝔱}^{0}(X) and Spec(X) as topological spaces with respect to the topology associated with topologizing subcategories and establish some general properties of these spaces. Appendix 3 contains some observations on supports of objects and the Krull filtrations. In Appendix 4, we apply the results of Section 9 to compare closed points of the S-spectrum Spec^{−}(X) and Spec(X). Closed points of Spec(X) play a special role due to their significance for representation theory and algebraic geometry. The S-spectrum is oftenly easier to compute than Spec(X) due to its better functorial properties. We show that, although the S-spectrum is, usually, considerably larger than Spec(X), their closed points are in natural bijective correspondence in many (if not all) cases of interest.

Section 10 of this work was inspired by a conversation with O. Gabber who directed my attention to the spectrum Spec_{𝔠}^{0}(X). I am happy to thank him for that and for other helpful remarks and discussions. A part of this text was written during my visiting the Max Planck Institut für Mathematik in Bonn in Summer of 2004. I would like to thank the Institute for hospitality and always excellent working atmosphere. The work was partially supported by the NSF grant DMS-0070921.

1. Preliminaries.

1.1. ’Spaces’ and morphisms of ’spaces’. ’Spaces’ here are spaces of noncommutative algebraic geometry. In the simplest (or the most abstract) setting, they are represented by categories. Morphisms of ’spaces’ are functors regarded as inverse image functors. We denote by C_{X} the category representing a ’space’ X and by f^{∗} a functor C_{Y} ⟶ C_{X} representing a morphism X ⟶^{f} Y. Formally, ’spaces’ are objects of the category Cat^{op} opposite to the category Cat. The bicategory of ’spaces’ is the bicategory Cat^{op}. The category of ’spaces’ is the category |Cat|^{o} having same objects as Cat^{op}. Morphisms from X to Y are isomorphism classes of (inverse image) functors C_{Y} ⟶ C_{X}.

The ’spaces’ of this work are represented by abelian categories and morphisms have additive inverse image functors.

1.2. Localizations and conservative morphisms. Let X be a ’space’ and Σ a family of arrows of the category C_{X}. We denote by Σ^{−1}X the object of |Cat|^{o} such that the corresponding category coincides with (the standard realization of) the category of fractions ot C_{X} for Σ (cf. [GZ], 1.1): C_{Σ^{−1}X} = Σ^{−1}C_{X}. We call Σ^{−1}X the ’space’ of

fractions of X for Σ. The canonical localization functor C_{X} ⟶^{q_{Σ}^{∗}} Σ^{−1}C_{X} is regarded as an inverse image functor of a morphism, Σ^{−1}X ⟶^{q_{Σ}} X.

For any morphism f : X ⟶ Y in |Cat|^{o}, we denote by Σ_{f} the family of all morphisms 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 f_{𝔠}^{∗} : Σ_{f}^{−1}C_{Y} ⟶ 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, or, equiv-

alently, 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. Topologizing subcategories of an abelian category. Fix an abelian category C_{X}. A full subcategory 𝕋 of C_{X} is called topologizing if it is closed under finite coproducts and subquotients taken in C_{X}. We denote by 𝔗(X) the preorder (with respect to the inclusions) of topologizing subcategories of the category C_{X}.

The Gabriel product, 𝕊 • 𝕋, of the pair of subcategories 𝕊, 𝕋 of C_{X} is the full subcategory of C_{X} spanned by all objects M of C_{X} such that there exists an exact sequence

0 ⟶ L ⟶ M ⟶ N ⟶ 0

with L ∈ Ob𝕋 and N ∈ Ob𝕊. It follows that 0 • 𝕋 = 𝕋 = 𝕋 • 0 for any strictly full subcategory 𝕋. The Gabriel product of two topologizing subcategories is a topologizing subcategory, and its restriction to topologizing categories is associative; i.e. topologizing subcategories form a monoid with respect to the Gabriel multiplication.

Note that the canonical dualization isomorphism T (X) ⟶^{∼} T (X^{o}), S ⟼ S^{op}, is an inti-isomorphism of monoids (T (X), •) ⟶ (T (X^{o}), •), that is (𝕋 • 𝕊)^{op} = 𝕊^{op} • 𝕋^{op}.

1.3.1. The preorder ≻. 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.

Let [M] denote the smallest topologizing subcategory containing the object M. One can show that M ≻ N iff [N] ⊆ [M]. This gives a description of [M]: its objects are those L ∈ ObC_{X} for which M ≻ L.

1.3.2. Lemma. The smallest topologizing subcategory of C_{X} containing a family of objects S coincides with ⋃_{N∈S_{Σ}}[N], where S_{Σ} denotes the family of all finite coproducts of

objects of S.

Proof. Clearly, ⋃_{N∈S_{Σ}}[N] is contained in every topologizing subcategory containing the

family S. It is closed under taking subquotients, because each [N] has this property. It is closed under finite coproducts, because if N_{1}, N_{2} ∈ S_{Σ} and N_{i} ≻ M_{i}, i = 1, 2, then N_{1} ⊕ N_{2} ≻ M_{1} ⊕ M_{2}.

For any subcategory (or a class of objects) S, we denote by [S] the smallest topologizing subcategory containing S.

1.3.3. Lemma. Let 𝕋, 𝕊 be topologizing subcategories of an abelian category C_{X}.

(a) The smallest topologizing subcategory, [𝕋, 𝕊], of C_{X} containing 𝕋 and 𝕊 coincides with

⋃_{(L,M)∈Ob(𝕊×𝕋)}[L ⊕ M].  (1)

(b) Every 𝕊-torsion free object of [𝕋, 𝕊] belongs to 𝕋.

Proof. (a) The assertion follows from 1.3.2.

(b) It follows from the definition of the Gabriel product that every 𝕊-torsion free object of 𝕋 • 𝕊 belongs to 𝕋, and [𝕋, 𝕊] ⊆ 𝕋 • 𝕊, hence the assertion.

We leave to the reader the formulation of the analogous assertion for an arbitrary family of topologizing subcategories.

1.4. Thick subcategories. A topologizing subcategory 𝕋 of the category C_{X} is called thick if 𝕋•𝕋 = 𝕋. In other words, a full subcategory 𝕋 of C_{X} is thick iff the following condition holds: the object M in an exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 belongs to the subcategory 𝕋 iff M′ and M′′ are objects of 𝕋.

We denote by 𝔗𝔥(X) the preorder of thick subcategories of C_{X}. Given a thick subcategory T of C_{X}, we shall denote by X/T the quotient ’space’ defined by C_{X/T} = C_{X}/T .

1.5. Serre subcategories. We recall 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. This construction has the following properties [R, III.2.3.2.1]:

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

1.5.1. Note. (a) One can see that for any subcategory 𝕋 of the category C_{X}, the associated Serre subcategory 𝕋^{−} is the largest topologizing subcategory of C_{X} such that every its nonzero object has a nonzero subobject from 𝕋.

(b) It follows from this description that if the subcategory 𝕋 is closed under taking quotient objects (e.g. 𝕋 is a topologizing subcategory), then 𝕋^{−} is the largest topologizing subcategory of C_{X} which has the zero intersection with 𝕋^{⊥}. Here 𝕋^{⊥} denotes, as usually, the right orthogonal to 𝕋, i.e. the full subcategory of C_{X} generated by all objects M such that C_{X}(L, M) = 0 for any L ∈ Ob𝕋.

1.5.2. The property (sup) and coreflective subcategories. Recall that the 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.

1.5.3. Lemma. Any coreflective thick subcategory of an abelian category C_{X} is a Serre subcategory. If C_{X} has the property (sup), then any Serre subcategory of C_{X} is coreflective.

Proof. See [R, III.2.4.4].

1.5.4. 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}).

1.5.5. Proposition. Let C_{X} be a category with the property (sup). Then for every thick subcategory 𝕋 of C_{X}, every object of 𝕋^{−} is the supremum of its subobjects which belong to 𝕋.

Proof. Since C_{X} has the property (sup), the supremums of objects from 𝕋 form a thick coreflective, hence Serre, subcategory. Therefore it coincides with 𝕋^{−}.

1.6. 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] (or, equivalently, L ≻ M) for any nonzero subobject L of M. In other words, 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).

The spectrum Spec(X) of the ’space’ X is the family of topologizing subcategories {[M] | M ∈ Spec(X)} endowed with the specialization preorder ⊇.

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.

1.6.1. Proposition. (a) The inclusion Simple(X) ↪ Spec(X) induces an embedding 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) If M is a simple object, then Ob[M] consists of all objects isomorphic to coproducts of finite number of copies of M. In particular, if M and N are simple objects, then [M] ⊆ [N] iff M ≃ N.

(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].

Notice that the notion of a simple object of an abelian category is selfdual, i.e. Simple(X) = Simple(X^{o}), where X^{o} is the dual ’space’ defined by C_{X^{o}} = C_{X}^{op}. In particular, the map M ⟼ [M] induces an embedding of isomorphism classes of simple objects of C_{X} into the intersection Spec(X) ⋂ Spec(X^{o}).

1.6.2. Proposition. If the category C_{X} has enough objects of finite type, then the set of closed points of Spec(X) coincides with Spec(X) ⋂ Spec(X^{o}).

Proof. Since every nonzero object of C_{X} has a nonzero subobject of finite type, Spec(X) consists of [M] such that M is of finite type and belongs to Spec(X). On the other hand, if M is of finite type and [M] belongs to Spec(X^{o}), then [M] = [M_{1}], where M_{1} is a simple quotient of M. Hence the assertion.

1.7. Local ’spaces’. A nonzero object P of an abelian category C_{X} is called quasifinal if N ≻ P for every nonzero object N of C_{X}. 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 the ’space’ X (and the category C_{X}) local if C_{X} has a quasi-final object. If X is local, then Spec(X) has the (unique) smallest element which coincides with

the smallest nonzero topologizing subcategory of C_{X} and is a unique closed point of the spectrum (in the topology associated with the preorder ⊇).

1.7.1. 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 every quasi-final object of C_{X} is isomorphic to a direct sum of a finite number of copies of a simple object.

Proof. In fact, if M is a simple object in C_{X}, then [M] is a closed point of Spec(X). If X is local, this closed point is unique. Therefore, M is a quasi-final object and every object of [M] is a finite coproduct of copies of M (see the argument of 1.6.1).

1.8. The complete spectrum and the S-spectrum. The complete spectrum of X is the preorder (with respect to ⊆) of all thick subcategories P of the category C_{X} such that X/P is a local ’space’. We denote it by Spec^{1}(X).

The S-spectrum of X is defined as the subpreorder of Spec^{1}(X) formed by Serre subcategories, i.e. Spec^{−}(X) = Spec^{1}(X) ⋂ 𝔖𝔢(X) = {P ∈ Spec^{1}(X) | P = P^{−}}.

For any M ∈ ObC_{X}, we denote by ⟨M⟩ the full subcategory of C_{X} generated by all objects N such that N ⊁ M. Notice that [M] ⊆ [L] iff ⟨M⟩ ⊆ ⟨L⟩.

1.8.1. Proposition. The map [P] ⟼ ⟨P ⟩ induces a monomorphism of preorders Spec(X) ⟶ Spec^{−}(X).

Proof. See [R, III.2.3.3 and III.3.3.1].

1.8.2. The S-spectrum and the Gabriel spectrum. We call a ’space’ X locally noetherian if C_{X} is a locally noetherian abelian category. It is argued in [R, Ch. VI] that if X is locally noetherian, then the elements of the S-spectrum are in bijective 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}.

1.8.2. Remark. If C_{X} = R − mod, where R is a commutative noetherian ring, 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).

2. A general construction of spectra.

Fix a category ℌ. Let ℌ_{0} denote the full subcategory of ℌ whose objects are initial objects of ℌ. Thus, ℌ_{0} is either empty, or a groupoid. Let ℌ^{1} denote the full subcategory of ℌ defined by Obℌ^{1} = Obℌ − Obℌ_{0}.

2.1. Definition. We call ℌ local if the category ℌ^{1} has an initial object.

2.1.1. Note. It follows that if ℌ is local, than ℌ has initial objects, i.e. ℌ_{0} ≠ ∅.

2.2. The spectrum 𝔖𝔭𝔢𝔠^{1}(ℌ). We denote by 𝔖𝔭𝔢𝔠^{1}(ℌ) the full subcategory of the category ℌ generated by all x ∈ Obℌ such that the category x\ℌ is local. We call 𝔖𝔭𝔢𝔠^{1}(ℌ) the local spectrum of ℌ.

In other words, an object x of ℌ belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ) iff there exists an object x^{⋆} of ℌ and an arrow x ⟶^{γ_{x}} x^{⋆} such that γ_{x} is not an isomorphism and if x ⟶^{f} y is not an

isomorphism, then there exists a unique arrow x^{⋆} ⟶^{f̄} y such that f = f̄∘γ_{x}. The morphism x ⟶^{γ_{x}} x^{⋆} (in particular, the object x^{⋆}) is determined by these conditions uniquely up to isomorphism.

2.2.1. Note. It follows from this definition and 2.1.1 that ℌ is local iff it has initial objects and they belong to 𝔖𝔭𝔢𝔠^{1}(ℌ).

2.2.2. Functorial properties. Let ℌ ⟶^{F} ℌ̃ be a functor. For any x ∈ Obℌ, the functor F induces a functor x\ℌ ⟶^{F_{x}} F (x)\ℌ̃. Suppose that the functor F is such that F_{x} is an equivalence of categories for every x ∈ Obℌ. Then F induces a functor 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟶ 𝔖𝔭𝔢𝔠^{1}(ℌ̃).

A typical example is the functor

y\ℌ ⟶^{f_{⋆}} z\ℌ, (y, y →^{g} v) ⟼ (z, z →^{gf} v),

corresponding to a morphism z ⟶^{f} y, or the canonical functor y\ℌ ⟶ ℌ.

2.3. Supports. For any x ∈ Obℌ, we denote by 𝔖𝔲𝔭𝔭_{ℌ}(x) the full subcategory of ℌ generated by all y ∈ Obℌ such that ℌ(x, y) = ∅. We call 𝔖𝔲𝔭𝔭_{ℌ}(x) the support of x in ℌ.

2.3.1. Proposition. (a) For any two objects, x and y, of the category ℌ, there exists an arrow x → y iff 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y).

(b) Let {x_{i} | i ∈ J} be a set of objects of ℌ such that there exists a coproduct, ∐_{i∈J} x_{i}.

Then

𝔖𝔲𝔭𝔭_{ℌ}( ∐_{i∈J} x_{i}) = ⋃_{i∈J} 𝔖𝔲𝔭𝔭_{ℌ}(x_{i}).  (1)

Proof. (a) If there exists a morphism x ⟶ y and ℌ(x, z) = ∅, then, obviously, ℌ(y, z) = ∅, hence 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y).

If ℌ(x, y) = ∅, i.e. y ∈ 𝔖𝔲𝔭𝔭_{ℌ}(x), then, since y ∉ Ob𝔖𝔲𝔭𝔭_{ℌ}(y), the inclusion 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y) does not hold.

(b) Since ℌ(∐_{i∈J} x_{i}, z) ≃ ∏_{i∈J} ℌ(x_{i}, z), ℌ(∐_{i∈J} x_{i}, z) = ∅ iff ℌ(x_{i}, z) = ∅ for some i ∈ J, whence the equality (1).

2.3.2. Support in 𝔖𝔭𝔢𝔠^{1}(ℌ). For any x ∈ Obℌ, we denote the intersection 𝔖𝔲𝔭𝔭_{ℌ}(x) ⋂ 𝔖𝔭𝔢𝔠^{1}(ℌ) by 𝔖𝔲𝔭𝔭_{ℌ}^{1}(x) and call it the support of x in 𝔖𝔭𝔢𝔠^{1}(ℌ). Evidently, 2.3.1(b) is still true if 𝔖𝔲𝔭𝔭_{ℌ}(x) is replaced by 𝔖𝔲𝔭𝔭_{ℌ}^{1}(x), as well as a half of 2.3.1(a): if ℌ(x, y) is not empty, then 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y).

2.4. The spectrum 𝔖𝔭𝔢𝔠^{0}(ℌ). We denote by 𝔖𝔭𝔢𝔠^{0}(ℌ) the full subcategory of ℌ generated by x ∈ Obℌ such that 𝔖𝔲𝔭𝔭_{ℌ}(x) is not empty and has a final object, x̂.

2.4.1. Proposition. Let ℌ be local. Then initial objects of ℌ^{1} belong to 𝔖𝔭𝔢𝔠^{0}(ℌ).

Proof. Let ℌ_{0} be the full subcategory (groupoid) of ℌ generated by all initial objects of ℌ. If x is an initial object of the category ℌ^{1}, then 𝔖𝔲𝔭𝔭_{ℌ}(x) coincides with ℌ_{0}.

In fact, suppose that there is an arrow, x ⟶^{f} y, for some y ∈ Obℌ_{0}. Since y is an initial object of the category ℌ, there exists a unique morphism y ⟶^{g} x. By the universal property of y, the composition y ⟶^{fg} y is the identical morphism. Since x is an initial object of the category ℌ^{1}, the composition x ⟶^{gf} x is the identical morphism too. This means that the morphism x ⟶^{f} y is an isomorphism which contradicts to the fact that x is not an initial object of the category ℌ.

Thus, ℌ_{0} is a subcategory of 𝔖𝔲𝔭𝔭_{ℌ}(x). Since for every z ∈ Obℌ^{1} = Obℌ − Obℌ_{0} there is a (unique) morphism x ⟶ z, the subcategory 𝔖𝔲𝔭𝔭_{ℌ}(x) is contained in ℌ_{0}; i.e. 𝔖𝔲𝔭𝔭_{ℌ}(x) = ℌ_{0}.

Since ℌ_{0} is a connected groupoid, every object of ℌ_{0} is final.

2.4.4. Lemma. A choice for every x ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ) of a final object, x̂, of the category 𝔖𝔲𝔭𝔭_{ℌ}(x) extends to a functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ.

Proof. In fact, if x, y ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ), and there is a morphism x ⟶ y, then 𝔖𝔲𝔭𝔭_{ℌ}(x) ⊆ 𝔖𝔲𝔭𝔭_{ℌ}(y). Therefore there exists a unique morphism x̂ ⟶ ŷ.

2.4.5. Remark. Notice that the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ is faithful iff ℌ is a preorder, i.e. for any pair of objects, x, y, of ℌ, there is at most one morphism x ⟶ y.

2.4.6. Proposition. Suppose the category ℌ is a preorder with finite coproducts (i.e. supremums of pairs of objects). Then the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ takes values in

𝔖𝔭𝔢𝔠^{1}(ℌ), i.e. it induces a functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{θ_{ℌ}} 𝔖𝔭𝔢𝔠^{1}(ℌ).

Proof. For any x ∈ Ob𝔖𝔭𝔢𝔠^{0}(ℌ), the final object, x̂, of the category 𝔖𝔲𝔭𝔭_{ℌ}(x) belongs to 𝔖𝔭𝔢𝔠^{1}(ℌ). More expicitly, we claim that the canonical coprojection, x̂ ⟶ x ⊔ x̂, is an initial object of the category (x̂\ℌ)^{1}.

In fact, let x̂ ⟶^{g} y be a morphism. Then one of two things happens: either y ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), or not. If y ∈ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), then, since x̂ is a final object of the category 𝔖𝔲𝔭𝔭_{ℌ}(x), there is a unique morphism y ⟶^{h} x̂. It follows from the universal property of x̂ that h ∘ g = id_{x}. By hypothesis, ℌ is a preorder, in particular, h is a monomorphism. Therefore, h is an isomorphism inverse to g.

If y ∉ Ob𝔖𝔲𝔭𝔭_{ℌ}(x), then there exists an arrow x ⟶ y which, together with x̂ ⟶^{g} y, determines (and is determined by) a morphism (x ⊔ x̂, x̂ → x ⊔ x̂) ⟶ (y, x̂ →^{g} y). Since ℌ is a preorder, this is all we need.

2.5. Relative spectra. Let 𝔊 ⟶^{F} ℌ be a functor. We define the relative spectra, 𝔖𝔭𝔢𝔠^{1}(𝔊, F) and 𝔖𝔭𝔢𝔠^{0}(𝔊, F), via cartesian squares

\begin{matrix}𝔖𝔭𝔢𝔠^{1}(𝔊, F) & ⟶^{θ_{F}^{1}} & 𝔊 & & 𝔖𝔭𝔢𝔠^{0}(𝔊, F) & ⟶^{ϑ_{F}} & 𝔊 \\ π_{1}^{F} ↓ & & ↓ F & and & π_{0}^{F} ↓ & & ↓ F \\ 𝔖𝔭𝔢𝔠^{1}(ℌ) & ⟶^{θ_{ℌ}^{1}} & ℌ & & 𝔖𝔭𝔢𝔠^{0}(ℌ) & ⟶^{ϑ_{ℌ}} & ℌ \end{matrix}  (1)

(in the bicategorical sense, i.e. the squares quasi-commute), where 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ is the canonical functor of 2.4.4.

Explicitly, objects of the category 𝔖𝔭𝔢𝔠^{1}(𝔊, F) are triples (z, x; ϕ), where z is an object of 𝔖𝔭𝔢𝔠^{1}(ℌ), x ∈ Ob𝔊, and ϕ is an isomorphism z ⟶^{∼} F (x). Morphisms from (z, x; ϕ) to (z′, x′; ϕ′) are given by pairs of arrows, z ⟶^{g} z′ and x ⟶^{h} x′ such that the diagram

z ⟶^{g} z′
ϕ ↓≀  ≀↓ ϕ′
F (x) ⟶^{F (h)} F (x′)

commutes. The projections 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟵^{π_{1}^{F}} 𝔖𝔭𝔢𝔠^{1}(𝔊, F) ⟶^{θ_{F}^{1}} 𝔊 in the left diagram (1) are defined by π_{1}^{F} (z, x; ϕ) = z and θ_{F}^{1} (z, x; ϕ) = x.

Similarly, objects of the category 𝔖𝔭𝔢𝔠^{0}(𝔊, F) are triples (z, x; ψ), where z is an object of 𝔖𝔭𝔢𝔠^{0}(ℌ), x ∈ Ob𝔊, and ψ is an isomorphism ϑ_{ℌ}(z) ⟶^{∼} F (x).

2.5.1. Proposition. Let i be 0 or 1. The map (𝔊, F) ⟼ 𝔖𝔭𝔢𝔠^{i}(𝔊, F) extends to a pseudo-functor 𝔖𝔭𝔢𝔠^{i} : Cat/ℌ ⟶ Cat.

Proof. The assertion follows from the universal property of cartesian squares.

2.5.2. Proposition. Suppose ℌ is a preorder with finite coproducts. Then for every functor 𝔊 ⟶^{F} ℌ, there is a canonical functor

𝔖𝔭𝔢𝔠^{0}(𝔊, F) ⟶^{ϑ_{(𝔊,F)}} 𝔖𝔭𝔢𝔠^{1}(𝔊, F).  (2)

The family {ϑ_{(𝔊,F)} | (𝔊, F) ∈ ObCat/ℌ} is a morphism of pseudo-functors,

𝔖𝔭𝔢𝔠^{0} ⟶^{ϑ} 𝔖𝔭𝔢𝔠^{1}.  (3)

Proof. Since ℌ is a preorder with finite coproducts, the functor 𝔖𝔭𝔢𝔠^{0}(ℌ) ⟶^{ϑ_{ℌ}} ℌ takes values in 𝔖𝔭𝔢𝔠^{1}(ℌ), hence it factors through the embedding 𝔖𝔭𝔢𝔠^{1}(ℌ) ⟶ ℌ (see 2.4.6). By the universal property of cartesian squares, there exists a unique functor (2) such that θ_{F}^{1} ∘ ϑ_{(𝔊,F)} = ϑ_{F} and π_{1}^{F} ∘ ϑ_{(𝔊,F)} = π_{0}^{F} (see the diagram (2)).

It is useful to have an explicit description of the functor (2) in terms of the descripiton of 𝔖𝔭𝔢𝔠^{0}(𝔊, F) and 𝔖𝔭𝔢𝔠^{1}(𝔊, F) given above. The functor ϑ_{(𝔊,F)} maps an object (z, x; ψ) of 𝔖𝔭𝔢𝔠^{0}(𝔊, F) to the object (ϑ_{ℌ}(z), x; ψ) of 𝔖𝔭𝔢𝔠^{1}(𝔊, F).

It follows from this description that ϑ = {ϑ_{(𝔊,F)} | (𝔊, F) ∈ ObCat/ℌ} is a morphism of pseudo-functors.

I. Spectra related to topologizing and thick subcategories.

Fix an abelian category C_{X}. Let 𝔗𝔥(X) (resp. 𝔗(X)) denote the preorder of all thick (resp. topologizing) subcategories of C_{X}. Applying the construction 2.5 to the inclusion

functor 𝔗𝔥(X) ⟶^{J_{X}^{𝔱}} 𝔗(X), we obtain two spectra,

Spec_{𝔱}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X), 𝔍_{X}^{𝔱}), i = 0, 1.

and (by 2.5.2) a canonical monomorphism from one to another, Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{1}(X). The category Spec_{𝔱}^{1}(X) is defined by the cartesian square

Spec_{𝔱}^{1}(X) ⟶ 𝔗𝔥(X)
↓  ↓
𝔖𝔭𝔢𝔠^{1}(𝔗(X)) ⟶ 𝔗(X)

in which the right vertical arrow and the lower horizontal arrow are inclusions (see 2.5). It follows from this description (or from the explicit description of relative spectra in 2.5) that objects of Spec_{𝔱}^{1}(X) are thick subcategories, P such that the intersection P^{𝔱} of all topologizing subcategories containing P properly does not coincide with P.

The spectrum Spec_{𝔱}^{0}(X) is defined by the cartesian square

\begin{matrix}Spec_{𝔱}^{0}(X) & ⟶ & 𝔗𝔥(X) \\ ↓ & & ↓ \\ 𝔖𝔭𝔢𝔠^{0}(𝔗(X)) & ⟶ & 𝔗(X) \end{matrix}  (1)

(see 2.5). By definition, objects of 𝔖𝔭𝔢𝔠^{0}(𝔗(X)) are topologizing subcategories, P, such that 𝔖𝔲𝔭𝔭_{𝔗(X)}(P) has a final object. This means, precisely, that the union, P̂, of all topologizing subcategories which do not contain P is a topologizing subcategory. The lower horizontal arrow of the diagram (1) maps an element P to P̂. Thus, Spec_{𝔱}^{0}(X) is naturally identified with the preorder of those topologizing subcategories P of C_{X} for which P̂ is a thick subcategory.

Similarly, the construction 2.5 applied to the incluison functor 𝔖𝔢(X) ⟶^{J_{X}^{𝔰}} 𝔗(X) produces two spectra

Spec_{𝔰,𝔱}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X), J_{X}^{𝔰}), i = 0, 1,

and a canonical injective map Spec_{𝔰,𝔱}^{0}(X) ⟶ Spec_{𝔰,𝔱}^{1}(X).

Here Spec_{𝔰,𝔱}^{1}(X) consists of all Serre subcategories P such that the intersection P^{𝔱} of topogizing subcategories properly containing P contains P properly too. The spectrum Spec_{𝔰,𝔱}^{0}(X) is identified with the preorder of all topogizing subcategories Q of C_{X} such that the union Q̂ of all topologizing subcategories of C_{X} which do not contain Q is a Serre subcategory. It follows from these definitions that these four spectra are related by a commutative diagram

Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{1}(X)
↑  ↑
Spec_{𝔰,𝔱}^{0}(X) ⟶ Spec_{𝔰,𝔱}^{1}(X)

(2) of injective morphisms.

In the next two sections, it is shown, among other facts, that Spec_{𝔰,𝔱}^{0}(X) coincides with the spectrum Spec(X) and the canonical map Spec_{𝔰,𝔱}^{0}(X) ⟶ Spec_{𝔰,𝔱}^{1}(X) is an isomorphism.

3. The decomposition of the spectrum Spec_{𝔱}^{1}(X).

Fix an abelian category C_{X}. For any subcategory S of C_{X}, we denote by S^{𝔱} the intersection of all topologizing subcategories of C_{X} properly containing S. The spectrum Spec_{𝔱}^{1}(X) is formed by thick subcategories P such that P ≠ P^{𝔱} (see above). We set

Spec_{𝔱}^{1,0}(X) = {P ∈ Spec_{𝔱}^{1}(X) | P^{𝔱} ∩ P^{⊥} = 0}

and

Spec_{𝔱}^{1,1}(X) = {P ∈ Spec_{𝔱}^{1}(X) | P^{𝔱} ∩ P^{⊥} ≠ 0},

where P^{⊥} is the right orthogonal to P. Since the subcategory P is closed under taking quotients, P^{⊥} coincides with the full subcategory of C_{X} formed by all P-torsion free objects (see 1.5.1(a)). Clearly Spec_{𝔱}^{1}(X) is the disjoint union of Spec_{𝔱}^{1,1}(X) and Spec_{𝔱}^{1,0}(X):

Spec_{𝔱}^{1}(X) = Spec_{𝔱}^{1,1}(X) ∐ Spec_{𝔱}^{1,0}(X).  (1)

3.1. Remarks. (a) It follows from definitions that

Spec_{𝔱}^{1,1}(X) = {P ∈ 𝔗𝔥(X) | P^{𝔱} ∩ P^{⊥} ≠ 0}.

(b) The notion of Spec_{𝔱}^{1}(X) is selfdual, i.e. Spec_{𝔱}^{1}(X) is naturally isomorphic to Spec_{𝔱}^{1}(X^{o}), where X^{o} is the dual ’space’: C_{X^{o}} = C_{X}^{op}. This is the consequence of the selfduality of the notions of a thick and topologizing subcategories.

The definition of P_{0} = P^{𝔱} ∩ P^{⊥} is not selfdual (because of P^{⊥}), in particular, the decomposition Spec_{𝔱}^{1}(X) = Spec_{𝔱}^{1,1}(X) ∐ Spec_{𝔱}^{1,0}(X) is not selfdual.

3.2. Proposition. (i) A thick subcategory P from Spec_{𝔱}^{1}(X) belongs to Spec_{𝔱}^{1,1}(X) iff it is a Serre subcategory. Thus,

Spec_{𝔱}^{1,1}(X) = {P ∈ 𝔗𝔥(X) | P^{−} = P ⫋ P^{𝔱}},
Spec_{𝔱}^{1,0}(X) = {P ∈ 𝔗𝔥(X) | P ⫋ P^{𝔱} ⊆ P^{−}}.

(ii) The map [P] ⟼ ⟨P ⟩ induces an isomorphism

Spec(X) ⟶^{∼} Spec_{𝔱}^{1,1}(X).

The inverse isomorphism assigns to every element P of Spec_{𝔱}^{1,1}(X) the topologizing subcategory [P^{𝔱} ⋂ P^{⊥}].

Proof. (i) Suppose that P = P^{−} ⫋ P^{𝔱}. Then (P^{𝔱} − P) ⋂ P^{−} = 0 which means that every nonzero object of P^{𝔱} − P has a nonzero subquotient which has no P-torsion. Since P^{𝔱} is a topologizing subcategory, this subquotient belongs to P^{𝔱} ⋂ P^{⊥}. In particular, this shows that P^{𝔱} ⋂ P^{⊥} ≠ 0, i.e. P is an element of Spec_{𝔱}^{1,1}(X), which proves the inclusion Spec_{𝔱}^{1}(X) ⋂ 𝔖𝔢(X) ⊆ Spec_{𝔱}^{1,1}(X). Here 𝔖𝔢(X) denotes the preorder of all Serre subcategories of C_{X}.

Conversely, if P is a thick subcategory which is not a Serre subcategory, i.e. P ⫋ P^{−}, then P^{𝔱} ⊆ P^{−}. Since P^{−} ⋂ P^{⊥} = 0, the inclusion P^{𝔱} ⊆ P^{−} implies that P^{𝔱} ⋂ P^{⊥} = 0, i.e. P ∉ Spec_{𝔱}^{1,1}(X). This shows that Spec_{𝔱}^{1,1}(X) = Spec_{𝔱}^{1}(X) ⋂ 𝔖𝔢(X).

(ii) (a) The image of Spec(X) by the map [P] ⟼ ⟨P ⟩ is contained in Spec_{𝔱}^{1,1}(X) because ⟨P ⟩ is a Serre (hence thick) subcategory and P ∈ ⟨P ⟩^{𝔱} ⋂⟨P ⟩^{⊥} for every P in Spec(X).

(b) Let P ∈ Spec_{𝔱}^{1,1}(X). If 𝕋 is a topologizing subcategory of C_{X} such that P_{0} = P^{𝔱} ⋂ P^{⊥} ⊈ 𝕋, then 𝕋 ⊆ P.

In fact, suppose 𝕋 ⊈ P. Then the topologizing subcategory [𝕋, P] spanned by 𝕋 and P contains properly P, hence it contains P^{𝔱}. By 1.3.3(b), every P-torsion free object of [𝕋, P] belongs to 𝕋; in particular, P_{0} = P^{𝔱} ⋂ P^{⊥} ⊈ 𝕋.

(c) Let P ∈ Spec_{𝔱}^{1,1}(X). Every nonzero object of P_{0} = P^{𝔱} ⋂ P^{⊥} belongs to Spec(X). Let L be a nonzero object of P_{0} and L_{1} a nonzero subobject of L. Then [L_{1}] ⊆ [L]. If

[L_{1}] ⊈ [L], then it follows from (b) above that [L_{1}] ⊆ P, or, equivalently, L_{1} ∈ ObP. This contradicts to the assumption that the object L is P-torsion free.

(d) Let P ∈ Spec_{𝔱}^{1,1}(X). Then P = ⟨L⟩ for any nonzero object of P_{0} = P^{𝔱} ⋂ P^{⊥}. Let L be a nonzero object of P_{0}. Since L does not belong to the Serre subcategory

⟨L⟩, by (b), we have the inclusion ⟨L⟩ ⊆ P. On the other hand, if ⟨L⟩ ⫋ P, then L ∈ ObP which is not the case. Therefore P = ⟨L⟩.

(e) The topologizing subcategory [P_{0}] coincides with the subcategory [L] for any nonzero object L of P_{0}.

Clearly [L] ⊆ [P_{0}] for any L ∈ ObP_{0}. By (b), if P_{0} ⊈ [L], then [L] ⊆ P, hence L = 0. Since, by (c), every nonzero object of P_{0} belongs to Spec(X), this shows that [P_{0}] is

an element of Spec(X).

(f) It follows from the argument above that the map

Spec(X) ⟶ Spec_{𝔱}^{1,1}(X), [P] ⟼ ⟨P ⟩,

is inverse to the map Spec_{𝔱}^{1,1}(X) ⟶ Spec(X) which assigns to every P the topologizing subcategory [P_{0}].

3.3. Note. Suppose that the category C_{X} has the property (sup). Then for any P ∈ Spec_{𝔱}^{1,0}(X), the subcategory P^{𝔱} − P does not have objects of finite type.

In fact, by 3.2(i), P ⊆ P^{𝔱} ⊆ P^{−}. By 2.4.7, objects of P^{−} are supremums of their subobjects from P. Thus, every object of P^{−} of finite type belongs to P.

4. The spectrum Spec_{𝔱}^{0}(X).

For any subcategory (or a family of objects, or an object) B of C_{X}, we denote by B̂ the union of all topologizing subcategories of C_{X} which do not contain B.

4.1. Lemma. Let B be a subcategory and 𝕋 a topologizing subcategory of C_{X}. Then B ⊆ 𝕋 iff B̂ ⊆ 𝕋̂.

Proof. If B ⊆ 𝕋 and 𝕊 is a topologizing subcategory of C_{X} such that B ⊈ 𝕊, then 𝕋 ⊈ 𝕊. This shows that the inclusion B ⊆ 𝕋 implies that B̂ ⊆ 𝕋̂.

Conversely, let B̂ ⊆ 𝕋̂. If B ⊈ 𝕋, then 𝕋, being a topologizing subcategory, is contained in B̂. Together with the inclusion B̂ ⊆ 𝕋̂, this implies that 𝕋 ⊆ 𝕋̂ which is impossible by definition of 𝕋̂. Therefore, B ⊆ 𝕋.

By definition, the spectrum Spec_{𝔱}^{0}(X) is the preorder (with respect to ⊆) of all topologizing subcategories T of C_{X} such that T̂ is a thick subcategory.

4.2. Note. One can see that Spec_{𝔱}^{0}(X) is selfdual, i.e. Spec_{𝔱}^{0}(X^{o}) is naturally isomorphic to Spec_{𝔱}^{0}(X).

4.3. Proposition. (a) The spectrum Spec_{𝔱}^{0}(X) consists of all topologizing subcategories P of C_{X} having the property:

if P ⊆ 𝕋•𝕊 for some topologizing subcategories 𝕊 and 𝕋, then either P ⊆ 𝕊, or P ⊆ 𝕋.

(b) The map T ⟼ T̂ induces a monomorphism

Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{1}(X).  (1)

(c) If T ∈ Spec_{𝔱}^{0}(X), then T = [L] for every L ∈ Ob(T − T̂).

Proof. (a) Suppose P ∈ Spec_{𝔱}^{0}(X), i.e. P̂ is a thick subcategory. If 𝕊 and 𝕋 are topologizing subcategories which do not contain P, then they both are contained in P̂ and, since the latter is thick, 𝕋 • 𝕊 ⊆ P̂.

Conversely, if for any pair of topologizing subcategories 𝕊, 𝕋 of P̂ their Gabriel product 𝕋 • 𝕊 is a subcategory of P̂, then P̂ is thick, i.e. P belongs to Spec_{𝔱}^{0}(X).

(b) The assertion (b) is a special case of the general nonsense fact 2.4.6. It is useful to specialize the argument. Let T ∈ Spec_{𝔱}^{0}(X), and let 𝕊 be a topologizing subcategory of C_{X} properly containing T̂. Then it follows from the definition of T̂ that T ⊆ 𝕊, or, equivalently, [T , T̂] ⊆ 𝕊. This shows that (T̂)^{𝔱} = [T , T̂]. Since T ⊈ T̂, the subcategory T̂ is properly contained in (T̂)^{𝔱}, i.e. T̂ belongs to Spec_{𝔱}^{1}(X). The injectivity of the map (1) follows from 4.1.

(c) If B is a subcategory of T , then, by 4.1, B̂ ⊆ T̂. If, in addition, B ⊈ T̂ and T̂ is a topologizing subcategory, then T̂ ⊆ B̂, hence T̂ = B̂. If B is a topologizing subcategory, the latter equality means that T = B. Taking B = [L] for any object L from T − T̂, we obtain the equality T = [L].

4.3.1. Note. Let (G, ≤) be a preordered monoid. We call a non-unit element x of G prime if the set {s ∈ G | x ≥ s} is a submonoid of G.

It follows from 4.3(a) that Spec_{𝔱}^{0}(X) is the set of all prime elements of the preordered monoid ((𝔗(X), •), ⊆) of topologizing subcategories of C_{X}.

4.4. Proposition. The decomposition Spec_{𝔱}^{1}(X) = Spec_{𝔱}^{1,1}(X) ∐ Spec_{𝔱}^{1,0}(X) induces, via the canonical map Spec_{𝔱}^{0}(X) ⟶ Spec_{𝔱}^{1}(X) (cf. 4.3(b)), a decomposition

Spec_{𝔱}^{0}(X) = Spec(X) ∐ Spec_{𝔱}^{0,0}(X)  (2)

Proof. For every object L of Spec(X), the category L̂ = ˆ[L] coincides with the subcategory ⟨L⟩, because if a topologizing subcategory 𝕋 does not contain L, it is contained in ⟨L⟩. By 3.2(ii), the map [L] ⟼ ⟨L⟩ is an isomorphism Spec(X) ⟶^{∼} Spec_{𝔱}^{1,1}(X). Therefore, the preimage of Spec_{𝔱}^{1,1}(X) in Spec_{𝔱}^{0}(X) coincides with Spec(X).

Thus, the map (1) in 4.3 is the coproduct of the isomorphism Spec(X) ⟶^{∼} Spec_{𝔱}^{1,1}(X) and the map

Spec_{𝔱}^{0,0}(X) ⟶ Spec_{𝔱}^{1,0}(X), T ⟼ T̂.  (3)

It follows that Spec_{𝔱}^{0,0}(X) consists of all topologizing subcategories T such that T̂ belongs to 𝔗𝔥(X) − 𝔖𝔢(X).

4.4.1. Note. Proposition 4.4 gives the following description of Spec(X): it consists of all topologizing subcategories P of C_{X} such that the union P̂ of all topologizing subcategories which do not contain P is a Serre subcategory.

4.4.2. Corollary. Suppose that C_{X} is a noetherian category (i.e. every object of C_{X} is noetherian). Then Spec_{𝔱}^{1}(X) = Spec_{𝔱}^{1,1}(X) and Spec_{𝔱}^{0}(X) = Spec(X).

Proof. Let C_{X} be an arbitrary abelian category, and let T be a full subcategory of C_{X} closed under finite coproducts and quotients. Then every noetherian object of C_{X} has the largest subobject which belongs to T .

In fact, if M is a noetherian object, then the set S_{T} (M) of subobjects of M which belong to T has a maximal element, M_{T}. If N is any other element of S_{T} (M), then the image of the natural morphism M_{T} ⊕ N ⟶ M belongs to T (thanks to the assumptions on T ) and contains M_{T}. Due to the maximality of M_{T}, the image of M_{T} ⊕ N ⟶ M coincides with M_{T}, hence N is a subobject of M_{T}.

This shows that if all objects of the category C_{X} are noetherian, than the subcategory T is coreflective. In particular, every topologizing and every thick subcategory of C_{X} is coreflective. By 2.4.5, thick coreflective subcategories are Serre subcategories. The assertion follows now from 3.2 and 4.4.

4.4.3. Examples. The conditions of 4.4.2 are fulfilled if C_{X} is the category of finitely generated modules over a left noetherian ring, or the category of quasi-coherent sheaves on a noetherian scheme, or the category of coherent D-modules on a noetherian scheme.

4.5. Representatives of Spec_{𝔱}^{0}(X) and representatives of Spec(X). Let Spec_{𝔱}^{0}(X) denote the family of all objects M of C_{X} such that [M] ∈ Spec_{𝔱}^{0}(X). In other words, an object M belongs to Spec_{𝔱}^{0}(X) iff the union ˆ[M] of all topologizing subcategories of C_{X} which do not contain M is a thick subcategory. We consider Spec_{𝔱}^{0}(X) together with the specialization preorder ≻, which is precisely the preorder induced by the specialization preorder ⊇ on Spec_{𝔱}^{0}(X).

The decomposition

Spec_{𝔱}^{0}(X) = Spec(X) ∐ Spec_{𝔱}^{0,0}(X)

induces the decomposition

Spec_{𝔱}^{0}(X) = Spec(X) ∐ Spec_{𝔱}^{0,0}(X).

4.5.1. Proposition. Let C_{X} have the property (sup). Then an object M of finite type belongs to Spec_{𝔱}^{0}(X) iff it belongs to Spec(X).

Proof. Let M be an object of Spec_{𝔱}^{0}(X). Suppose that M ∈ ˆ[M]^{−}. Then, by 2.4.7, M is the supremum of its subobjects from ˆ[M]. Therefore, if M is of finite type, it belongs to

ˆ[M] which is impossible by the definition of ˆ[M]. This shows that ˆ[M] = ˆ[M]^{−}, i.e. ˆ[M] is a Serre subcategory. Therefore, by 3.2(i), [M] ∈ Spec(X).

4.5.2. Corollary. Let C_{X} have the property (sup) and enough objects of finite type (i.e. every nonzero object of C_{X} has a nonzero subobject of finite type). Then Spec_{𝔱}^{0,0}(X) is a subset of all [M] ∈ Spec_{𝔱}^{0}(X) such that the subcategory [M]− ˆ[M] does not contain objects of finite type. Or, equivalently, Spec(X) = {[M] ∈ Spec_{𝔱}^{0}(X) | M is of finite type}.

4.5.3. Corollary. Let C_{X} = R − mod for an associative ring R. The set of all left ideals m such that R/m belongs to Spec_{𝔱}^{0}(X) coincides with the left spectrum, Spec_{ℓ}(R) of the ring R.

Proof. By 4.5.2, if the module R/m belongs to Spec_{𝔱}^{0}(X), then it belongs to Spec(X). By [R, III.4.2], the left spectrum Spec_{ℓ}(R) consists of all left ideals 𝔪 such that the quotient module R/𝔪 belongs to Spec(X).

The reader is encouraged to look into Appendix 2 for more details about the spectra Spec_{𝔱}^{0}(X) and Spec(X).

5. The complete spectrum and the general construction.

5.1. Invariant topologizing subcategories and localizations. Fix an abelian category C_{X}. Let 𝕋 and 𝕊 be topologizing subcategories of C_{X}. We say that the subcategory 𝕋 is 𝕊-invariant if 𝕋 = 𝕊 • 𝕋 • 𝕊.

It follows from this definition that for any topologizing subcategory 𝕋, the smallest 𝕋-invariant topologizing subcategory is the thick envelope 𝕋^{∞} = ⋃_{n≥1} 𝕋^{•n} of 𝕋, which is the

smallest thick subcategory containing 𝕋. In particular, 𝕋 is 𝕋-invariant iff it is thick.

On the other hand, 𝕋 is 𝕊-invariant iff 𝕋 is 𝕊^{∞}-invariant. In particular, if 𝕋 is 𝕊invariant, then 𝕊^{∞} ⊆ 𝕋.

If 𝕋_{1} is 𝕋_{0}-invariant and 𝕋_{2} is 𝕋_{1}-invariant, then 𝕋_{2} is 𝕋_{0}-invariant, because in this case we have: 𝕋_{2} ⊆ 𝕋_{0} • 𝕋_{2} • 𝕋_{0} ⊆ 𝕋_{1} • 𝕋_{2} • 𝕋_{1} = 𝕋_{2}.

Given a thick subcategory 𝕊 of the category C_{X}, let T (X, 𝕊) denote the preorder (with respect to ⊆) of 𝕊-invariant topologizing subcategories. Note that the inclusion T (X, 𝕊) ↪ T (X) has a left inverse (a left adjoint), T (X) ⟶ T (X, 𝕊) which maps every topologizing subcategory 𝕋 to the 𝕊-invariant topologizing subcategory 𝕊 • 𝕋 • 𝕊.

5.1.1. Lemma. Let 𝕊 be a thick subcategory of the category C_{X} and C_{X} ⟶^{q^{∗}} C_{X}/𝕊 the localization functor. Then for any topologizing subcategory 𝕋 of C_{X}, the topologizing subcategory [q^{∗}(𝕋)] of C_{X}/𝕊 spanned by q^{∗}(T) is generated by all objects of C_{X}/𝕊 isomorphic to objects of q^{∗}(T), and its preimage in C_{X}, q^{∗^{−1}}([q^{∗}(𝕋)]), coincides with 𝕊 • 𝕋 • 𝕊. In other words, q^{∗^{−1}}([q^{∗}(𝕋)]) is the smallest 𝕊-invariant subcategory of C_{X} containing 𝕋.

Proof. It follows that [q^{∗}(𝕋)] = ⋃_{M∈Ob𝕋}[q^{∗}(M)]. Let q^{∗}(M) ≻ q^{∗}(L), i.e. there exists

a diagram q^{∗}(M)^{⊕n} ⟵ q^{∗}(N) ⟶ q^{∗}(L) in which the left arrow is a monomorphism and the right arrow is an epimorphism. To this diagram, there corresponds a diagram M^{⊕n} ⟵ N′ ⟶ N ⟵ N′′ ⟶ L such that q^{∗} maps the arrows N′ ⟶ N ⟵ N′′ to isomorphisms. Completing N′ ⟶ N ⟵ N′′ to a cartesian square

N′ ⟶ N
↑ ↑
K′ ⟶ N′′

and taking the compositions of the arrows M^{⊕n} ⟵ N′ ⟵ K′ and K′ ⟶ N′′ ⟶ L, we obtain a diagram, M^{⊕n} ⟵^{𝔧′} K′ ⟶^{𝔢′} L such that q^{∗}(𝔧′) is a monomorphism and q^{∗}(𝔢′) is an epimorphism. Representing arrows 𝔧′ and 𝔢′ as compositions of an epimorphism and a monomorphism, we obtain the diagram

M^{⊕n} ⟵^{𝔧} K ⟵^{s′} K′ ⟶^{e′} L′ ⟶^{t′} L

in which 𝔧 is a monomorphism, e′ is an epimorphism, and q^{∗}(s′), q^{∗}(t′) are isomorphisms.

Taking a push-forward of the pair of arrows K ⟵^{s′} K′ ⟶^{e′} L′, we obtain a commutative diagram

M^{⊕n} ⟵^{𝔧} K ⟶^{𝔢} L_{1}
s′↑  ↑ t
K′ ⟶^{e′} L′  ⟶^{t′} L

in which 𝔧 is a monomorphism, 𝔢 is an epimorphism, and q^{∗}(t), q^{∗}(t′) are isomorphisms. Therefore q^{∗}(L_{1}) ≃ q^{∗}(L) and M ≻ L_{1}. The latter implies that if M ∈ Ob𝕋, then L_{1} ∈ Ob𝕋.

5.1.2. Corollary. Let 𝕊 be a thick subcategory of C_{X} and C_{X} ⟶^{q^{∗}} C_{X/𝕊} = C_{X}/𝕊 the localization functor. The map 𝕋 ⟼ q^{∗^{−1}}(𝕋) induces an isomorphism

T (X/𝕊) ⟶^{∼} T (X, 𝕊)

between the preorder of topologizing subcategories of the quotient category C_{X/𝕊} = C_{X}/𝕊 the preorder of 𝕊-invariant topologizing subcategories of C_{X}.

5.2. The complete spectrum. Fix an abelian category C_{X}. Recall that the complete spectrum, Spec^{1}(X), of X is formed by thick subcategories P of C_{X} such that the quotient category C_{X/P} = C_{X}/P is local. The latter means that the category C_{X/P} has the smallest nonzero topologizing subcategory.

For every thick subcategory 𝕊 of C_{X}, let 𝕊^{⊛} denote the intersection of all 𝕊-invariant topologizing subcategories of C_{X} properly containing 𝕊.

5.2.1. Proposition. A thick subcategory P of the category C_{X} belongs to Spec^{1}(X) iff there exists the smallest P-invariant topologizing subcategory properly containing P. In other words, Spec^{1}(X) = {P ∈ 𝔗𝔥(X) | P ≠ P^{⊛}}.

Proof. The assertion follows from 5.1.2.

5.2.2. Corollary. Spec_{𝔱}^{1}(X) ⊆ Spec^{1}(X).

5.2.3. The preorder 𝔗̃(X) and the complete spectrum. Let 𝔗̃(X) denote the preorder whose objects are topologizing subcategories of C_{X} and morphisms are identical morphisms and inclusions 𝕊 ↪ 𝕋 such that 𝕋 is 𝕊-invariant. It follows from the discussion at the beginning of 5.1 that the composition of morphisms is a morphism.

Notice that 𝔗𝔥(X) is a subpreorder of 𝔗̃(X), because if 𝕊 ↪ 𝕋 and 𝕋 is thick, then 𝕋 ⊆ 𝕊 • 𝕋 • 𝕊 ⊆ 𝕋 • 𝕋 • 𝕋 = 𝕋, that is 𝕋 is 𝕊-invariant. In particular, the preorder 𝔖𝔢(X) of Serre subcategories is a subpreorder of 𝔗̃(X).

5.2.3.1. Lemma. For every topologizing subcategory 𝕊 of C_{X}, there is a natural isomorphism of preorders 𝕊\𝔗̃(X) ⟶^{∼} 𝔗̃(X/𝕊^{∞}), where 𝕊^{∞} is the smallest thick subcategory containing 𝕊.

Proof. The assertion follows from 5.1.2.

Applying the general construction to the inclusion functor 𝔗𝔥(X) \underrightarrow{_{J_{X}}} 𝔗̃(X), we obtain the spectra 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X), 𝔍_{X}), i = 0, 1. and a canonical monomorphism

𝔖𝔭𝔢𝔠^{0}(𝔗𝔥(X), J̃_{X}) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}).

5.2.3.2. Proposition. 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) = Spec^{1}(X).

Proof. The preorder 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) is defined by the cartesian square

𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) ⟶ 𝔗𝔥(X)
↓  ↓
𝔖𝔭𝔢𝔠^{1}(𝔗̃(X))  ⟶ 𝔗̃(X)

In other words, the objects of 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) are thick subcategories P of C_{X} such that there exists the smallest P-invariant topologizing subcategory of C_{X} properly containing P. Comparing this description with 5.2.1, one can see that 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) coincides with Spec^{1}(X).

5.2.4. Note. The ’abslolute’ spectrum, 𝔖𝔭𝔢𝔠^{1}(𝔗̃(X)), of the preorder 𝔗̃(X) is the (obviously, disjoint) union of 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) = Spec^{1}(X) and 𝔗̃(X) − 𝔗𝔥(X).

In fact, if P ∈ Spec^{1}(X), then there exists the smallest P-invariant topologizing subcategory P^{⊛} of C_{X} properly containing P. The pair (P, P ↪ P^{⊛}) is, evidently, the initial object of the category P\𝔗̃(X). If 𝕋 is a topologizing, but not thick, subcategory of C_{X}, then (𝕋, 𝕋 ↪ 𝕋^{∞}) is the initial object of the category 𝕋\𝔗̃(X); hence 𝕋 is an object of 𝔖𝔭𝔢𝔠^{1}(𝔗̃(X)).

5.2.5. The spectrum Spec^{0}(X). We denote 𝔖𝔭𝔢𝔠^{0}(𝔗𝔥(X), J̃_{X}) by Spec^{0}(X). It follows that objects of Spec^{0}(X) are topologizing subcategories T such that the union T̂ of all topologizing subcategories which are not T -invariant belongs to Spec^{1}(X). In particular, T̂ is thick, hence it coincides with the union of all thick subcategories of C_{X} which do not contain T . This discussion together with the inclusion Spec_{𝔱}^{1}(X) ⊆ Spec^{1}(X) of 5.2.2 shows that there is a commutative diagram

Spec_{𝔱}^{0}(X) ⟶ Spec^{0}(X)
↓  ↓
Spec_{𝔱}^{1}(X) ⟶ Spec^{1}(X)

(1)

whose horizontal arrows are inclusions and vertical arrows are injective.

5.3. A canonical decomposition of the complete spectrum and the Sspectrum. For every element P of Spec^{1}(X), we set P_{⊛} = P^{⊛} ⋂ P^{⊥}; i.e. P_{⊛} is the full subcategory of P^{⊛} generated by all P-torsion free objects of P^{⊛}. Similarly to Section 3, we have a decomposition

Spec^{1}(X) = Spec^{1,1}(X) ∐ Spec^{1,0}(X),  (2)

where

Spec^{1,0}(X) = {P ∈ 𝔗𝔥(X) | P_{⊛} = 0} and Spec^{1,1}(X) = {P ∈ 𝔗𝔥(X) | P_{⊛} ≠ 0}.

Recall that the S-spectrum, Spec^{−}(X), of the ’space’ X is formed by all Serre subcategories P of C_{X} such that the quotient ’space’ X/P is local. In other words, Spec^{−}(X) = Spec^{1}(X) ⋂ 𝔖𝔢(X), where 𝔖𝔢(X) is the preorder of all Serre subcategories of C_{X}.

We have the following analogue of 3.2(i):

5.3.1. Proposition. A thick subcategory P from Spec^{1}(X) belongs to Spec^{1,1}(X) iff it is a Serre subcategory. Thus,

Spec^{1,1}(X) ={P ∈ 𝔗𝔥(X) | P^{−} = P ⫋ P^{⊛}} = Spec^{−}(X)
Spec^{1,0}(X) ={P ∈ 𝔗𝔥(X) | P ⫋ P^{⊛} ⊆ P^{−}}.

Proof. Suppose P = P^{−} ⫋ P^{⊛}. Then every nonzero object of P^{⊛} − P has a nonzero subquotient which has no P-torsion. Since P^{⊛} is a topologizing subcategory, this subquotient belongs to P^{⊛} ⋂ P^{⊥}. In particular, this shows that P^{⊛} ⋂ P^{⊥} ≠ 0, that is P is an element of Spec^{1,1}(X), which proves the inclusion Spec^{1}(X) ⋂ 𝔖𝔢(X) ⊆ Spec^{1,1}(X). Here 𝔖𝔢(X) denotes the preorder of all Serre subcategories of C_{X}.

Conversely, if P is a thick subcategory which is not a Serre subcategory, i.e. P ⫋ P^{−}, then P^{⊛} ⊆ P^{−}, because P^{−} (as any thick subcategory containing P) is P-invariant. Since P^{−} ⋂ P^{⊥} = 0, the inclusion P^{⊛} ⊆ P^{−} implies that P^{⊛} ⋂ P^{⊥} = 0, i.e. P ∉ Spec^{1,1}(X). This shows that Spec^{1,1}(X) = Spec^{1}(X) ⋂ 𝔖𝔢(X) = Spec^{−}(X).

5.4. The preorder 𝔗̃(X) and the S-spectrum. General pattern associate with

the inclusion functor 𝔖𝔢(X) \underrightarrow{_{J_{X}^{𝔰}}} 𝔗̃(X) the spectra 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X), J_{X}^{𝔰}), i = 0, 1, and a canonical monomorphism

𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X), J̃_{X}^{𝔰}) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X), J̃_{X}^{𝔰}).

5.4.1. Proposition. 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X), J̃_{X}^{𝔰}) = Spec^{−}(X).

Proof. By 5.2.3.2, 𝔖𝔭𝔢𝔠^{1}(𝔗𝔥(X), J̃_{X}) coincides with the complete spectrum Spec^{1}(X) This implies that 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X), J̃_{X}^{𝔰}) is the S-spectrum Spec^{−}(X) of the ’space’ X.

6. The spectrum Spec_{⋊}(X).

For every subcategory (or a family of objects) B of the category C_{X}, let B_{⋊} denote the union of all thick subcategories of C_{X} which are left orthogonal to B.

Clearly, B_{⋊} ⊆ D_{⋊} if D ⊆ B, and (B_{⋊})^{⊥} is the largest subcategory among the subcategories B′ of C_{X} such that B′_{⋊} = B_{⋊}.

6.1. Lemma. If the subcategory B is closed under taking subobjects, then B_{⋊} is a Serre subcategory.

Proof. In fact, suppose that B_{⋊}^{−} ≠ B_{⋊}, and let M be an object of B_{⋊}^{−} − B_{⋊}.

The condition M ∉ ObB_{⋊} means that for some object M′ of [M]_{•}, there is a nonzero morphism N ⟶^{g} M′ with N ∈ ObB. The image M′′ of the morphism g is a nonzero object of B_{⋊}^{−}. In particular, it has a nonzero subobject L from B_{⋊}. Thus, we have a cartesian square

N ×_{M′′} L ⟶ L
↓  ↓
N  ⟶ M′′

whose vertical arrows are monomorphisms and horizontal arrows are epimorphisms. Since B is closed under taking subobjects, in particular N ×_{M′′} L is an object of B, we run into a contradiction.

6.2. The functor 𝔏_{⋊} and the related spectrum. Let 𝔗_{≤}(X) denote the preorder (with respect to ⊆) of full subcategories of C_{X} closed under taking subobjects and finite coproducts. Let 𝔏_{⋊} denote the morphism of preorders 𝔗_{≤}(X) ⟶ 𝔖𝔢(X) which maps every object B of 𝔗_{≤}(X) to the Serre subcategory B_{⋊} (defined in 6.1). And let 𝔏′_{⋊} be the composition of 𝔏_{⋊} and the inclusion 𝔖𝔢(X) ↪ 𝔗̃(X). The morphism 𝔏′_{⋊} gives rise to the spectrum 𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏_{⋊}) defined via the cartesian square

\begin{matrix}𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏′_{⋊}) & ⟶ & 𝔗_{≤}(X) \\ ↓ & & ↓ 𝔏′_{⋊} \\ 𝔖𝔭𝔢𝔠^{1}(𝔗̃(X)) & ⟶ & 𝔗̃(X) \end{matrix}  (1)

Since the morphism 𝔏′_{⋊} takes values in the preorder 𝔖𝔢(X) of Serre subcategories, the spectrum 𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏′_{⋊}) coincides with the spectrum 𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏_{⋊}) defined by the cartesian square

\begin{matrix}𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏_{⋊}) & ⟶ & 𝔗_{≤}(X) \\ ↓ & & ↓ 𝔏_{⋊} \\ Spec^{−}(X) & ⟶ & 𝔖𝔢(X) \end{matrix}  (2)

In other words, objects of 𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏_{⋊}) are naturally identified with objects B of 𝔗_{≤}(X) such that the Serre subcategory B_{⋊} belongs to the S-spectrum Spec^{−}(X).

6.3. The spectrum Spec_{⋊}(X). We denote by Spec_{⋊}(X) the preorder (with respect to the inclusion) of full nonzero subcategories B of the category C_{X} such that B = (B_{⋊})^{⊥} ⋂(B_{⋊})^{⊛} and B_{⋊} ∈ Spec^{−}(X).

It follows from the discussion above that Spec_{⋊}(X) ⊆ 𝔖𝔭𝔢𝔠^{1}(𝔗_{≤}(X), 𝔏_{⋊}).

6.4. Proposition. The map B ⟼ B_{⋊} induces an isomorphism

Spec_{⋊}(X) ⟶ Spec^{−}(X)

with the inverse map P ⟼ P_{⊛} = P^{⊛} ∩ P^{⊥}.

Proof. Let P ∈ Spec^{−}(X). The subcategory P_{⊛} = P^{⊛} ∩ P^{⊥} is closed under taking subobjects, hence, by 6.1, (P_{⋆})_{⋊} is a Serre subcategory. This Serre subcategory contains P, because all objects of P_{⊛} are right orthogonal to P. If (P_{⊛})_{⋊} would contain P properly, then it would contain P_{⊛}, which is impossible. Therefore (P_{⊛})_{⋊} = P. This shows that P ⟼ P_{⊛} maps Spec^{−}(X) to Spec_{⋊}(X).

On the other hand, if B belongs to Spec_{⋊}(X), i.e. 0 ≠ B = (B_{⋊})^{⊥} ⋂(B_{⋊})^{⊛}, then B_{⋊} ∈ Spec^{−}(X).

6.5. Representatives of points of Spec_{⋊}(X). For every subcategory A of C_{X}, we denote by [A) the smallest full subcategory of C_{X} which contains A and is closed under taking subobjects and finite coproducts.

For every M ∈ C_{X}, we write [M) instead of [M, id_{M}). We denote by Spec_{⋊}(X) the family of all objects M of C_{X} such that [M)_{⋊} ∈ Spec^{−}(X). Objects of Spec_{⋊}(X) are regarded as representatives of the corresponding points of Spec_{⋊}(X).

It follows that Spec(X) ⊆ Spec_{⋊}(X) and the map

Spec_{⋊}(X) ⟶ Spec^{−}(X), M ⟼ [M)_{⋊},  (3)

is an extension of the map Spec(X) ⟶ Spec^{−}(X), P ⟼ ⟨P ⟩.

We define a preorder ⋟ on Spec_{⋊}(X) by M ⋟ N ⇔ [N)_{⋊} ⊆ [M)_{⋊}. If M and N are objects of Spec(X), then [N)_{⋊} = ⟨N⟩ and [M)_{⋊} = ⟨M⟩. Therefore in this case M ⋟ N ⇔ ⟨N⟩ ⊆ ⟨M⟩ ⇔ [N] ⊆ [M] ⇔ M ≻ N; i.e. the restriction of the preorder ⋟ to Spec(X) coincides with the specialization preorder ≻ on Spec(X).

6.5.1. Proposition. Suppose that P ∈ Spec^{1}(X) is such that the localization functor

C_{X} ⟶^{q_{P}^{∗}} C_{X}/P has a right adjoint, q_{_{P} ∗} (in particular, P ∈ Spec^{−}(X)) and the quotient category C_{X}/P has a simple object, L. Then [q_{_{P} ∗}(L)) = P_{⊛} = P^{⊛} ∩ P^{⊥}.

Proof. In fact, for every L ∈ ObC_{X}/P, the object q_{_{P} ∗}(L) is P-torsion free, i.e. it belongs to P^{⊥}, and q_{P}^{∗}q_{_{P} ∗}(L) ≃ L. Therefore, if L is a quasi-final object, then q_{_{P} ∗}(L) belongs to P_{⊛} = P^{⊛} ∩ P^{⊥}. In particular, [q_{_{P} ∗}(L)) ⊆ P_{⊛}. In order to prove the inverse inclusion, notice that any nonzero object M of P_{⊛} is a subobject of q_{_{P} ∗}(q_{P}^{∗}(M)); in particular, M is an object of the subcategory [q_{_{P} ∗}(q_{P}^{∗}(M))). Since M is a nonzero object of P_{⊛}, its image q_{P}^{∗}(M) is a quasi-final object of C_{X}/P. Therefore q_{P}^{∗}(M) is isomorphic to the coproduct of a finite set of copies of the simple object L, which implies that q_{_{P} ∗}q_{P}^{∗}(M) is isomorphic to the coproduct of a finite number of copies of q_{_{P} ∗}(L). It follows that the subcategories [q_{_{P} ∗}(q_{P}^{∗}(M))) and [q_{_{P} ∗}(L)) coincide. In particular, M is an object of [q_{_{P} ∗}(L)). This proves the inverse inclusion P_{⊛} ⊆ [q_{_{P} ∗}(L)).

6.5.2. The case of Grothendieck categories. Suppose that C_{X} is a Grothendieck category. Then the localization functor at any Serre subcategory has a right adjoint. If C_{X} has Gabriel-Krull dimension (cf. A3.7), then for every P ∈ Spec^{−}(X), the quotient category C_{X}/P has a unique (up to isomorphism) simple object, L_{P}.

The map P ⟼ q_{_{P} ∗}(L_{P}) takes values in Spec_{⋊}(X) and is a section of the map

Spec_{⋊}(X) ⟶ Spec^{−}(X), M ⟼ [M)_{⋊}.

Thus, in the case when C_{X} has a Gabriel-Krull dimension, the map P ⟼ q_{_{P} ∗}(L_{P}) is a canonical (defined uniquely up to isomorphism) choice of representatives of points of the spectrum Spec_{⋊}(X). For a general Grothendieck category C_{X}, this gives a canonical choice of a representative for every point Q of Spec_{⋊}(X) such that the category C_{X}/Q_{⋊} has simple objects.

7. Functorialities.

7.1. Lemma. Let 𝕋 be a topologizing subcategory of an abelian category C_{X} and |𝕋| a ’space’ defined by C_{|𝕋|} = 𝕋.

(a) For every S ∈ 𝔗𝔥(|𝕋|), there exists the biggest and the smallest thick subcategory of C_{X}, resp. S_{−} and S_{+}, such that S_{−} ∩ 𝕋 = S = S_{+} ∩ 𝕋.

(b) If S is a Serre subcategory of 𝕋, then S_{+} is a Serre subcategory of C_{X}, and S^{−} ∩ 𝕋 = S.

Proof. (a) Let 𝔗(X; 𝕋, S) denote the preorder of all topologizing subcategories B of C_{X} such that B∩𝕋 ⊆ S. Notice that 𝔗(X; 𝕋, S) is closed under the Gabriel multiplication: if B′, B′′ are elements of 𝔗(X; 𝕋, S) and 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence such that M′ ∈ ObB′, M′′ ∈ ObB′′ and M ∈ Ob𝕋, then M′ and M′′ are objects of S, hence M is an object of S. Therefore, the union S_{+} of all B from 𝔗(X; 𝕋, S) is a thick subcategory of C_{X} such that S_{+} ∩ 𝕋 ⊆ S. Since S is a subcategory of S_{+}, the intersection S_{+} ∩ 𝕋 coincides with S. This also implies that the intersection S_{−} ∩ 𝕋 ⊆ S of the smallest thick subcategory of C_{X} containing S with 𝕋 coincides with S.

(b) Suppose now that S is a Serre subcategory of 𝕋. Then S_{+}^{−} ∩ 𝕋 = S.

In fact, let M be an object of S_{+}^{−} ∩ 𝕋; and let L be a nonzero subquotient of M. Since 𝕋 is closed with respect to taking subquotients, there exists a diagram in M ⟵ K ⟶ L in 𝕋 whose left arrow is a monomorphism and the right arrow is an epimorphism. By definition of S_{+}^{−}, the object L has a nonzero subobject which belongs to S_{+}, hence it belongs to S_{+} ∩ 𝕋 = S. This shows that M is an object of S^{−}. Therefore S_{+}^{−} ∩ 𝕋 = S_{𝕋}^{−}. Here S_{𝕋}^{−} denotes the Serre envelope of S in 𝕋. In particular, if S is a Serre subcategory of 𝕋 (i.e. S = S_{𝕋}^{−}), then S_{+}^{−} ∩ 𝕋 = S = S^{−} ∩ 𝕋.

For a subcategory 𝕋 of C_{X}, we set

U^{1}(𝕋) = {P ∈ Spec^{1}(X) | 𝕋 ⊆ P}
V^{1}(𝕋) = Spec^{1}(X) − U^{1}(𝕋) = {P ∈ Spec^{1}(X) | 𝕋 ⊈ P}.
U^{−}(𝕋) = U^{1}(𝕋) ∩ 𝔖𝔢(X) and V^{−}(𝕋) = V^{1}(𝕋) ∩ 𝔖𝔢(X)

(1)

The following assertion can be extracted from [R, Ch.6]. For the reader’s convenience, we give here a (mostly different) proof.

7.2. Proposition. Let 𝕋 be a topologizing subcategory of C_{X} and 𝕋^{∞} the smallest thick subcategory of C_{X} containing 𝕋.

(a) V^{1}(𝕋) = V^{1}(𝕋^{∞}) and U^{1}(𝕋) = U^{1}(𝕋^{∞}). The map

U^{1}(𝕋) ⟶ Spec^{1}(X/𝕋^{∞}), P ⟼ P/𝕋^{∞},  (2)

is an isomorphism.

(b) V^{−}(𝕋) = V^{−}(𝕋^{−}) and U^{−}(𝕋) = U^{1}(𝕋^{−}). The map

U^{−}(𝕋) ⟶ Spec^{−}(X/𝕋^{−}), P ⟼ P/𝕋^{−},  (3)

is an isomorphism.

(c) For every P ∈ V^{1}(𝕋), the intersection P ∩ 𝕋 is an element of Spec^{1}(|𝕋|), and the map

V^{1}(𝕋) ⟶ Spec^{1}(|𝕋|), P ⟼ P ∩ 𝕋,  (4)

is an isomorphism. The inverse map is given by P̃ ⟼ P̃_{+} (see 7.1). Here |𝕋| is defined by C_{|𝕋|} = 𝕋.

(d) For every P ∈ V^{−}(𝕋), the intersection P ∩ 𝕋 is an element of Spec^{1}(|𝕋|), and the map (2) induces an isomorphism V^{−}(𝕋) ⟶ Spec^{−}(|𝕋|).

Proof. (a) Since elements of Spec^{1}(X) are thick subcategories, U^{1}(𝕋) = U^{1}(𝕋^{∞}), hence V^{1}(𝕋) = V^{1}(𝕋^{∞}). Clearly P ⟼ P/𝕋^{∞} maps U^{1}(𝕋^{∞}) to Spec^{1}(X/𝕋^{∞}). The inverse map assigns to every element P̃ of Spec^{1}(X/𝕋^{∞}) the thick subcategory q^{∗^{−1}}(P̃), where q^{∗} is the localization functor C_{X} ⟶ C_{X/𝕋^{∞}}.

(b) The argument is similar to that of (a).

(c) Let P ∈ V^{1}(𝕋), that is P is a thick subcategory of C_{X} such that X/P is local and 𝕋 ⊈ P. The composition of the inclusion functor 𝕋 ↪ C_{X} and the localization C_{X} ⟶^{q_{P}^{∗}} C_{X/P} is an exact functor with the kernel P ∩ 𝕋, hence it factors uniquely through the localization 𝕋 ⟶ 𝕋/(𝕋 ∩ P). Thus, we have a commutative diagram

\begin{matrix}𝕋 ∩ P & ⟶ & 𝕋 & ⟶^{q^{∗}} & 𝕋/(𝕋 ∩ P) \\ ↓ & & ↓ & & ↓ 𝔧^{∗} \\ P & ⟶ & C_{X} & ⟶^{q_{P}^{∗}} & C_{X/P} \end{matrix}  (5)

in which the two left vertical arrows are inclusion functors. It follows that the right arrow in (5) is a fully faithful functor which induces an equivalence between 𝕋/(𝕋 ∩ P) and a nonzero topologizing subcategory of C_{X/P}. Every nonzero topologizing subcategory of a local category is local. Therefore 𝕋/(𝕋 ∩ P) is a local category. The latter means that 𝕋 ∩ P ∈ Spec^{1}(|𝕋|).

(c1) Let now P̃ be an element of Spec^{1}(|𝕋|). By 7.1, there is the biggest topologizing subcategory P̃_{+} of C_{X} (which happens to be thick) such that P̃ = 𝕋 ∩ P̃_{+}. We claim that P̃_{+} is an element of Spec^{1}(X).

In fact, let B be a topologizing P̃_{+}-invariant subcategory of C_{X} properly containing P̃_{+}. Then B ∩ 𝕋 contains P̃ properly; because if B ∩ 𝕋 = P̃, then B ⊆ P̃_{+}. Clearly, the subcategory B ∩ 𝕋 is P̃-invariant. Therefore it contains the subcategory P̃^{⊛}. This shows that (P̃_{+})^{⊛} = P̃_{+} • P̃^{⊛} • P̃_{+} is the smallest P̃_{+}-invariant topologizing subcategory of C_{X} properly containing P̃_{+}; so that P̃_{+} belongs to Spec^{1}(X). Thus, the map

Spec^{1}(|𝕋|) ⟶ V^{1}(𝕋), P̃ ⟼ P̃_{+},

is a right inverse to the map (4). It remains to show that the map (4) is injective.

(c2) Let P̃ ∈ Spec^{1}(|𝕋|); and let P_{1} and P_{2} be elements of V^{1}(𝕋) such that P_{1} ∩ 𝕋 = P̃ = P_{2} ∩ 𝕋. Replacing X by X/(P_{1} ∩ P_{2}), we can (and will) assume that P̃ = 0. In particular, |𝕋| is local. Let M be a quasi-final object of 𝕋. Notice that the composition

of the embedding 𝕋 ↪ C_{X} with the localization functor C_{X} ⟶^{q_{P_{1}}^{∗}} C_{X/P_{1}} is a fully faithful functor 𝕋 ⟶ C_{X/P_{1}} which induces an equivalence between 𝕋 and a nonzero topologizing subcategory of the category C_{X/P_{1}} (see the diagram (5)). Therefore the image q_{P_{1}}^{∗}(M) of the object M is a quasi-final object of C_{X/P_{1}}. This implies that P_{1} = ⟨M⟩.

In fact, P_{1} ⊆ ⟨M⟩, because M ∉ ObP_{1}. On the other hand, if P_{1} ≠ ⟨M⟩, then ⟨M⟩ (being a thick subcategory containing P_{1}, hence P_{1}-invariant) contains the object M which cannot happen, because M ∈ Spec(|𝕋|) ⊆ Spec(X).

By the same reason, P_{2} = ⟨M⟩, hence P_{2} = P_{1}.

(d) The assertion follows from (c) and 7.1(b). Details are left to the reader.

7.3. Corollary. Let P_{1} and P_{2} be elements of Spec^{1}(X) such that P_{1}^{⊛} ∩ P_{2} ⊆ P_{1}. Then P_{2} ⊆ P_{1}.

Proof. The assertion follows from 7.2(c) applied to 𝕋 = P^{⊛}.

For a subcategory 𝕋 of C_{X}, we set

U_{⋊}(𝕋) ={P_{⊛} ∈ Spec_{⋊}(X) | 𝕋 ∩ P_{⊛} = 0}
V_{⋊}(𝕋) =Spec_{⋊}(X) − U_{⋊}(𝕋) = {P_{⊛} ∈ Spec_{⋊}(X) | 𝕋 ∩ P ≠ 0}.

(6)

7.4. Proposition. Let 𝕋 be a topologizing subcategory of C_{X}.

(a) V_{⋊}(𝕋) = V_{⋊}(𝕋^{−}) and U_{⋊}(𝕋) = U_{⋊}(𝕋^{−}). The map

U_{⋊}(𝕋) ⟶ Spec_{⋊}(X/𝕋^{−})  (7)

which assigns to an element P of U_{⋊}(𝕋) the strictly full subcategory of C_{X/𝕋^{−}} generated by the image of P, is an isomorphism.

(b) For every P ∈ V_{⋊}(𝕋), the intersection P ∩ 𝕋 is an element of Spec_{⋊}(|𝕋|), and the map

V_{⋊}(𝕋) ⟶ Spec_{⋊}(|𝕋|), P ⟼ P ∩ 𝕋,  (8)

is an isomorphism.

Proof. The assertion follows from 7.2 and 6.4. Details are left to the reader.

7.5. Proposition. Let P_{1} and P_{2} be elements of Spec^{−}(X). Then either P_{2} ⊆ P_{1}, or P_{1}^{⊛} ∩ P_{1}^{⊥} ∩ P_{2} ≠ 0.

Proof. Replacing X by X/(P_{1} ∩ P_{2}), we assume that P_{1} ∩ P_{2} = 0. If P_{1} is nonzero, then (by 7.2(c)) |P_{1}| is local. Let M_{1} be a quasi-final object of P_{1}. The argument of 7.2(c) shows that P_{2} = ⟨M_{1}⟩. If P_{2} is nonzero, then, by the same reason, |P_{2}| is local and P_{1} = ⟨M_{2}⟩ for a quasi-final object M_{2} of P_{2}. Therefore, P_{1}^{𝔱} = [[M_{2}], ⟨M_{2}⟩] = [[M_{2}], P_{1}] is the smallest topologizing subcategory P^{𝔱} properly containing P_{1}. By 1.3.3, P_{1}^{𝔱} ∩ P_{1}^{⊥} is contained in P_{2}.

8. Spectra related with localizations and their canonical decompositions.

Spectra discussed in this section were first introduced in [R5]. Here we follow the general pattern dictated by [R6] (Section 2) and similar to those of the previous sections.

8.1. The thick spectra. Fix an abelian category C_{X}. The thick spectra, Spec_{𝔗𝔥}^{0}(X) and Spec_{𝔗𝔥}^{1}(X), of the ’space’ X are the spectra of the preorder (𝔗𝔥(X), ⊆), i.e. they are defined by

Spec_{𝔗𝔥}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔗𝔥(X)), i = 0, 1,

and there is a canonical morphism

Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X).  (1)

For any subcategory B of C_{X}, let B^{⋆} denote the intersection of all thick subcategories of C_{X} containing properly B, and let B̂ denote the union of all thick subcategories which do not contain B. The spectra Spec_{𝔗𝔥}^{1}(X) and Spec_{𝔗𝔥}^{0}(X) and the morphism (1) can be described as follows.

Objects of Spec_{𝔗𝔥}^{1}(X) are thick subcategories 𝕋 of C_{X} such that 𝕋^{⋆} ≠ 𝕋.

The spectrum Spec_{𝔗𝔥}^{0}(X) is formed by all thick subcategories Q such that the union Q̂ of all thick subcategories of C_{X} which do not contain Q is a thick subcategory.

The canonical morphism Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X) maps every object Q of Spec_{𝔗𝔥}^{0}(X) to the corresponding thick subcategory Q̂.

8.2. Proposition. (a) If T_{1}, T_{2} are thick subcategories, then T_{1} ⊆ T_{2} iff T̂_{1} ⊆ T̂_{2} (b) The canonical map

Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X), Q ⟼ Q̂,  (4)

is injective. In particular, the induced maps

Spec_{𝔗𝔥}^{0,1}(X) ⟶ Spec_{𝔗𝔥}^{1,1}(X) and Spec_{𝔗𝔥}^{0,0}(X) ⟶ Spec_{𝔗𝔥}^{1,0}(X).  (5)

are injective.

Proof. (a) The implication T_{1} ⊆ T_{2} ⇒ T̂_{1} ⊆ T̂_{2} holds for any pair subcategories T_{1}, T_{2}. If the category T_{2} is thick and T_{1} ⊈ T_{2}, then T_{2} ⊆ T̂_{1}. In particular, T̂_{1} ⊈ T̂_{2}, whence

the assertion (a).

(b) It follows from (a) that the map T ⟼ T̂ induces a monomorphism of preorders Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X).

8.3. The canonical decompositions of the thick spectra. We represent the thick spectrum Spec_{𝔗𝔥}^{1}(X) as the disjoint union of Spec_{𝔗𝔥}^{1,1}(X) and Spec_{𝔗𝔥}^{1,0}(X), where

Spec_{𝔗𝔥}^{1,1}(X) ={T ∈ 𝔗𝔥(X) | T^{⋆} ∩ T^{⊥} ≠ 0}
Spec_{𝔗𝔥}^{1,0}(X) ={T ∈ Spec_{𝔗𝔥}^{1}(X) | T^{⋆} ∩ T^{⊥} = 0}.

(1)

8.3.1. Proposition. A thick subcategory T from Spec_{𝔗𝔥}^{1}(X) belongs to Spec_{𝔗𝔥}^{1,1}(X) iff it is a Serre subcategory. Thus,

Spec_{𝔗𝔥}^{1,1}(X) ={P ∈ 𝔗𝔥(X) | P^{−} = P ⫋ P^{⋆}}
Spec_{𝔗𝔥}^{1,0}(X) ={P ∈ 𝔗𝔥(X) | P ⫋ P^{⋆} ⊆ P^{−}}.

(2)

Proof. The argument is similar to that of 3.2(i).

8.3.2. The decomposition of Spec_{𝔗𝔥}^{0}(X). The decomposition (1) induces, via the map Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X), a decomposition

Spec_{𝔗𝔥}^{0}(X) = Spec_{𝔗𝔥}^{0,1}(X) ∐ Spec_{𝔗𝔥}^{0,0}(X).  (3)

8.4. The spectra Spec_{𝔰}^{0}(X) and Spec_{𝔰}^{1}(X). Applying the general construction to

the inclusion functor 𝔖𝔢(X) ⟶^{J_{X}^{𝔰}} 𝔗𝔥(X), we obtain the spectra 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X), J_{X}^{𝔰}), i = 0, 1, together with the canonical monomorphism

𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X), J_{X}^{𝔰}) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X), J_{X}^{𝔰}).

We shall write Spec_{𝔰}^{i}(X) instead of 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X), J_{X}^{𝔰}). Consider the canonical commutative diagram

Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔰}^{1}(X)
↓  ↓
Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{1}(X)

(6)

8.4.1. Proposition. The diagram (6) induces the commutative diagram of isomorphisms

Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔰}^{1}(X)
≀↓  ↓≀
Spec_{𝔗𝔥}^{0,1}(X) ⟶ Spec_{𝔗𝔥}^{1,1}(X)

(7)

whose vertical arrows are isomorphisms.

Proof. It follows from the general definitions that Spec_{𝔰}^{1}(X) is naturally identified with the preorder of all Serre subcategories P of C_{X} such that P ≠ P^{⋆}. By 8.3.1, this preorder coincides with Spec_{𝔗𝔥}^{1,1}. The rest is a consequence of the definition of Spec_{𝔰}^{0}(X). Details are left to the reader.

Our next objective is to describe the component Spec_{𝔗𝔥}^{0,1}(X) = Spec_{𝔰}^{0}(X) of the spectrum Spec_{𝔗𝔥}^{0}(X).

8.4.2. Spec_{𝔰}(X). We denote by Spec_{𝔰}(X) the class of all nonzero objects L of C_{X} such that [L_{1}]_{•} = [L]_{•} for any nonzero subobject L_{1} of L. Equivalently, Spec_{𝔰}(X) consists of objects L such that if there exists a nonzero morphism N ⟶ L, then L ∈ Ob[M]_{•}. We consider Spec_{𝔰}(X) together with the preorder ≽ defined by L ≽ M iff [M]_{•} ⊆ [L]_{•}.

8.4.3. Proposition. The spectrum Spec_{𝔰}^{0}(X) coincides with {[L]_{•} | L ∈ Spec_{𝔰}(X)}.

Proof. (a) If a nonzero object L is such that [L_{1}]_{•} = [L]_{•} for any nonzero subobject L_{1} of L, then [L]_{•} ∈ Spec_{𝔗𝔥}^{0,1}(X), i.e. L̂ ∈ Spec_{𝔗𝔥}^{1,1}(X).

(a1) In fact, if 𝕋 is a thick subcategory, then L ∉ Ob𝕋 iff L belongs to 𝕋^{⊥}. Because if there is a nonzero arrow N ⟶^{f} L with N ∈ Ob𝕋, then 𝕋 ⊇ [N]_{•} ⊇ [im(f)]_{•} = [L]_{•}. This shows that L is right orthogonal to L̂.

(a2) If L ∈ ObL̂^{−}, then, since L ≠ 0, it has a nonzero subobject, M, which belongs to L̂. But, this is impossible, because, according to (b1), L is right orthogonal to L̂. Thus, L ∉ ObL̂^{−} which means that L̂ = L̂^{−}, i.e. L̂ is a Serre subcategory.

(b) Notice that Spec_{𝔗𝔥}^{0,1}(X) consists precisely of all T ∈ Spec_{𝔗𝔥}^{0}(X) such that the intersection T ∩ T̂^{⊥} is nonzero.

In fact, if T̂ is a Serre subcategory, then every nonzero object of T − T̂ has a nonzero T̂-torsion free subquotient. The latter is an object of T ∩ T̂^{⊥}.

(c) If T ∈ Spec_{𝔗𝔥}^{0}(X), then T = [L]_{•} for every object L of T − T̂.

Since L ∉ ObT̂ and T̂ is a thick subcategory, we have the inclusion T̂ ⊆ ˆ[L]_{•} = L̂. The inverse inclusion follows from the inclusion [L]_{•} ⊆ T and 8.2(a).

(d) If L is a nonzero object of T ∩ T̂^{⊥}, then every nonzero object, L_{1}, of L belongs to T ∩ T̂^{⊥}. By (c), [L]_{•} = T = [L_{1}]_{•}. This shows that L belongs to Spec_{𝔰}(X), hence, by (a), T = [L]_{•} is an element of Spec_{𝔰}^{0}(X).

8.5. The spectrum Spec_{⋊}^{𝔰}(X). Starting with the functor

𝔗_{≤}(X) ⟶^{𝔏_{⋊}} 𝔖𝔢(X), B ⟼ B_{⋊},

(see 6.2) and the embedding Spec_{𝔰}^{1}(X) ⟶ 𝔖𝔢(X), we come, mimiking 6.2 and 6.3, to the preorder Spec_{⋊}^{𝔰}(X) of all full nonzero subcategories B of the category C_{X} such that B = (B_{⋊})^{⊥} ⋂(B_{⋊})^{⋆} and B_{⋊} ∈ Spec_{𝔗𝔥}^{1,1}(X).

8.5.1. Proposition. The map B ⟼ B_{⋊} induces an isomorphism

Spec_{⋊}^{𝔰}(X) ⟶^{∼} Spec_{𝔗𝔥}^{1,1}(X)  (8)

with the inverse map P ⟼ P_{⋆} = P^{⋆} ∩ P^{⊥}.

Proof. The argument is similar to the proof of 6.4. Details are left to the reader.

8.6. Connections with the other spectra. The inclusion maps

Spec_{𝔱}^{1,1}(X) ↪ Spec^{−}(X) ↪ Spec_{𝔗𝔥}^{1,1}(X)

are a part of the commutative diagram

Spec_{𝔱}^{1,1}(X) ⟶ Spec^{−}(X) ⟶ Spec_{𝔗𝔥}^{1,1}(X) ⟵^{∼} Spec_{𝔰}^{1}(X)
≀↑  ≀↑  ↑≀  ↑
Spec(X) ⟶^{α} Spec_{⋊}(X) ⟶^{β} Spec_{⋊}^{𝔰}(X) ⟵ Spec_{𝔰}^{0}(X)

(9)

in which four arrows are isomorphisms and the remaining arrows are injective. Here the map Spec(X) ⟶^{α} Spec_{⋊}(X) assigns to each element P of Spec(X) the intersection

(⟨P⟩ • P • ⟨P⟩) ∩ ⟨P⟩^{⊥}, and Spec_{⋊}(X) ⟶^{β} Spec_{𝔗𝔥}^{0,1}(X) transforms any element P_{⊛} = P^{⊛} ∩ P^{⊥} of Spec_{⋊}(X) (– the image of P ∈ Spec^{−}(X)) into the subcategory P^{⋆} ∩ P^{⊥}. The vertical isomorphisms are defined by resp.

P ⟼ [P^{𝔱} ∩ P^{⊥}], P ⟼ P_{⊛} = P^{⊛} ∩ P^{⊥} and P ⟼ P_{⋆} = P^{⋆} ∩ P^{⊥}.

8.7. Spectra defined by the preorder of Serre subcategories.

Let ℌ be the preorder 𝔖𝔢(X) of all Serre subcategories of the category C_{X}. Thus, we have two spectra and an embedding:

𝔖𝔭𝔢𝔠^{0}(𝔖𝔢(X)) ⟶ 𝔖𝔭𝔢𝔠^{1}(𝔖𝔢(X)).

We change the notations setting Spec_{𝔖𝔢}^{i}(X) = 𝔖𝔭𝔢𝔠^{i}(𝔖𝔢(X)), i = 0, 1.

It follows from the definitions that objects of Spec_{𝔖𝔢}^{1}(X) are all Serre subcategories P such that the intersection P^{𝔰} of all Serre subcategories of C_{X} properly containing P does not coincide with P. The spectrum Spec_{𝔖𝔢}^{0}(X) is formed by all Serre subcategories Q such that the union ⟨Q⟩_{𝔰} of all Serre subcategories of C_{X} which do not contain Q is a Serre subcategory: ⟨Q⟩_{𝔰} = ⟨Q⟩_{𝔰}^{−}.

8.7.1. Proposition. There are natural injective morphisms of preorders

Spec_{𝔰}^{i}(X) ⟶ Spec_{𝔖𝔢}^{i}(X), i = 0, 1,

such that the diagram

Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖𝔢}^{0}(X)
↓  ↓
Spec_{𝔰}^{1}(X) ⟶ Spec_{𝔖𝔢}^{1}(X)

(10)

commutes.

Proof. The morphism Spec_{𝔰}^{1}(X) ⟶ Spec_{𝔖𝔢}^{1}(X) is the inclusion. The morphism Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖𝔢}^{0}(X) assigns to each object Q of Spec_{𝔰}^{0}(X) the associated Serre subcategory Q^{−}.

The diagram (10) can be combined with the diagram (9) above. In particular, we have a commutative diagram

Spec(X) ⟶ Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖𝔢}^{0}(X)
≀↓  ↓  ↓
Spec_{𝔱}^{1,1}(X) ⟶ Spec_{𝔰}^{1}(X) ⟶ Spec_{𝔖𝔢}^{1}(X)

(11)

whose lower horizontal arrows are embeddings. The diagram (11) can be, in turn, extended to a commutative diagram including other spectra of this work. This exercise is left to the reader. If C_{X} is a category with Gabriel-Krull dimension, then it follows from (the argument of) [R5, 8.7.1] that Spec^{−}(X) = Spec_{𝔰}^{1}(X) = Spec_{𝔖𝔢}^{1}(X).

II. Local properties of spectra.

9. Spectra and (co)covers. Local properties of the spectra with respect to finite (co)covers.

Fix an abelian category C_{X}. We call a set {T_{i} | i ∈ J} of thick subcategories of C_{X} a cocover of X, or a thick cocover of X if ⋂_{i∈J} T_{i} = 0. The latter condition means that the

corresponding family of exact localizations {C_{X} ⟶^{u_{i}^{∗}} C_{X}/T_{i} | i ∈ J} is conservative (i.e. it reflects isomorphisms), i.e. the morphisms {X/T_{i} ⟶^{u_{i}} X | i ∈ J} form a cover of X.

9.1. Proposition. Let a set {T_{i} | i ∈ J} of thick subcategories of an abelian category C_{X} be a cocover, i.e. ⋂_{i∈J} T_{i} = 0. Then for every P ∈ Spec_{𝔱}^{1,1}(X), there exists an i ∈ J

such that T_{i} ⊆ P.

Proof. Let P ∈ Spec_{𝔱}^{1,1}(X), and let Q be the image of P in Spec(X); that is P = Q̂. It follows that T_{i} ⊈ P = Q̂ iff Q ⊆ T_{i}. Since ⋂_{i∈J} T_{i} = 0, there are i ∈ J such that Q ⊈ T_{i}, or, equivalently, T_{i} ⊆ Q̂ = P.

9.2. Corollary. For any thick cocover {T_{i} | i ∈ J} of X, there is an embedding Spec_{𝔱}^{1,1}(X) ↪ ⋃_{i∈J} Spec_{𝔱}^{1,1}(X/T_{i}). Here Spec_{𝔱}^{1,1}(X) and Spec_{𝔱}^{1,1}(X/T_{i}) are identified with their images in Spec_{𝔗𝔥}^{1}(X).

Proof. It follows from 9.1 that Spec_{𝔱}^{1,1}(X) = ⋃_{i∈J} U_{𝔱}^{1,1}(T_{i}), where U_{𝔱}^{1,1}(T_{i}) = {P ∈ Spec_{𝔱}^{1,1}(X) | T_{i} ⊆ P}. By A2.2.1.1, for each i ∈ J, there is a natural embedding U_{𝔱}^{1,1}(T_{i}) ⟶ Spec_{𝔱}^{1,1}(X/T_{i}), hence the assertion.

9.3. Proposition. Let a set {T_{i} | i ∈ J} of thick subcategories of an abelian category C_{X} be a cocover. Then for every Q ∈ Spec_{𝔗𝔥}^{0}(X), there exists i ∈ J such that Q ⊈ T_{i}.

If J is finite, then for every P ∈ Spec_{𝔗𝔥}^{1}(X), there exists i ∈ J such that T_{i} ⊆ P. Proof. The first assertion is true by a trivial reason: elements of Spec_{𝔗𝔥}^{0}(X) are

nonzero subcategories and ⋂_{i∈J} T_{i} = 0.

Let P ∈ Spec_{𝔗𝔥}^{1}(X). Notice that T_{i} ⊈ P iff the thick subcategory T_{i} ⊔ P spanned by T_{i} and P contains P properly, hence it contains P^{⋆}. Therefore, if T_{i} ⊈ P for all i ∈ J, then P^{⋆} ⊆ ⋂_{i∈J}(T_{i} ⊔ P). Since J is finite, it follows from A1.4.1 that

P^{⋆} ⊆ ⋂_{i∈J}(T_{i} ⊔ P) = ( ⋂_{i∈J} T_{i}) ⊔ P = 0 ⊔ P = P,

which is impossible because P^{⋆} ≠ P.

9.4. Corollary. If {T_{i} | i ∈ J} is a finite thick cocover of X, then

Spec_{𝔗𝔥}^{1}(X) = ⋃_{i∈J} Spec_{𝔗𝔥}^{1}(X/T_{i}).  (1)

Here Spec_{𝔗𝔥}^{1}(X/T_{i}) is identified with its image in Spec_{𝔗𝔥}^{1}(X).

Proof. The fact follows from 9.3.

9.5. Proposition. Let {T_{i} | i ∈ J} be a finite thick cocover of X. Then

Spec^{1}(X) = ⋃_{i∈J} Spec^{1}(X/T_{i}).  (2)

If C_{X} has the property (sup), then

Spec^{−}(X) = ⋃_{i∈J} Spec^{−}(X/T_{i}^{−}),

Spec_{𝔖𝔢}^{1}(X) = ⋃_{i∈J} Spec_{𝔖𝔢}^{1}(X/T_{i}^{−}).

(3)

Proof. (a) The equality (2) follows from 9.4, because Spec^{1}(X) ⊆ Spec_{𝔗𝔥}^{1}(X).

(b) By A1.4.1, the equality ⋂_{i∈J} T_{i} = 0 implies that ⋂_{i∈J} T_{i}^{−} = 0, i.e. the set {T_{i}^{−} | i ∈

J} of Serre subcategories form a cocover. Therefore, the inclusions

Spec_{𝔖𝔢}^{1}(X) ⊆ ⋃_{i∈J} Spec_{𝔖𝔢}^{1}(X/T_{i}^{−}) and Spec^{1}(X) ⊆ ⋃_{i∈J} Spec^{1}(X/T_{i})

follows from 9.3 and the inclusions Spec^{1}(X) ⊆ Spec_{𝔖𝔢}^{1}(X) ⊆ Spec_{𝔗𝔥}^{1}(X).

(b1) If T is a coreflective thick subcategory of C_{X}, then a thick subcategory S containing T is a Serre subcategory iff S/T is a Serre subcategory.

Let u^{∗} denote the localization functor C_{X} ⟶ C_{X}/T . Let M be an object of S^{−} − T , and L̃ a nonzero subquotient of the image u ∗ (M) of M in C_{X}/T^{−}. By a standard argument, L̃ is isomorphic to u ∗ (L), where L is a subquotient of M. Since T^{−} is a coreflective subcategory (thanks to the property (sup)), we can choose L to be T^{−}-torsion free. Since M ∈ ObS^{−}, the object L, being a nonzero subquotient of M, has a nonzero subobject L_{1} from S which is T^{−}-torsion free. Therefore, u^{∗}(L_{1}) is a nonzero subobject of L̃ which belongs to S/T^{−}. This shows that u_{i}^{∗}(M) is an object of (S/T^{−})^{−}. But, (S/T^{−})^{−} = S/T^{−} by hypothesis, hence M belongs to S.

(b2) Since C_{X} has the property (sup), every Serre subcategory of C_{X} is coreflective, in particular, T_{i}^{−}, i ∈ J, are coreflective subcategories. The assertion follows now from (b1) above: Spec_{𝔖𝔢}^{1}(X/T_{i}^{−}) realized as a subset of Spec_{𝔗𝔥}^{1}(X) consists of Serre subcategories, hence is contained in Spec_{𝔖𝔢}^{1}(X). Similarly with Spec^{−}(X/T_{i}^{−}) realized as a subset of Spec^{1}(X).

9.6. Local property of Spec_{𝔱}^{1,1}(X).

9.6.1. Proposition. Let {T_{i} | i ∈ J} be a finite set of thick subcategories of the category C_{X} such that ⋂_{i∈J} T_{i} = 0. The following conditions on a thick subcategory P of

C_{X} are equivalent:

(a) P ∈ Spec_{𝔱}^{1,1}(X),

(b) P ∈ Spec_{𝔖𝔢}^{1}(X) and P/T_{i} ∈ Spec_{𝔱}^{1,1}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P. If the category C_{X} has the property (sup), then the conditions (a) and (b) are equiv-

alent to the condition (c) P ∈ Spec_{𝔗𝔥}^{1}(X) and P/T_{i} ∈ Spec_{𝔱}^{1,1}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P.

Proof. (a) ⇒ (b). Let P ∈ Spec_{𝔱}^{1,1}(X), and let T be a thick subcategory of C_{X} contained in P. Then (T • P^{𝔱} • T )/T is the smallest topologizing subcategory of C_{X}/T properly containing P/T , and the localization functor C_{X} ⟶^{u^{∗}} C_{X}/T maps nonzero objects of P_{𝔱} = P^{𝔱} ∩ P^{⊥} to nonzero objects of (P/T )_{𝔱} = (P/T )^{𝔱} ⋂(P/T )^{⊥}.

(b) ⇒ (a). Let u_{i}^{∗} denote the localization functor C_{X} ⟶ C_{X}/T_{i}. Set J_{P} = {j ∈ J | T_{j} ⊆ P}. For every i ∈ J_{P}, we denote by Q̃_{i} the intersection u_{i}^{∗^{−1}}((P/T_{i})^{𝔱}) ⋂ P^{⊥} and by Q_{i} the topologizing subcategory [Q̃_{i}] spanned by Q̃_{i}. By assumption, Q̃_{i} ≠ 0 for each i ∈ J_{P}, hence Q_{i} ⊈ P. The latter implies that, for every j ∈ J_{P}, the topologizing subcategory [u_{j}^{∗}(Q_{i} • P)] contains (P/T_{j})^{𝔱}, or, equivalently, u_{j}^{∗^{−1}}((P/T_{i})^{𝔱}) ⊆ T_{j} • Q_{i} • P. Therefore,

Q̃_{j} = u_{j}^{∗^{−1}}((P/T_{j})^{𝔱}) ⋂ P^{⊥} ⊆ (T_{j} • Q_{i} • P) ⋂ P^{⊥} = (T_{j} • Q_{i}) ⋂ P^{⊥} ⊆ T_{j} • Q_{i}

which implies the inclusion Q_{j} ⊆ T_{j} • Q_{i} for every (i, j) ∈ J_{P} × J_{P}, hence

Q_{j} ⊆ ⋂_{i∈J_{P}}(T_{j} • Q_{i}) = T_{j} • ( ⋂_{i∈J_{P}} Q_{i}).

Here the equality is due to the finiteness of J_{P}.

It follows from the inclusion Q_{j} ⊆ T_{j} • ( ⋂_{i∈J_{P}} Q_{i}) that ⋂_{i∈J_{P}} Q_{i} ≠ 0, because other-

wise Q_{j} ⊆ T_{j} • 0 = T_{i}, which is impossible, since T_{j} ⊆ P and Q_{j} ⊈ P.

There are two cases: J = J_{P} and J ≠ J_{P}. Consider each of them.

(i) Let J_{P} = J. We set Q = ⋂_{i∈J_{P}} Q_{i} and claim that Q is an element of Spec(X)

corresponding to P, that is P = ⟨Q⟩.

In fact, let S is a topologizing subcategory of C_{X} which is not contained in P. Then P is properly contained in S • P and, therefore, u_{i}^{∗^{−1}}((P/T_{i})^{𝔱}) ⊆ T_{i} • S • P for each i ∈ J. This implies that u_{i}^{∗^{−1}}((P/T_{i})^{𝔱}) ⋂ P^{⊥} ⊆ T_{i} • S • P ⋂ P^{⊥} ⊆ T_{i} • S. Therefore,

Q̃ = ⋂_{i∈J} u_{i}^{∗^{−1}}((P/T_{i})^{𝔱}) ⋂ P^{⊥} ⊆ ⋂_{i∈J}(T_{i} • S) = ( ⋂_{i∈J} T_{i}) • S = 0 • S = S,

which implies that Q = [Q̃] ⊆ S.

(ii) Consider now the second case: J_{P} ≠ J, i.e. J^{P} = J − J_{P} is non-empty. This case can be reduced to the first case as follows.

1) Set C_{V_{P}} = ⋂_{j∈J^{P}} T_{j}. Notice that C_{V_{P}} ⊈ P.

In fact, if i ∈ J^{P} = J −J_{P}, then T_{i} ⊈ P. Therefore, for every j ∈ J_{P}, the topologizing subcategory [u_{j}^{∗}(T_{i} • P)] contains (P/T_{j})^{𝔱}, or, equivalently, u_{j}^{∗^{−1}}((P/T_{j})^{𝔱}) ⊆ T_{j} • T_{i} • P, which implies that Q̃_{j} = u_{j}^{∗^{−1}}((P/T_{j})^{𝔱}) ⋂ P^{⊥} ⊆ (T_{j} • T_{i}) ⋂ P^{⊥}. Thanks to the finiteness of J^{P}, we obtain:

Q̃_{j} ⊆ ( ⋂_{i∈J^{P}}(T_{j} • T_{i})) ⋂ P^{⊥} = (T_{j} • ( ⋂_{i∈J^{P}} T_{i})) ⋂ P^{⊥}.  (4)

The inclusion ⋂_{i∈J^{P}} T_{i} ⊆ P implies (together with (4)) that Q̃_{j} ⊆ T_{j} ⊆ P, which is

impossible. So that ⋂_{i∈J^{P}} T_{i} ⊈ P.

2) Since C_{V_{P}} ⊈ P, the intersection P_{0} = C_{V_{P}} ⋂ P is an element of Spec_{𝔖𝔢}^{1}(V_{P}). Notice that {T_{i} ∩ C_{V_{P}} = T̃_{i} | i ∈ J_{P}} is a cocover of V_{P}, i.e. ⋂_{j∈J_{P}} T̃_{j} = 0. It remains to

notice that P_{0}/T̃_{j} ∈ Spec_{𝔱}^{1,1}(V_{P}/T̃_{j}) for each j ∈ J_{P}.

In fact, the localization functor C_{X} ⟶ C_{X}/T_{j} induces an equivalence of C_{V_{P}} /T̃_{j} and the topologizing subcategory (T_{j} • C_{V_{P}} • T_{j})/T_{j} of C_{X}/T_{j}. The subcategory P_{0}/T̃_{j} of C_{V_{P}} /T̃_{j} is the preimage of the intersection of the P/T_{j} ∈ Spec_{𝔱}^{1,1}(X/T_{j}) with the topologizing subcategory (T_{j} • C_{V_{P}} • T_{j})/T_{j}, hence it belongs to Spec_{𝔱}^{1,1}(V_{P}/T̃_{j}).

3) Thus, the ’space’ V_{P}, the cocover {T̃_{i} | i ∈ J_{P}}, and the point P_{0} = P ⋂ C_{V_{P}} of the spectrum Spec_{𝔖𝔢}^{1}(V_{P}) satisfy the conditions (b) with all T̃_{i} being subcategories of P_{0}. By 2) above, P_{0} belongs to the spectrum Spec_{𝔱}^{1,1}(V_{P}), and P_{0} = ⟨Q̃_{0}⟩_{V_{P}} = ⟨Q_{0}⟩_{V_{P}} , where Q_{0} is the smallest topologizing subcategory of (C_{V_{P}}, hence) C_{X} containing Q̃_{0}. Therefore, Q_{0} is a point of the spectrum Spec_{𝔱}^{0}(X) and ⟨Q_{0}⟩_{X} = P.

Obviously, (b) ⇒ (c) without additional conditions on the category C_{X}. Suppose now that C_{X} has the property (sup).

(c) ⇒ (b). It follows from (c) that P ∈ Spec_{𝔰}^{1}(X), i.e. P is a Serre subcategory. In fact, by A1.4.1, the equality ⋂_{i∈J} T_{i} = 0 implies that ⋂_{i∈J} T_{i}^{−} = 0. In other words,

{T_{i}^{−} | i ∈ J} is a finite cocover, which implies, by the local property of Spec_{𝔗𝔥}^{1}(X) that T_{i}^{−} ⊆ P for some i ∈ J. Notice that if T_{i}^{−} ⊆ P, then P/T_{i}^{−} belongs to Spec_{𝔱}^{1,1}(X/T_{i}^{−}). This follows from the fact that P/T_{i} is, by the condition (c), an element of Spec_{𝔱}^{1,1}(X/T_{i}), and the spectrum Spec_{𝔱}^{1,1} is functorial with respect to localizations (see the proof of (a) ⇒ (b) above), in particular, with respect to C_{X}/T_{i} ⟶ C_{X}/T_{i}^{−}.

Therefore, P/T_{i}^{−} is a Serre subcategory of the quotient category C_{X}/T_{i}^{−}. Thanks to the property (sup), the Serre subcategory T_{i}^{−} is coreflective. By the argument 9.5(b1), this implies that P is a Serre subcategory of C_{X}.

9.6.2. Note. Proposition 9.6.1 is a stronger statement than [R4, 6.3] in all respects. The equivalence (a) ⇔ (b) is essentially the assertion of [R4, 6.3], but the argument presented here is valid for arbitrary abelian categories, while the proof of [R4, 6.3] used the property (sup). The equivalence (a) and (b) to (c) (when C_{X} has the property (sup)) is a new observation (which could of be made in [R4]).

9.7. Spec_{𝔱}^{1,1}(X) and Spec^{−}(X). Let 𝔘 = {U_{i} ⟶^{u_{i}} X | i ∈ J} be a set of morphisms

such that their inverse image functors C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}} are exact localizations. We call 𝔘 a cover if the family {u_{i}^{∗} | i ∈ J} of inverse image functors is conservative, i.e. it reflects isomorphisms. We set

Spec_{℘}^{1,1}(𝔘) = {P ∈ Spec^{−}(X) | if Ker(u_{i}^{∗}) ⊆ P, then P/Ker(u_{i}^{∗}) ∈ Spec_{𝔱}^{1,1}(U_{i})}.

There is a canonical embedding Spec_{𝔱}^{1,1}(X) ⟶ Spec_{℘}^{1,1}(𝔘). By 9.6.1, if the cover 𝔘 is finite then this embedding is an isomorphism.

Recall that the associated points of the object M in Spec(X) are elements Q of Spec(X) such that M has a nonzero subobject from Q ∩ Q̂^{⊥}.

9.7.1. Proposition. Let C_{X} be an abelian category with property (sup). Let 𝔘 = {U_{i} ⟶^{u_{i}} X | i ∈ J} be a finite cover of the ’space’ X such that all morphisms U_{ij} = U_{i} ∩ U_{j} ⟶^{u_{ij}} U_{i} are continuous. Let P_{i} ∈ Spec_{𝔱}^{1,1}(U_{i}), and let L_{i} be an object of Spec(U_{i}) such that P_{i} = ⟨L_{i}⟩.

The following conditions are equivalent:

(a) P = u_{i}^{∗^{−1}}(P_{i}) belongs to Spec_{𝔱}^{1,1}(X); i.e. P = ⟨M⟩ for some object M of Spec(X);

(b) for any j ∈ J such that u_{ij}^{∗}(L_{i}) ≠ 0, the object u_{_{ji}∗}u_{ij}^{∗}(L_{i}) of C_{U_{j}} has an associated point; i.e. it has a subobject L_{j} which belongs to Spec(U_{j});

(c) P/Ker(u_{j}^{∗}) = P_{j} belongs to Spec_{𝔱}^{1,1}(U_{j}) for all j such that Ker(u_{j}^{∗}) ⊆ P.

Proof. (a)⇒(c) follows from 3.2(ii) and the functoriality of Spec (hence Spec_{𝔱}^{1,1}) with respect to localizations.

(c)⇒(a) follows from 9.6.1.

(b)⇒(c). Suppose that Ker(u_{j}^{∗}) ⊆ P, or, equivalently, u_{ij}^{∗}(L_{i}) ≠ 0. Then P_{j} =

P/Ker(u_{j}^{∗}) is a point of Spec^{1}(U_{j}). Let U_{i} ⟵^{u_{ij}} U_{i} ∩ U_{j} = U_{ij} ⟶^{u_{ji}} U_{j} be the canonical embeddings. Since L_{i} ∈ Spec(U_{i}) and u_{ij}^{∗}(L_{i}) ≠ 0, it follows that u_{ij}^{∗}(L_{i}) ∈ Spec(U_{ij}).

Let L_{j} be a nonzero subobject of u_{_{ji}∗}u_{ij}^{∗}(L_{i}), and L_{j} ∈ Spec(U_{j}). Then u_{ji}^{∗}(L_{j}) is a nonzero subobject of u_{ij}^{∗}(L_{i}). Therefore, since u_{ij}^{∗}(L_{i}) belongs to Spec(U_{ij}), the objects u_{ji}^{∗}(L_{j}) and u_{ij}^{∗}(L_{i}) are equivalent. Notice that, it follows from P_{i} = ⟨L_{i}⟩ that P_{i}/Ker(u_{ij}^{∗}) = ⟨u_{ij}^{∗}(L_{i})⟩. But, P_{i}/Ker(u_{ij}^{∗}) = P_{j}/Ker(u_{ji}^{∗}) = P/Ker(u_{ji}^{∗}u_{j}^{∗}) and, by the argument above, ⟨u_{ij}^{∗}(L_{i})⟩ = ⟨u_{ji}^{∗}(L_{j})⟩. Together with the fact that L_{j} is an object of Spec(U_{j}), this shows that P_{j} = ⟨L_{j}⟩.

(a)⇒(b). Suppose that P = u_{i}^{∗^{−1}}(P_{i}) belongs to Spec_{𝔱}^{1,1}(X); i.e. P = ⟨M⟩ for some object M of Spec(X). Let L̃_{i} be a P-torsion free object of C_{X} such that u_{i}^{∗}(L̃_{i}) ≃ L_{i}. The relation u_{i}^{∗}(M) ≻ L_{i} means that there exists a diagram M^{⊕n} ⟵^{𝔧} K ⟶^{𝔢} L_{1} ⟶^{g} L̃_{i} in which 𝔢 is an epimorphism, the arrows 𝔧 and g are nonzero monomorphisms; in particular, M ≻ L_{1}. Notice that L_{1} ≻ M, i.e. M and L_{1} are equivalent. In fact, u_{i}^{∗}(L_{1}) is a nonzero subobject of L_{i}. Since the latter belongs to Spec(U_{i}), they are equivalent. Therefore, u_{i}^{∗}(L_{1}) is equivalent to u_{i}^{∗}(M). The relation u_{i}^{∗}(L_{1}) ≻ u_{i}^{∗}(M) is expressed by a diagram

L_{1}^{⊕m} ⟵^{𝔧′} K̃ ⟶^{𝔢′} M_{1} ⟶^{h} M in which 𝔢′ is an epimorphism and 𝔧′ and h are nonzero monomorphisms. Since M ∈ Spec(X), M_{1} is equivalent to M, hence the relation L_{1} ≻ M_{1} which is explicit in the diagram above, implies that L_{1} ≻ M. Thus L_{1} ∈ ObSpec(X).

By the functoriality of Spec with respect to exact localizations, u_{j}^{∗}(L_{1}) = L_{j} belongs to Spec(U_{j}). Since L_{1} is P-torsion free, the adjunction arrow L_{j} = u_{j}^{∗}(L_{1}) ⟶ u_{ji∗}u_{ji}^{∗}u_{j}^{∗}(L_{1}) is a monomorphism. On the other hand,

u_{ji∗}u_{ji}^{∗}u_{j}^{∗}(L_{1}) ≃ u_{ji∗}u_{ij}^{∗}u_{i}^{∗}(L_{1}) ⟶ u_{ji∗}u_{ij}^{∗}u_{i}^{∗}(L̃_{i}) ≃ u_{ji∗}u_{ij}^{∗}(L_{i}),  (5)

where the arrow in the middle is the image of the monomorphism L_{1} ⟶ L̃_{i}. Since all functors in the diagram (5) are left exact, this arrow is a monomorphism. Altogether gives the desired monomorphism L_{j} ⟶ u_{ji∗}u_{ij}^{∗}(L_{i}).

9.7.2. Example. Let C_{X} be the category of quasi-coherent sheaves on a quasicompact quasi-separated scheme X = (X , O_{X}). Let {U_{i} ↪ X | i ∈ J} be an affine cover and C_{U_{i}} the category of quasi-coherent sheaves on (U_{i}, O_{U_{i}}). Then all morphisms U_{i} ∩ U_{j} ⟶ U_{i} are continuous and the equivalent conditions (a), (b), (c) hold for every point P_{i} ∈ Spec_{𝔱}^{1,1}(U_{i}). This reflects the fact that Spec(U_{i}) is naturally identified with U_{i} and is an open subset of the spectrum Spec(X) ≃ Spec_{𝔱}^{1,1}(X). It follows from 9.6.1 that Spec_{𝔱}^{1,1}(X) = ⋃_{i∈J} Spec_{𝔱}^{1,1}(U_{i}). So, Proposition 9.7.1 becomes trivial in the case of

commutative schemes. It is non-trivial and meaningful in the case of noncommutative schemes, even in the case of D-schemes.

9.7.3. Example: simple holonomic D-modules. Let C_{X} be the category of holonomic D-modules on a smooth quasi-compact scheme X = (X , O_{X}). Let {U_{i} ↪ X | i ∈ J} be an affine cover of X, and let C_{U_{i}} be the category of holonomic D-modules on the affine subscheme (U_{i}, O_{U_{i}}). Then all morphisms U_{i} ∩ U_{j} ⟶ U_{i} are continuous and the equivalent conditions (a), (b), (c) hold for every simple object L_{i} of C_{U_{i}}. The latter is due to the fact that direct and inverse image functors of open immersions preserve holonomicity. Thanks to the fact that all holonomic D-modules are of finite length, the ’space’ X (i.e. the category C_{X}) has the Gabriel-Krull dimension zero, hence elements of Spec(X) are in a bijective correspondence with isomorphism classes of holonomic simple objects. Therefore, it follows from 9.7.1 and 9.6.1 that Spec_{𝔱}^{1,1}(X) = ⋃_{i∈J} Spec_{𝔱}^{1,1}(U_{i}). Thus, the problem of

the description of simple holonomic modules on a smooth quasi-compact scheme is local: it can be reduced to the affine case.

Consider, for instance, the cover of the flag variety G/B of a reductive algebraic connected group G over ℂ (or any other algebraicly closed field of zero characteristic) by translations U_{w}, w ∈ W, of the big Schubert cell (here, as usual, W denotes the Weyl group of G. Then for any w ∈ W, the category C_{U_{w}} is equivalent to the category A_{n} − mod of left modules over the Weyl algebra A_{n}. So the problem of a classification of holonomic D-modules on G/B is reduced to the problem of classification of holonomic modules on the affine n-dimensional space 𝔸^{n}, that is holonomic A_{n}-modules.

9.7.4. Proposition. Let C_{X} have the property (sup), and let {T_{i} | i ∈ J} be a finite set of Serre subcategories of the category C_{X} such that ⋂_{i∈J} T_{i} = 0. Suppose that

Spec_{𝔱}^{1,1}(X/T_{i}) = Spec^{−}(X/T_{i}) for all i ∈ J. Then Spec_{𝔱}^{1,1}(X) = Spec^{−}(X); i.e. the map

Spec(X) ⟶ Spec^{−}(X), P ⟼ ⟨P⟩,

is an isomorphism.

Proof. By 9.6.1, Spec_{𝔱}^{1,1}(X) coincides with

{P ∈ Spec^{−}(X) | P/T_{i} ∈ Spec_{𝔱}^{1,1}(X/T_{i}) if T_{i} ⊆ P}.

By 9.5, Spec^{−}(X) = ⋃_{i∈J} Spec^{−}(X/T_{i}). In particular,

Spec^{−}(X) = {P ∈ Spec^{−}(X) | P/T_{i} ∈ Spec^{−}(X/T_{i}) if T_{i} ⊆ P}.

Hence the assertion.

9.7.5. Example. Let C_{X} be the category of quasi-coherent sheaves on a smooth quasi-compact scheme X = (X , O_{X}) of dimension n. Let C_{𝔄} be the category of D-modules on X and C_{𝔄} ⟶^{u_{∗}} C_{X} the pull-back functor corresponding to the embedding of the structure sheaf O_{X} into the sheaf D_{X} of differential operators on X. Let 𝔘 = {U_{i} ⟶^{u_{i}} X | i ∈ J} be an affine finite cover of X such that each U_{i} is isomorphic to the affine space 𝔸^{n}. Then Spec^{−}(𝔄) = ⋃_{i∈J} Spec^{−}(|A_{n} − mod|), where A_{n} is the n-th Weyl algebra.

9.7.5.1. The case of a curve. Suppose n = 1, i.e. X is a curve. Then Spec_{𝔱}^{1,1}(X) and Spec^{−}(X) coincide.

In fact, the equality holds when C_{X} is the category of left modules over the first Weyl algebra A_{1}. This follows from the fact that A_{1} has Gabriel-Krull dimension one, hence Spec^{−}(X) consists of closed points and one generic point.

In the general case, the equality follows from this and 9.7.4.

9.7.5.2. Corollary. Let C_{𝔄} be the category U(sl_{2}) − mod_{0} of U(sl_{2})-modules with the trivial central character. Then Spec^{−}(𝔄) = Spec(𝔄).

Proof. The fact is true if the base field is of positive characteristic, because then U(sl_{2}) is finite-dimensional over its center.

Suppose that the base field is of characteristic zero. The category C_{𝔄} = U(sl_{2})−mod_{0} is equivalent to the category D(ℙ^{1}) of D-modules on the one-dimensional projective space. The assertion follows from 9.7.4.

9.8. Reconstruction of quasi-compact schemes.

9.8.1. Geometric center of a ’space’. Let C_{X} be an abelian category. Fix a topology τ on Spec(X). The map Õ_{X,τ} which assigns to every open subset W of Spec(X) the center of the quotient category C_{X}/S_{W}, where S_{W} = ⋂_{Q∈W} Q̂, is a presheaf

on (Spec(X), τ). Recall that the center of the category C_{Y} is the (commutative) ring of endomorphisms of its identical functor. If C_{Y} is a category of left modules over a ring R, then the center of C_{Y} is naturally isomorphic to the center of R.

We denote by O_{X,τ} the associated sheaf. The ringed space ((Spec(X), τ), O_{X,τ}) is called the geometric center of the ’space’ X. If τ is the Zariski topology, then we write simply (Spec(X), O_{X}) and call this ringed space the Zariski geometric center of X. Recall that open sets in Zariski topology are sets of the form U(𝕋) = {Q ∈ Spec(X) | Q ⊈ 𝕋}, where 𝕋 is an arbitrary bireflective topologizing subcategory of C_{X}. Recall that ’bireflective’ means that the inclusion functor 𝕋 ↪ C_{X} has right and left adjoints.

Let X = (X , O_{X}) be a ringed topological space and U = (U, O_{U}) ⟶^{𝔧} (X , O_{X}) an open immersion. Then the morphism 𝔧 has an exact inverse image functor 𝔧^{∗} and a fully faithful direct image functor 𝔧_{∗}. This implies that Ker(𝔧^{∗}) is a Serre subcategory of the category O_{X} − Mod of sheaves of O_{X}-modules and the unique functor

O_{X} − Mod/Ker(𝔧^{∗}) ⟶ O_{U} − Mod

induced by 𝔧^{∗} is an equivalence of categories [Gab, III.5].

Suppose now that X = (X , O_{X}) is a scheme and Qcoh_{X} the category of quasi-coherent sheaves on X. The inverse image functor 𝔧^{∗} of the immersion 𝔧 maps quasi-coherent sheaves to quasi-coherent sheaves. Let u^{∗} denote the functor Qcoh_{X} ⟶ Qcoh_{U} induced by 𝔧^{∗}. The funtor u^{∗}, being the composition of the exact full embedding of Qcoh_{X} into O_{X} − Mod and the exact functor 𝔧^{∗}, is exact; hence it is represented as the composition of an exact localization Qcoh_{X} ⟶ Qcoh_{X}/Ker(u^{∗}) and a uniquely defined exact functor Qcoh_{X}/Ker(u^{∗}) ⟶ Qcoh_{U}. If the direct image functor 𝔧_{∗} of the immersion 𝔧 maps quasi-coherent sheaves to quasi-coherent sheaves, then it induces a fully faithful functor Qcoh_{U} ⟶^{u_{∗}} Qcoh_{X} which is right adjoint to u^{∗}. In particular, the canonical functor Qcoh_{X}/Ker(u^{∗}) ⟶ Qcoh_{U} is an equivalence of categories.

The reconstruction of a scheme X from the category Qcoh_{X} of quasi-coherent sheaves on X is based on the existence of an affine cover {U_{i} ⟶^{u_{i}} X | i ∈ J} such that the canonical functors Qcoh_{X}/Ker(u_{i}^{∗}) ⟶ Qcoh_{U_{i}}, i ∈ J, are category equivalences. It follows from the discussion above (or from [GZ, I.2.5.2]) that this is garanteed if the inverse image functor Qcoh_{X} ⟶^{u_{i}} Qcoh_{U_{i}} has a fully faithful right adjoint.

9.8.2. Proposition. Let X = (X , O_{X}) be a quasi-compact scheme such that there exists an affine cover {U_{i} ⟶^{u_{i}} X | i ∈ J} such that the canonical functors

Qcoh_{X}/Ker(u_{i}^{∗}) ⟶ Qcoh_{U_{i}}, i ∈ J,

are category equivalences. Then

(a) The scheme X is isomorphic to the Zariski geometric center ((Spec(X), τ_{𝔷}), O_{X}) of the ’space’ X, where C_{X} = Qcoh_{X}, τ_{𝔷} is the Zariski topology on X and O_{X} is the sheaf of commutative rings defined in 9.6.4.

(b) For every open immersion U ⟶^{u} X such that Qcoh_{X}/Ker(u^{∗}) ⟶ Qcoh_{U} is a category equivalence, Ker(u^{∗}) is a Serre subcategory of Qcoh_{X}. In particular, Ker(u_{i}^{∗}) is a Serre subcategory for all i ∈ J.

Proof. (a) Set C_{U_{i}} = Qcoh_{U_{i}} and T_{i} = Ker(u_{i}^{∗}). Since X is quasi-compact, we can and will assume that J is finite. The condition that {U_{i} ⟶^{u_{i}} X | i ∈ J} is a cover means precisely that ⋂_{i∈J} T_{i} = 0.

(a1) Let x be a point of the underlying space X of the scheme X. Let ℑx̄ be the defining ideal of the closure x̄ of the point x and Mx̄ the quotient sheaf O/ℑx̄. Set J_{x} = {i ∈ J | Mx̄ ∉ ObT_{i}}. We claim that Q_{x} = [Mx̄] is an element of Spec(X).

For every i ∈ J_{x}, the object u_{i}^{∗}(Mx̄) of the category C_{U_{i}} belongs to Spec(U_{i}) and Spec(U_{i}), because C_{U_{i}} is (equivalent to) the category of modules over a ring. Therefore, [u_{i}^{∗}(Q_{x})] = [u_{i}^{∗}(Mx̄)] is an element of Spec(U_{i}). By 9.6.1, Q_{x} ∈ Spec(X).

(a2) Conversely, let Q be an element of Spec(X). Let Q ⊈ T_{i}, or, equivalently, T_{i} ⊆ Q̂. By the functoriality of Spec(X) under exact localizations, [u_{i}^{∗}(Q)] is an element of Spec(U_{i}). Since U_{i} is affine, Spec(U_{i}) is in bijective correspondence with the underlying space U_{i} of the subscheme U_{i} = (U_{i}, O_{U_{i}}); in particular, to the element [u_{i}^{∗}(Q)] there corresponds a point x of U_{i} which we identify with its image in X. Notice that the point x does not depend on the choice of i ∈ J_{Q} = {j ∈ J | T_{j} ⊆ Q̂}. This gives a map Spec(X) ⟶ X which is inverse to the map X ⟶ Spec(X) constructed in (a1) above. These maps are homeomorphisms in the case if the cover consists of one element, i.e. the scheme is affine. The general case follows from the commutative diagrams

Spec(U_{i}) ⟶^{∼} U_{i}
↓ ↓
Spec(X) ⟶ X ,  i ∈ J.

(5)

in which vertical arrows are open immersions and the upper horizontal arrow is a homeomorphism; hence the lower horizontal arrow is a homeomorphism.

(a3) The diagrams (5) extend to the commutative diagrams of ringed spaces

(Spec(U_{i}), O_{U_{i}}) ⟶^{∼} (U_{i}, O_{U_{i}})
↓ ↓
(Spec(X), O_{X}) ⟶ (X , O_{X})  i ∈ J.

(6)

in which Spec_{𝔠}^{0}(U_{i}), O_{U_{i}}) and (Spec_{𝔠}^{0}(X), O_{X}) are Zariski geometric centra of resp. U_{i} and X, vertical arrows are open immersions and upper horizontal arrow is an isomorphism. Therefore the lower horizontal arrow is an isomorphism.

(b) Let x be a point of U_{i}, which we identify with its image in the underlying space X of the scheme X. Let Q_{x} denote the corresponding element of Spec(X). It follows that Q_{x} ⊈ T_{i}, or, equivalently, T_{i} ⊆ Q̂_{x}. Thus, T_{i} ⊆ ⋂_{x∈U_{i}} Q̂_{x}. We claim that T_{i} = ⋂_{x∈U_{i}} Q̂_{x}. In

fact, if M is an object of C_{X} − T_{i}, then u_{i}^{∗}(M) ≠ 0. Since the category C_{U_{i}} is equivalent to the category of modules over a ring, every nonzero object of C_{U_{i}} has a non-empty support. In particular, there is a point x ∈ U_{i} which belongs to the support of u_{i}^{∗}(M). The latter means that u_{i}^{∗}(M) ∉ Ob(Q̂_{x}/T_{i}), or, what is the same, M ∉ ObQ̂_{x}.

Since each Q̂_{x} is a Serre subcategory and the intersection of any family of Serre subcategories is a Serre subcategory, T_{i} is a Serre subcategory.

9.8.3. Remarks. (i) A comment to the assertion 9.8.2(b): if C_{Y} is a Grothendieck category and T is a Serre subcategory than the localization functor C_{Y} ⟶ C_{Y} /T has a right adjoint.

(ii) The quasi-compactness of the scheme in 9.8.2 is an essential requirement. If the scheme is not quasi-compact, the spectrum Spec(X) might be not sufficiently big to reconstruct the underlying space. This was observed by O. Gabber who produced an example of a scheme which is not isomorphic to the ringed space (Spec(X), O_{X}) associated with its category of quasi-coherent sheaves. He also mentioned that his example does not work for slightly different version of a spectrum. This version of a spectrum is nothing else, but the spectrum related to coreflective topologizing subcategories studied in the next section.

10. The spectra related to coreflective topologizing subcategories.

10.1. The spectra Spec_{𝔠}^{0}(X) and Spec_{𝔠}^{1}(X). These are spectra obtained via an application of the construction of the relative spectra (see 2.5) to the inclusion functor 𝔗𝔥_{𝔠}(X) ↪ 𝔗_{𝔠}(X) from the preorder 𝔗𝔥_{𝔠}(X) of coreflective thick subcategories to the preorder 𝔗_{𝔠}(X) of coreflective topologizing subcategories of the category C_{X}. Objects of the spectrum Spec_{𝔠}^{1}(X) are coreflective thick subcategories P such that the intersection P^{𝔠} of all coreflective topologizing subcategories properly containing P contains P properly too. The spectrum Spec_{𝔠}^{0}(X) is formed by coreflective topologizing subcategories Q of C_{X} such that the union ⟨Q⟩ of all coreflective subcategories of C_{X} which do not contain Q is a coreflective thick subcategory. The canonical injective morphism Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔠}^{1}(X) maps Q to ⟨Q⟩. It follows from the definition of ⟨Q⟩ that every coreflective topologizing subcategory properly containing ⟨Q⟩ contains Q, hence the smallest coreflective subcategory [⟨Q⟩, Q]_{𝔠} containing Q and ⟨Q⟩ coincides with ⟨Q⟩^{𝔠}. The injectivity of the map Q ⟼ ⟨Q⟩ is a consequence of the following fact which is going to be used more than once.

10.1.1. Lemma. If Q_{1}, Q_{2} are elements of T_{𝔠}(X), then Q_{1} ⊆ Q_{2} iff Q̂_{1} ⊆ Q̂_{2}.

Proof. The argument is the same as in 4.1.

10.1.2. Proposition. Let C_{X} be an abelian category with the property (sup).

(a) The canonical morphism

Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔠}^{1}(X), Q ⟼ ⟨Q⟩,  (1)

is an isomorphism.

(b) There are natural injective morphisms

Spec(X) ⟶ Spec_{𝔠}^{0}(X) and Spec_{𝔱}^{1,1}(X) ⟶ Spec_{𝔠}^{1}(X)  (2)

such that the diagram

Spec(X) ⟶ Spec_{𝔠}^{0}(X)
≀↓  ↓≀
Spec_{𝔱}^{1,1}(X) ⟶ Spec_{𝔠}^{1}(X)

commutes.

(c) If C_{X} has enough objects of finite type, then the morphisms (2) are isomorphisms.

Proof. (a) For every P ∈ Spec_{𝔠}^{1}(X), the intersection P^{𝔠} ∩ P^{⊥} is nonzero, because P is a Serre subcategory. The claim is that the coreflective topologizing subcategory [P_{∗}]_{𝔠} spanned by the subcategory P_{∗} = P^{𝔠} ∩ P^{⊥} belongs to the spectrum Spec_{𝔠}^{0}(X) and ⟨[P_{∗}]_{𝔠}⟩ = P. The map

Spec_{𝔠}^{1}(X) ⟶ Spec_{𝔠}^{0}(X), P ⟼ ⟨[P_{∗}]_{𝔠}⟩,  (3)

is inverse to the map (1) above.

(a1) Notice that ⟨[P_{∗}]_{𝔠}⟩ = ⟨P_{∗}⟩ because a coreflective topologizing subcategory does not contain [P_{∗}]_{𝔠} iff it does not contain P_{∗}. Therefore, our claim is that ⟨P_{∗}⟩ = P.

(a2) If T is a coreflective topologizing subcategory of C_{X} which is not contained in P, then P_{∗} = P^{𝔠} ∩ P^{⊥} ⊆ T .

In fact, if T ⊈ P, then the coreflective topologizing subcategory T • P contains P properly, hence it contains P^{𝔠}. Notice that every P-torsion free object of T • P belongs to T . In particular, P_{∗} ⊆ T .

(a3) It follows from (a1) that if T ∈ 𝔗_{𝔠}(X) is such that P_{∗} ⊈ T , then T ⊆ P. This means that ⟨P_{∗}⟩ ⊆ P. On the other hand, P_{∗} ⊈ P and P is a Serre subcategory; in particular, it is coreflective and topologizing; hence the inverse inclusion, P ⊆ ⟨P_{∗}⟩.

(a4) Since the map (1) is injective and has a right inverse, P ⟼ [P_{∗}]_{𝔠}, it is bijective.

(b) Thanks to the property (sup), a thick subcategory of the category C_{X} is coreflective iff it is a Serre subcategory. In particular, since elements of Spec_{𝔱}^{1,1}(X) are Serre subcategories, Spec_{𝔱}^{1,1}(X) ⊆ 𝔗𝔥_{𝔠}(X). A Serre subcategory P belongs to Spec_{𝔱}^{1,1}(X) iff the intersection P^{𝔱} of topologizing subcategories properly containing P contains P properly. Therefore, P^{𝔠} contains P properly. The map Spec_{𝔱}^{1,1}(X) ⟶ Spec_{𝔠}^{1}(X) is the inclusion.

Let Q ∈ Spec(X), and let [Q]_{𝔠} be the smallest coreflective topologizing subcategory of C_{X} containing Q. Clearly [Q]_{𝔠} ⊈ Q̂, hence Q̂ ⊆ ⟨[Q]_{𝔠}⟩. On the other hand, if T is a coreflective topologizing subcategory of C_{X} such that [Q]_{𝔠} ⊈ T , then Q ⊈ T , or, equivalently, T ⊆ Q̂. This shows the inverse inclusion, ⟨[Q]_{𝔠}⟩ ⊆ Q̂. The equality ⟨[Q]_{𝔠}⟩ = Q̂, together with the fact that Q̂ is a Serre subcategory, shows that [Q]_{𝔠} ∈ Spec_{𝔠}^{0}(X) for every Q ∈ Spec(X). The map Spec(X) ⟶ Spec_{𝔠}^{0}(X) assigns to every element Q of Spec(X) the coreflective topologizing subcategory [Q]_{𝔠} spanned by Q.

(c) Let Q be an object of Spec_{𝔠}^{0}(X). One can see that ⟨Q⟩ = ⟨M⟩ for every object M of Q − ⟨Q⟩: the inclusion ⟨M⟩ ⊆ ⟨Q⟩ is due to the fact that M ∈ ObQ and the inverse inclusion holds because M ∉ Ob⟨Q⟩. This implies that Q = [M]_{𝔠} for any object M of Q − ⟨Q⟩. In particular, Q = [M]_{𝔠} for any nonzero object M of Q ∩ ⟨Q⟩^{⊥}.

Suppose that the category C_{X} has enough objects of finite type, i.e. every nonzero object of C_{X} has a nonzero subobject of finite type. In particular, any nonzero object of the subcategory Q ∩ ⟨Q⟩^{⊥} has a nonzero subobject L. Since L belongs to Q ∩ ⟨Q⟩^{⊥} and is nonzero, [L]_{𝔠} = Q. We claim that L is an object of Spec(X), which implies that the topologizing subcategory [L] generated by L belongs to Spec(X).

In fact, let N be a nonzero subobject of L. Then [N]_{𝔠} = Q = [L]_{𝔠}. In particular, L is an object of the coreflective topologizing subcategory of C_{X} spanned by N. Objects of the subcategory [N]_{𝔠} are precisely objects of the category C_{X} which are supremums of their subobjects from [N]. In particular, L is a supremum of its subobjects from [N]. Since subobjects of L which belong to the topologizing subcategory [N] form a filtered system and L is of finite type, it follows that L is isomorphic to one of its subobjects from [N], i.e. L ∈ Ob[N]. This proves that L belongs to Spec(X).

10.2. The spectra Spec_{𝔠}^{i}(X) and Spec_{𝔖𝔢}^{i}(X). Recall that Spec_{𝔖𝔢}^{i}(X), i = 0, 1, are the spectra of the preorder 𝔖𝔢(X) of Serre subcategories of C_{X} (see 8.7): points of Spec_{𝔖𝔢}^{1}(X) are Serre subcategories P of C_{X} such that the intersection P^{𝔰} of all Serre subcategories of C_{X} properly containing P does not coincide with P; and Spec_{𝔖𝔢}^{0}(X) is formed by Serre subcategories Q such that the union ⟨Q⟩_{𝔰} of Serre subcategories not containing Q is a Serre subcategory.

10.2.1. Proposition. There are natural injective morphisms

Spec_{𝔠}^{i}(X) ⟶ Spec_{𝔖𝔢}^{i}(X) i = 0, 1,

such that the diagram

Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔖𝔢}^{0}(X)
≀↓  ↓
Spec_{𝔠}^{1}(X) ⟶ Spec_{𝔖𝔢}^{1}(X)

(1)

commutes.

Proof. The spectrum Spec_{𝔠}^{1}(X) is contained in the spectrum Spec_{𝔖𝔢}^{1}(X), because if P^{𝔠} ≠ P, then (P^{𝔠})^{−} is the smallest Serre subcategory properly containing P, hence P belongs to Spec_{𝔖𝔢}^{1}(X). The map Spec_{𝔠}^{0}(X) ⟶ Spec_{𝔖𝔢}^{0}(X) assigns to every Q ∈ Spec_{𝔠}^{0}(X) the Serre subcategory Q^{−} spanned by Q (cf. 1.5).

10.2.2. Extended spectra. Extended spectra are obtained via adjoining to the original spectra a marked point. In the case of Spec_{𝔠}^{0}(X) and Spec_{𝔖𝔢}^{0}(X), this marked point might be realized the zero subcategory. In the case of the spectra Spec_{𝔠}^{1}(X) and Spec_{𝔖𝔢}^{1}(X), the marked point is realized as the empty subcategory which reflects the equalities ⟨0⟩ = ∅ = ⟨0⟩_{𝔰}.

Morphisms between the original spectra determine morphisms between the corresponding extended spectra mapping marked points to marked points. In particular, the commutative diagram (1) extends to the commutative diagram

Spec_{𝔠}^{0}(X)_{⋆} ⟶ Spec_{𝔖𝔢}^{0}(X)_{⋆}
≀↓  ↓
Spec_{𝔠}^{1}(X)_{⋆} ⟶ Spec_{𝔖𝔢}^{1}(X)_{⋆}

(1_{⋆})

10.2.2.1 Proposition. There are natural maps

Spec_{𝔖𝔢}^{i}(X)_{⋆} ⟶ Spec_{𝔠}^{i}(X)_{⋆}, i = 0, 1,

such that the diagram

Spec_{𝔠}^{0}(X)_{⋆} ⟶ Spec_{𝔖𝔢}^{0}(X)_{⋆} ⟶ Spec_{𝔠}^{0}(X)_{⋆}
≀↓  ↓  ↓≀
Spec_{𝔠}^{1}(X)_{⋆} ⟶ Spec_{𝔖𝔢}^{1}(X)_{⋆} ⟶ Spec_{𝔠}^{1}(X)_{⋆}

(2)

commutes and the compositions of its horizontal arrows are identical morphisms.

Proof. The map Spec_{𝔖𝔢}^{1}(X)_{⋆} ⟶ Spec_{𝔠}^{1}(X)_{⋆} assings to each P ∈ Spec_{𝔖𝔢}^{1}(X) the Serre subcategory ⟨P^{𝔠} ∩ P^{⊥}⟩ if P^{𝔠} ≠ P (i.e. if P ∈ Spec_{𝔠}^{1}(X)) and the marked point, ⟨0⟩ = ∅, if P^{𝔠} = P. The map Spec_{𝔖𝔢}^{1}(X)_{⋆} ⟶ Spec_{𝔠}^{1}(X)_{⋆} is uniquely defined by the commutativity of the right square in the diagram (2). It follows from the (argument of) 10.1.2 that the composition of the lower horizontal arrows in (2) is the identical map. The similar fact for the upper horizontal arrows is a consequence of this and the commutativity of the diagram (2).

10.3. Functorial properties of Spec_{𝔠}^{1}(X) and Spec_{𝔠}^{0}(X). For any topologizing subcategory 𝕋 of the category C_{X}, we set

U_{𝔠}^{1}(𝕋) = {P ∈ Spec_{𝔠}^{1}(X) | 𝕋 ⊆ P}
V_{𝔠}^{1}(𝕋) = Spec_{𝔠}^{1}(X) − U_{𝔠}^{1}(𝕋) = {P ∈ Spec_{𝔠}^{1}(X) | 𝕋 ⊈ P}.
U_{𝔠}^{0}(𝕋) = {Q ∈ Spec_{𝔠}^{0}(X) | Q ⊈ [𝕋]_{𝔠}} and
V_{𝔠}^{0}(𝕋) = Spec_{𝔠}^{0}(X) − U_{𝔠}^{0}(𝕋) = {Q ∈ Spec_{𝔠}^{0}(X) | Q ⊆ [𝕋]_{𝔠}}

(1)

10.3.1. Proposition. Let 𝕋 be a topologizing subcategory of the category C_{X}.

(a) The isomorphism

Spec_{𝔠}^{0}(X) ⟶^{∼} Spec_{𝔠}^{1}(X), Q ⟼ ⟨Q⟩,

(cf. 10.1.2) induces isomorphisms

U_{𝔠}^{0}(𝕋) ⟶^{∼} U_{𝔠}^{1}(𝕋) and V_{𝔠}^{0}(𝕋) ⟶^{∼} V_{𝔠}^{1}(𝕋).  (2)

(b) There are equalities V_{𝔠}^{i}(𝕋) = V_{𝔠}^{i}(𝕋^{−}) and U_{𝔠}^{i}(𝕋) = U_{𝔠}^{i}(𝕋^{−}), i = 0, 1.

(c) For every P ∈ V_{𝔠}^{1}(𝕋), the intersection P ∩ 𝕋 is an element of Spec^{1}(|𝕋|), where C_{|𝕋|} = 𝕋, and the map

V_{𝔠}^{1}(𝕋) ⟶ Spec_{𝔠}^{1}(|𝕋|), P ⟼ P ∩ 𝕋,  (3)

is an isomorphism. The inverse map is given by P̃ ⟼ P̃_{+} (see 7.1).

Similarly, the map Q ⟼ Q ∩ 𝕋 induces an isomorphism V_{𝔠}^{0}(𝕋) ⟶ Spec_{𝔠}^{0}(|𝕋|). (c^{bis}) If 𝕋 is coreflective, then the inverse isomorphism, Spec_{𝔠}^{0}(|𝕋|) ⟶ V_{𝔠}^{0}(𝕋), is

given by the identical map.

(d) The maps P ⟼ P/𝕋^{−} and Q ⟼ (𝕋^{−}•Q•𝕋^{−})/𝕋^{−} define injective morphisms resp.

U_{𝔠}^{1}(𝕋) ⟶ Spec_{𝔠}^{1}(X/𝕋^{−}) and U_{𝔠}^{0}(𝕋) ⟶ Spec_{𝔠}^{0}(X/𝕋^{−})  (4)

such that the diagram

U_{𝔠}^{0}(𝕋) ⟶ Spec_{𝔠}^{0}(X/𝕋^{−})
≀↓  ↓≀
U_{𝔠}^{1}(𝕋) ⟶ Spec_{𝔠}^{1}(X/𝕋^{−})

(5)

commutes.

Proof. (a) Let Q ∈ U_{𝔠}^{0}(𝕋), i.e. Q ∈ Spec_{𝔠}^{0}(X) and Q ⊈ [T ]_{𝔠}. This means precisely that ⟨Q⟩ ∈ Spec_{𝔠}^{0}(X) and T ⊆ ⟨Q⟩, i.e. Q is an element of U_{𝔠}^{0}(𝕋) iff ⟨Q⟩ is an element of U_{𝔠}^{1}(𝕋). The isomorphism V_{𝔠}^{0}(𝕋) ⟶^{∼} V_{𝔠}^{1}(𝕋) follows from this and the isomorphism Spec_{𝔠}^{0}(X) ⟶^{∼} Spec_{𝔠}^{1}(X).

(b) The equalities V_{𝔠}^{1}(𝕋) = V_{𝔠}^{1}(𝕋^{−}) and U_{𝔠}^{1}(𝕋) = U_{𝔠}^{1}(𝕋^{−}) follow from an observation that elements of Spec_{𝔠}^{1}(X) are Serre subcategories, and if P is a Serre subcategory, then 𝕋 ⊆ P iff 𝕋^{−} ⊆ P. The other two equalities follow from these isomorphisms (2) above.

(c) Let P ∈ V_{𝔠}^{1}(𝕋), i.e. P ∈ Spec_{𝔠}^{1}(X) and 𝕋 ⊈ P. The latter implies that [𝕋]_{𝔠} • P is a coreflective topologizing subctegory of C_{X} properly containing P. Therefore, it contains P^{𝔠}, and we have:

P^{𝔠} ∩ P^{⊥} ⊆ ([𝕋]_{𝔠} • P) ∩ P^{⊥} = [𝕋]_{𝔠} ∩ P^{⊥} ⊆ [𝕋]_{𝔠}.

In particular, the intersection Q̃_{𝕋} = 𝕋 ∩ P^{𝔠} ∩ P^{⊥} is nonzero. This implies that ⟨Q̃_{𝕋}⟩ = P (see the argument 10.1.2(c)). Notice that if S is a coreflective topologizing subcategory of 𝕋, then S coincides with the intersection of 𝕋 with the smallest coreflective topologizing subcategory of C_{X} containing S. Therefore, the union ⟨Q̃_{𝕋}⟩_{𝕋} of coreflective topologizing subctegories of 𝕋 which do not contain Q̃_{𝕋} coincides with the intersection ⟨Q̃_{𝕋}⟩ ∩ 𝕋. Thus, ⟨Q̃_{𝕋}⟩_{𝕋} = P ∩ 𝕋; in particular, P ∩ 𝕋 ∈ Spec_{𝔠}^{1}(|𝕋|) and the corresponding element of Spec_{𝔠}^{0}(|𝕋|) is [P^{𝔠} ∩ P^{⊥}]_{𝔠} ∩ 𝕋. In other words, it is obtained from P by applying the composition of the isomorphism Spec_{𝔠}^{1}(X) ⟶^{∼} Spec_{𝔠}^{0}(X) and the intersection with 𝕋, i.e. the diagram

V_{𝔠}^{0}(𝕋) ⟶ Spec_{𝔠}^{0}(|𝕋|)
≀↓  ↓≀
V_{𝔠}^{1}(𝕋) ⟶ Spec_{𝔠}^{1}(|𝕋|)

(6)

whose horizontal arrows are given by P ⟼ P ∩𝕋, commutes. It follows from the argument above that the map V_{𝔠}^{0}(𝕋) ⟶ Spec_{𝔠}^{0}(|𝕋|), Q ⟼ Q ∩ 𝕋, is an isomorphism with the inverse map which assigns to every element Q′ of Spec_{𝔠}^{0}(|𝕋|) the coreflective topologizing subcategory [Q′]_{𝔠} in C_{X} spanned by Q′. Therefore the lower horizontal arrow in (6) is an isomorphism too.

(c^{bis}) If 𝕋 is coreflective, then every coreflective topologizing subcategory of 𝕋 is a coreflective topologizing subcategory of C_{X}, hence the isomorphism

Spec_{𝔠}^{0}(|𝕋|) ⟶^{∼} V_{𝔠}^{0}(𝕋), Q′ ⟼ [Q′]_{𝔠},

discussed above becomes an identical map.

(d) Let q^{∗} denote the localization functor C_{X} ⟶ C_{X}/𝕋^{−}.

If P belongs to U_{𝔠}^{1}(𝕋), i.e. 𝕋 ⊆ P ⊊ P^{𝔠}, then
(P/𝕋^{−})^{𝔠} = [q^{∗}(P^{𝔠})]_{𝔠} = (𝕋^{−} • P^{𝔠} • 𝕋^{−})/𝕋^{−} ⊋ P/𝕋^{−}.

This shows that P/𝕋^{−} is an element of Spec_{𝔠}^{1}(X/𝕋^{−}).

If Q is the image of the element P in U_{𝔠}^{0}(𝕋) (see the assertion (a) above), then the coreflective subcategory [q^{∗}(Q)]_{𝔠} = (𝕋^{−} • Q • 𝕋^{−})/𝕋^{−} is the image of P/𝕋^{−} in Spec_{𝔠}^{0}(X/𝕋^{−}). The commutativity of the diagram (5) follows from the definition of its arrows.

10.4. The local property of the spectrum Spec_{𝔠}^{1}(X).

10.4.1. Proposition. Let {T_{i} | i ∈ J} be a set of Serre subcategories of the category C_{X} such that ⋂_{i∈J} T_{i} = 0; and let u_{i}^{∗} denote the localization functor C_{X} ⟶ C_{X}/T_{i}.

1) The following conditions on P ∈ Spec_{𝔖𝔢}^{1}(X) are equivalent:

(a) P ∈ Spec_{𝔠}^{1}(X),

(b) ⋂_{i∈J_{P}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⊈ P, where J_{P} = {j ∈ J | T_{j} ⊆ P}, and if J^{P} = J −J_{P} ≠ ∅,

then ⋂_{j∈J^{P}} T_{j} ⊈ P.

2) The conditions (a) and (b) imply the condition (c) P/T_{i} ∈ Spec_{𝔠}^{1}(X/T_{i}) for each i ∈ J_{P}.

If J is finite, then the conditions (a) and (b) are equivalent to the condition (c).

Proof. 1) Since P is a Serre subcategory, the condition ⋂_{i∈J} T_{i}^{−} = 0 implies that

J_{P} = {i ∈ J | T_{i} ⊆ P} is not empty.

In fact, if T_{i} ⊈ P for all i ∈ J, then T_{i} • P ⊇ P^{𝔠}, hence P^{𝔠} ⊆ ⋂_{i∈J}(T_{i} • P). But, by

A1.2.1, ⋂_{i∈J}(T_{i} • P) = ( ⋂_{i∈J} T_{i}) • P = 0 • P = P, hence P^{𝔠} = P, which contradicts to the

assumption that P ∈ Spec_{𝔠}^{1}(X), i.e. P ⊊ P^{𝔠}.

2) Notice that if S is a Serre subcategory, and 𝕋 a subcategory of C_{X} closed under taking subquotients, then 𝕋 ⊈ S iff 𝕋 ∩ S^{⊥} ≠ 0, because an object M of C_{X} does not belong to S iff it has a nonzero S-torsion free subquotient.

In particular, the condition (b) above can be written as follows:

(b) ⋂_{i∈J_{P}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ≠ 0, and ( ⋂_{j∈J^{P}} T_{j}) ⋂ P^{⊥} ≠ 0, if J^{P} = J − J_{P} ≠ ∅.

(a) ⇒ (b). Let P ∈ Spec_{𝔠}^{1}(X), i.e. P ≠ P^{𝔠}. If i ∈ J_{P}, that is T_{i} ⊆ P, then u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) = T_{i} • P^{𝔠} • T_{i}. Since T_{i} ⊆ P, the intersection (T_{i} • P^{𝔠} • T_{i}) ⋂ P^{⊥} coincides with (T_{i} • P^{𝔠}) ⋂ P^{⊥}.

Therefore

\begin{matrix}⋂_{i∈J_{P}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} = ⋂_{i∈J_{P}}((T_{i} • P^{𝔠}) ⋂ P^{⊥}) = \\ ( ⋂_{i∈J_{P}}(T_{i} • P^{𝔠})) ⋂ P^{⊥} = (( ⋂_{i∈J_{P}} T_{i}) • P^{𝔠}) ⋂ P^{⊥} ⊇ P^{𝔠} ⋂ P^{⊥} ≠ 0. \end{matrix}  (1)

Here we used the equality ⋂_{i∈J_{P}}(T_{i} • P^{𝔠}) = ( ⋂_{i∈J_{P}} T_{i}) • P^{𝔠} which is due, by A1.2.1, to the

fact that the subcategory P^{𝔠} is coreflective.

Notice that if J_{P} = J, that is T_{i} ⊆ P for all j ∈ J, then ⋂_{i∈J_{P}} T_{i} = 0, hence the last in-

clusion in (1) can be replaced by the equality, i.e. the intersection ⋂_{i∈J_{P}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} coincides with P^{𝔠} ∩ P^{⊥}.

Suppose that J^{P} = J − J_{P} is non-empty. If j ∈ J^{P}, that is T_{j} is not contained in P, then T_{j} • P is a coreflective topologizing subcategory properly containing both T_{j} and P, hence properly containing P. Therefore P^{𝔠} ⊆ T_{j} • P for all j ∈ J^{P}, or P^{𝔠} ⊆ ⋂_{j∈J^{P}}(T_{j} • P).

Since P is a coreflective subcategory, ⋂_{j∈J^{P}}(T_{j} • P) = ( ⋂_{j∈J^{P}} T_{j}) • P (see A1.2.1). Thus,

P^{𝔠} ⊆ ( ⋂_{j∈J^{P}} T_{j}) • P, which implies (actually, is equivalent to) that ⋂_{j∈J^{P}} T_{j} ⊈ P.

(b) ⇒ (a). There are two cases: J_{P} = J and J_{P} ≠ J.

(i) We start with the first case; i.e. we assume that T_{i} ⊆ P for all i ∈ J. Set Q̃ = ⋂_{i∈J} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} and Q = [Q̃]_{𝔠} – the smallest coreflective topologizing subcategory

of C_{X} containing Q̃. We claim that Q belongs to Spec_{𝔠}^{0}(X) and ⟨Q⟩ = P. Since, by condition (b), Q ⊈ P, it suffices to show that ⟨Q⟩ = P.

Let S be a coreflective topologizing subcategory of C_{X} which is not contained in P. Then P is properly contained in S • P and, therefore, u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⊆ T_{i} • S • P for each i ∈ J. This implies that u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ⊆ T_{i} • S • P ⋂ P^{⊥} ⊆ T_{i} • S. Therefore,

Q̃ = ⋂_{i∈J} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ⊆ ⋂_{i∈J}(T_{i} • S) = ( ⋂_{i∈J} T_{i}) • S = 0 • S = S,

so that Q = [Q̃]_{𝔠} ⊆ S.

(ii) Consider now the second case: J_{P} ≠ J, i.e. J^{P} = J − J_{P} is non-empty. This case can be reduced to the first case as follows.

Set C_{V_{P}} = ⋂_{j∈J^{P}} T_{j}. Since ⋂_{j∈J^{P}} T_{j} ⊈ P, the intersection P_{0} = C_{V_{P}} ⋂ P is an element

of Spec_{𝔖𝔢}^{1}(V_{P}). Notice that {T_{i} ∩ C_{V_{P}} = T̃_{i} | i ∈ J_{P}} is a cocover of V_{P}, i.e. ⋂_{j∈J_{P}} T̃_{j} = 0.

It remains to show that the condition Q̃ = ⋂_{i∈J} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ≠ 0 implies the anal-

ogous condition for the object P_{0} = C_{V_{P}} ⋂ P of Spec_{𝔖𝔢}^{1}(V_{P}) and the cover {T̃_{i} | i ∈ J_{P}}; that is

Q̃_{0} = ⋂_{i∈J} u_{i}^{∗^{−1}}((P_{0}/T̃_{i})^{𝔠}) ⋂ P_{0}^{⊥} ≠ 0.

In fact, let u_{i}^{∗} denote the localization functor C_{V_{P}} ⟶ C_{V_{P}} /T̃_{i}. Then

\begin{matrix}u_{i}^{∗^{−1}}((P_{0}/T̃_{i})^{𝔠}) = u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ C_{V_{P}} , and \\ u_{i}^{∗^{−1}}((P_{0}/T̃_{i})^{𝔠}) ⋂ P_{0}^{⊥} = u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ C_{V_{P}} ⋂ P^{⊥}. \end{matrix}

Therefore,

⋂_{i∈J} u_{i}^{∗^{−1}}((P_{0}/T̃_{i})^{𝔠}) ⋂ P_{0}^{⊥} = ⋂_{i∈J} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ⋂ C_{V_{P}} = Q̃ ⋂ C_{V_{P}} .  (2)

On the other hand, for every i ∈ J_{P}, there is an inclusion T_{i}•C_{V_{P}} •P ⊇ u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}), because C_{V_{P}} ⊈ P, which implies the inclusion u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ⊆ T_{i} • C_{V_{P}} . Taking the intersection, we obtain the inclusion

Q̃ ⊆ ⋂_{i∈J_{P}}(T_{i} • C_{V_{P}}) = ( ⋂_{i∈J_{P}} T_{i}) • C_{V_{P}} .  (3)

Notice that Q̃ is a full subcategory of C_{X} closed under taking subobjects. In particular, the equality Q̃ ⋂ C_{V_{P}} = 0 means precisely that every object of Q̃ is C_{V_{P}} -torsion free. The latter fact together with the inclusion Q̃ ⊆ ( ⋂_{i∈J_{P}} T_{i}) • C_{V_{P}} (see (3)) implies that

Q̃ ⊆ ⋂_{i∈J_{P}} T_{i} ⊆ P, which contradicts to the fact that Q̃ is a nonzero subcategory of P^{⊥}.

This contradiction shows that Q̃ ⋂ C_{V_{P}} ≠ 0, hence, by (2),

Q̃_{0} = ⋂_{i∈J} u_{i}^{∗^{−1}}((P_{0}/T̃_{i})^{𝔠}) ⋂ P_{0}^{⊥} ≠ 0.

(iii) Thus, the ’space’ V_{P}, the cocover {T̃_{i} | i ∈ J_{P}}, and the point P_{0} = P ⋂ C_{V_{P}} of the spectrum Spec_{𝔖𝔢}^{1}(V_{P}) satisfy the conditions (b) with all T̃_{i} being subcategories of P_{0}. By (i) above, P_{0} belongs to the spectrum Spec_{𝔠}^{1}(V_{P}), and P_{0} = ⟨Q̃_{0}⟩_{V_{P}} = ⟨Q_{0}⟩_{V_{P}} , where Q_{0} is the smallest coreflective topologizing subcategory of C_{V_{P}} containing Q̃_{0}. Therefore, [Q_{0}]_{𝔠} is a point of the spectrum Spec_{𝔠}^{0}(X) and ⟨Q_{0}⟩_{X} = P.

(b) ⇒ (c). The condition ⋂_{i∈J_{P}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} ≠ 0 implies that the intersection

u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} is nonzero for every i ∈ J_{P}. In particular, (P/T_{i})^{𝔠} does not coincide with P/T_{i}, which means that P/T belongs to the spectrum Spec_{𝔠}^{1}(X/T_{i}).

(c) ⇒ (b) (when J is finite). For every i ∈ J_{P}, let Q̃_{i} denote the intersection u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) ⋂ P^{⊥} and Q_{i} = [Q̃_{i}]_{𝔠} – the smallest coreflective topologizing subcategory of C_{X} containing Q̃_{i}. By assumption, Q̃_{i} ≠ 0 for each i ∈ J_{P}, hence Q_{i} ⊈ P. The latter implies that, for every j ∈ J_{P}, the coreflective topologizing subcategory spanned by u_{j}^{∗}(Q_{i} • P) contains (P/T_{j})^{𝔠}, or, equivalently, u_{j}^{∗^{−1}}((P/T_{i})^{𝔠}) ⊆ T_{j} • Q_{i} • P. Therefore,

Q̃_{j} = u_{j}^{∗^{−1}}((P/T_{j})^{𝔠}) ⋂ P^{⊥} ⊆ (T_{j} • Q_{i} • P) ⋂ P^{⊥} = (T_{j} • Q_{i}) ⋂ P^{⊥} ⊆ T_{j} • Q_{i}

which implies the inclusion Q_{j} ⊆ T_{j} • Q_{i} for every (i, j) ∈ J_{P} × J_{P}, hence the inclusion

Q_{j} ⊆ ⋂_{i∈J_{P}}(T_{j} • Q_{i}) = T_{j} • ( ⋂_{i∈J_{P}} Q_{i}).

Here the equality is due to the finiteness of J_{P}.

It follows from the inclusion Q_{j} ⊆ T_{j} • ( ⋂_{i∈J_{P}} Q_{i}) that ⋂_{i∈J_{P}} Q_{i} ≠ 0, because other-

wise Q_{j} ⊆ T_{j} • 0 = T_{i}, which is impossible, since T_{j} ⊆ P and Q_{j} ⊈ P.

If J = J_{P}, the condition (b) is fulfilled. If J ≠ J_{P}, we need to check that ⋂_{i∈J^{P}} T_{i} ⊈ P.

In fact, if i ∈ J^{P} = J − J_{P}, then T_{i} ⊈ P. Therefore, for every j ∈ J_{P}, the coreflective subcategory spanned by u_{j}^{∗}(T_{i} • P) contains (P/T_{j})^{𝔠}, or, equivalently, u_{j}^{∗^{−1}}((P/T_{j})^{𝔠}) ⊆ T_{j} • T_{i} • P, which implies that Q̃_{j} = u_{j}^{∗^{−1}}((P/T_{j})^{𝔠}) ⋂ P^{⊥} ⊆ (T_{j} • T_{i}) ⋂ P^{⊥}. Taking the intersection and using the finiteness of J^{P}, we obtain:

Q̃_{j} ⊆ ( ⋂_{i∈J^{P}}(T_{j} • T_{i})) ⋂ P^{⊥} = (T_{j} • ( ⋂_{i∈J^{P}} T_{i})) ⋂ P^{⊥}.  (4)

The inclusion ⋂_{i∈J^{P}} T_{i} ⊆ P implies (together with (4)) that Q̃_{j} ⊆ T_{j} ⊆ P, which is

impossible. So that ⋂_{i∈J^{P}} T_{i} ⊈ P.

10.4.2. Note. The reader had, probably, noticed that some parts of the proof of 10.4.1 are similar to some parts of the argument of 9.6.1. If one considers only the case of finite cocovers, one can follow the argument of 9.6.1 which is considerably shorter than the proof above.

10.4.3. Proposition. Let {T_{i} | i ∈ J} be a set of Serre subcategories of the category C_{X} such that ⋂_{i∈J} T_{i} = 0; and let u_{i}^{∗} denote the localization functor C_{X} ⟶ C_{X}/T_{i}.

1) The following conditions on a nonzero coreflective topologizing subcategory Q of C_{X} are equivalent:

(a) Q ∈ Spec_{𝔠}^{0}(X),

(b) [u_{i}^{∗}(Q)]_{𝔠} ∈ Spec_{𝔠}^{0}(X/T_{i}) for every i ∈ J such that Q ⊈ T_{i}.

Proof. The implication (a) ⇒ (b) follows from 10.3.1(d).

(b) ⇒ (a). Let J_{⟨Q⟩} denote the set of all i ∈ J such that Q ⊈ T_{i}, or, equivalently, T_{i} ⊆ ⟨Q⟩. Notice that J_{⟨Q⟩} is non-empty, because Q is nonzero and ⋂_{i∈J} T_{i} = 0.

Fix an i ∈ J_{⟨Q⟩} and set P_{i} = u_{i}^{∗^{−1}}(⟨u_{i}^{∗}(Q)⟩). It follows from the formula for P_{i} that Q ⊈ P_{i}. Notice that P_{i} = P_{j} for every j ∈ J_{⟨Q⟩}.

In fact, replacing C_{X} by C_{X′} = C_{X}/(P_{i} ∩ P_{j}) and P_{k} by P′_{k} = P_{k}/(P_{i} ∩ P_{j}), k = i, j, we can obtain that P′_{i} ∩ P′_{j} = 0. It follows from 10.3.1 that the condition (b) survives this operation. By 9.6.1, the image Q′ of Q in C_{X′} belongs to Spec_{𝔠}^{0}(X′). Therefore, P′_{i} = P′_{j}, which implies that P_{i} = P_{j}.

So, we write P instead of P_{i}. For every i ∈ J_{⟨Q⟩}, the subcategory (P/T_{i})^{𝔠} contains u_{i}^{∗}(Q), hence its preimage, u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}), contains Q. Since ⋂_{i∈J_{⟨Q⟩}} u_{i}^{∗^{−1}}((P/T_{i})^{𝔠}) contains Q, it is not contained in P. Similarly, if J^{⟨Q⟩} = J −J_{⟨Q⟩} is non-empty, then Q ⊆ ⋂_{i∈J^{⟨Q⟩}} T_{j},

hence Q ⊆ ⋂_{i∈J^{⟨Q⟩}} T_{j} ⊈ P. Thus, P satisfies the condition (b) of 10.4.1. Therefore, by

10.4.1, P ∈ Spec_{𝔠}^{1}(X). It remains to show that Q is an element of Spec_{𝔠}^{0}(X) corresponding to P.

It follows from the argument of 10.4.1 that the element of Spec_{𝔠}^{0}(X) corresponding to P is the coreflective topologizing subcategory [Q_{P}]_{𝔠} generated by Q_{P} = Q ∩ P^{⊥}. In particular, [Q_{P}]_{𝔠} ⊆ Q. Let M be an object of the subcategory Q. Since [Q_{P}]_{𝔠} is a coreflective subcategory of C_{X}, the object M has the biggest subobject M_{P} ↪ M with M_{P} ∈ Ob[Q_{P}]_{𝔠}. Consider the corresponding exact sequence

0 ⟶ M_{P} ⟶^{𝔧} M ⟶ N ⟶ 0.  (5)

For every i ∈ J_{⟨Q⟩}, the morphism u_{i}^{∗}(𝔧) is an isomorphism, because the images of [Q_{P}]_{𝔠} and Q in C_{X}/T_{i} coincide. This means that the object N in (5) belongs to T_{i} = Ker(u_{i}^{∗}) for each i ∈ J_{⟨Q⟩}. On the other hand, Q ⊆ T_{ν} for every ν ∈ J − J_{⟨Q⟩} (by definition of J_{⟨Q⟩}); in particular, N ∈ T_{ν} for all ν ∈ J − J_{⟨Q⟩}. Thus, N is an object of ⋂_{i∈J} T_{i} = 0, i.e.

N = 0, or, equivalently, the arrow M_{P} ⟶^{𝔧} M in (5) is an isomorphism. This proves the inverse inclusion, [Q_{P}]_{𝔠} ⊇ Q.

For every object M of C_{X}, we define the support of M in Spec_{𝔠}^{0}(X) by Supp_{𝔠}^{0}(M) = {Q ∈ Spec_{𝔠}^{0}(X) | Q ⊆ [M]_{𝔠}}.

10.5. Some consequences.

10.5.1. Proposition. Let {T_{i} | i ∈ J} be a finite set of Serre subcategories of an abelian category C_{X} such that ⋂_{i∈J} T_{i} = 0 and Spec_{𝔠}^{1}(X/T_{i}) = Spec_{𝔱}^{1,1}(X/T_{i}) for every

i ∈ J. Then Spec_{𝔠}^{1}(X) = Spec_{𝔱}^{1,1}(X). In particular, the canonical map

Spec(X) ⟶ Spec_{𝔠}^{0}(X), Q ⟼ [Q]_{𝔠},

is an isomorphism.

Proof. The inclusion Spec_{𝔱}^{1,1}(X) ⊆ Spec_{𝔠}^{1}(X) holds by 10.1.2(b). Let P ∈ Spec_{𝔠}^{1}(X). Then, by the implication (a) ⇒ (c) in 10.4.1, P/T_{i} ∈ Spec_{𝔠}^{1}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P. By hypothesis, Spec_{𝔠}^{1}(X/T_{i}) = Spec_{𝔱}^{1,1}(X/T_{i}) for all i ∈ J. Therefore, by the implication (b) ⇒ (a) in 9.6.1, P belongs to Spec_{𝔱}^{1,1}(X).

10.5.2. Corollary. Let {T_{i} | i ∈ J} be a finite set of Serre subcategories of an abelian category C_{X} such that ⋂_{i∈J} T_{i} = 0 and for every i ∈ J, the quotient category C_{X}/T_{i} has

enough objects of finite type. Then Spec_{𝔠}^{1}(X) = Spec_{𝔱}^{1,1}(X).

In particular, the canonical map Spec(X) ⟶ Spec_{𝔠}^{0}(X) is an isomorphism.

Proof. Since each quotient category C_{X}/T_{i} has enough objects of finite type, it follows from 10.1.2(c) that Spec_{𝔠}^{1}(X/T_{i}) = Spec_{𝔱}^{1,1}(X/T_{i}) for all i ∈ J. The assertion follows now from 10.5.1.

10.5.3. Affine and quasi-affine cocovers. A morphism X ⟶^{f} Y is called affine if it has a conservative (i.e. reflecting isomorphisms) direct image functor, C_{X} ⟶^{f_{∗}} C_{Y} (– a right adjoint to f^{∗}) which has, in turn, a right adjoint. We call a ’space’ X affine over a ring R, if there is an affine morphism X ⟶ Sp(R), where C_{Sp(R)} = R − mod. A ’space’ X is called affine if it is affine over ℤ. By [R4, 9.3.3], X is affine iff the category C_{X} has a projective generator of finite type. By a well-known theorem of Gabriel and Mitchell, the latter condition means precisely that the category C_{X} is equivalent to the category of modules over an associative ring.

We call set {T_{i} | i ∈ J} of thick subcategories of the category C_{X} an affine cocover of the ’space’ X if ⋂_{i∈J} T_{i} = 0 and X/T_{i} is affine for every i ∈ J.

10.5.4. Proposition. Let a finite set {T_{i} | i ∈ J} of Serre subcategories of C_{X} be a cocover of X (that is ⋂_{i∈J} T_{i} = 0) such that every quotient category C_{X}/T_{i} has a family of

generators of finite type. Then Spec_{𝔠}^{1}(X) = Spec_{𝔱}^{1,1}(X).

In particular, Spec_{𝔠}^{1}(X) = Spec_{𝔱}^{1,1}(X), if {T_{i} | i ∈ J} is an affine cocover of X.

Proof. In fact, quotients of an object of finite type is an object of finite type. Therefore, if a category C_{Y} has a family of generators of finite type, then every nonzero object of C_{Y} has a subobject of finite type. The assertion follows now from 10.5.3.

10.5.5. Remark. If C_{X} is a Grothendieck category and S is a Serre subcategory of C_{X}, then the localization functor C_{X} ⟶ C_{X}/S has a right adjoint, hence C_{X}/S is Grothendieck category (see [BD, Ch.6]). In particular, all C_{X}/T_{i} are Grothendieck categories. One can regard Grothendieck categories with a generator of finite type as a noncommutative version of a quasi-affine scheme.

Recall that quasi-affine commutative schemes are, by definition, quasi-compact open subschemes of affine commutative schemes.

10.6. Geometric centers of a ’space’ X associated with Spec_{𝔠}^{0}(X).

10.6.1. The geometric center associated with a topology on Spec_{𝔠}^{0}(X). Let τ be a topology on Spec_{𝔠}^{0}(X). For every open subset U, let Õ_{X}(U) denote the center of the quotient category C_{X}/⟨U⟩, where ⟨U⟩ = ⋂_{Q∈U}⟨Q⟩. Recall that the center of the category

C_{Y} is the (commutative) ring of the endomorphisms of the identical functor C_{Y} ⟶ C_{Y}.

The correspondence U ⟼ Õ_{X}(U) is a presheaf of commutative rings on the topological space (Spec_{𝔠}^{0}(X), τ). We denote by O_{X} the associated sheaf. The ringed space ((Spec_{𝔠}^{0}(X), τ), O_{X}) is called the geometric center of X associated with (Spec_{𝔠}^{0}(X), τ).

10.6.2. Zariski geometric center. A topologizing subcategory 𝕋 of C_{X} is called a Zariski topologizing subcategory if it is bireflective, i.e. the inclusion functor 𝕋 ↪ C_{X} has right and left adjoints. Subsets U_{𝔠}^{0}(𝕋) = {Q ∈ Spec_{𝔠}^{0}(X) | Q ⊈ 𝕋}, where 𝕋 runs through the preorder 𝔗_{𝔷}(X) of Zariski topologizing subcategories, are open sets of the Zariski topology on Spec_{𝔠}^{0}(X) (cf. A1.2.4.1). We call the corresponding geometric center the Zariski geometric center of X.

10.6.3. The topologies τ_{𝔠} and τ^{𝔠} on Spec_{𝔠}^{0}(X). The Zariski topology might be trivial or too coarse in the noncommutative case. For instance, it is trivial if C_{X} = R−mod, where the ring R is simple (i.e. does not have nonzero proper two-sided ideals), like, for instance, any Weyl algebra. Following A1.7, we introduce other topologies on Spec_{𝔠}^{0}(X) by fixing a set Ξ of coreflective topologizing subcategories of C_{X} and then taking subsets U_{𝔠}^{0}(𝕋) = {Q ∈ Spec_{𝔠}^{0}(X) | Q ⊈ 𝕋}, 𝕋 ∈ Ξ, as a base of open sets of a topology τ_{Ξ}.

Taking Ξ = Spec_{𝔠}^{0}(X), we obtain the topology τ_{𝔠} (compare with A1.7.2). Thus the sets V_{𝔠}^{0}(Q) = {Q′ ∈ Spec_{𝔠}^{0}(X) | Q′ ⊆ Q}, Q ∈ Spec_{𝔠}^{0}(X), form a base of closed subsets of this topology.

The topology τ^{𝔠} is determined by Ξ consisting of the subcategories [M]_{𝔠} spanned by objects which are locally of finite type. Here ’locally of finite type’ means that the localization of M at every point Q of Spec_{𝔠}^{0}(X) (i.e. its image in the quotient category C_{X}/⟨Q⟩) is an object of finite type. It seems that τ^{𝔠} is an appropriate version of Zariski topology for the spectrum Spec_{𝔠}^{0}(X). If C_{X} has enough objects of locally finite type, then the topology τ^{𝔠} is finer than the topology τ_{𝔠}.

10.7. Reconstruction of commutative schemes.

10.7.1. Proposition. Let C_{X} be the category of quasi-coherent sheaves on a scheme X = (X , O). Suppose that there is an affine cover {U_{i} ↪ X | i ∈ J} of the scheme X such that all immersions U_{i} ↪ X , i ∈ J, have a direct image functor. Then the geometric center (Spec_{𝔠}^{0}(X), O_{X}) is isomorphic to the scheme X.

Proof. The argument follows the lines of the proof of 9.8.2.

(a) The underlying space X of the scheme X is isomorphic to Spec_{𝔠}^{0}(X).

Let {U_{i} ↪ X | i ∈ J} be an affine cover of the scheme X. For each i ∈ J, we denote by C_{U_{i}} the category of quasi-coherent sheaves on the affine scheme (U_{i}, O_{U_{i}}) and

by T_{i} the kernel of the inverse image functor C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}}. This inverse image functor uniquely determines the equivalence of the quotient category C_{X}/T_{i} and C_{U_{i}}. The fact that {U_{i} ↪ X | i ∈ J} is a cover means precisely that ⋂_{i∈J} T_{i} = 0. The existence of a direct

image functor, C_{U_{i}} ⟶^{u_{i∗}} C_{X}, of the embedding U_{i} ↪ X implies that the subcategory T_{i} is coreflective: a right adjoint to the inclusion functor T_{i} ⟶ C_{X} assigns to every object M of C_{X} the kernel of the adjunction morphism M ⟶ u_{i∗}u_{i}^{∗}(M).

(a1) Let x be a point of the underlying space X of the scheme X. Let Ix̄ be the defining ideal of the closure x̄ of the point x and Mx̄ the quotient sheaf O/Ix̄. Set J_{x} = {i ∈ J | Mx̄ ∉ ObT_{i}}. We claim that Q_{x} = [Mx̄]_{𝔠} is an element of Spec_{𝔠}^{0}(X).

For every i ∈ J_{x}, the object u_{i}^{∗}(Mx̄) of the category C_{U_{i}} belongs to Spec(U_{i}) and Spec(U_{i}) = Spec_{𝔠}^{0}(U_{i}), because C_{U_{i}} is (equivalent to) the category of modules over a ring. Therefore, [u_{i}^{∗}(Q_{x})]_{𝔠} = [u_{i}^{∗}(Mx̄)]_{𝔠} is an element of Spec_{𝔠}^{0}(U_{i}). By 10.4.3, Q_{x} ∈ Spec_{𝔠}^{0}(X).

(a2) Conversely, let Q be an element of Spec_{𝔠}^{0}(X). Let Q ⊈ T_{i}, or, equivalently, T_{i} ⊆ ⟨Q⟩. By 10.4.3, [u_{i}^{∗}(Q)]_{𝔠} is an element of Spec_{𝔠}^{0}(U_{i}). Since U_{i} is affine, Spec_{𝔠}^{0}(U_{i}) is in bijective correspondence with the underlying space U_{i} of the subscheme (U_{i}, O_{U_{i}}); in particular, to the element [u_{i}^{∗}(Q)]_{𝔠} there corresponds a point x of U_{i} which we identify with its image in X. Notice that the point x does not depend on the choice of i ∈ J_{⟨Q⟩} = {j ∈ J | T_{j} ⊆ ⟨Q⟩}. This gives a map Spec_{𝔠}^{0}(X) ⟶ X which is inverse to the map X ⟶ Spec_{𝔠}^{0}(X) constructed in (a1) above. These maps are homeomorphisms in the case if the cover consists of one element, i.e. the scheme is affine. The general case follows from the commutative diagrams

Spec_{𝔠}^{0}(U_{i}) ⟶^{∼} U_{i}
↓  ↓
Spec_{𝔠}^{0}(X) ⟶ X ,  i ∈ J.

(6)

in which vertical arrows are open immersions and the upper horizontal arrow is a homeomorphism; hence the lower horizontal arrow is a homeomorphism.

(b) The diagrams (6) extend to the commutative diagrams of ringed spaces

(Spec_{𝔠}^{0}(U_{i}), O_{U_{i}}) ⟶^{∼} (U_{i}, O_{U_{i}})
↓  ↓
(Spec_{𝔠}^{0}(X), O_{X}) ⟶ (X , O)  i ∈ J.

(7)

in which Spec_{𝔠}^{0}(U_{i}), O_{U_{i}}) and (Spec_{𝔠}^{0}(X), O_{X}) are Zariski geometric centra of resp. U_{i} and X, vertical arrows are open immersions and upper horizontal arrow is an isomorphism. Therefore the lower horizontal arrow is an isomorphism.

Appendix 1. The noncommutative cosite of topologizing subcategories and topologies on spectra.

A1.1. The noncommutative finite cosite of topologizing subcategories. We regard topologizing subcategories of an abelian category C_{X} as ’closed sets’ of a ’finite topology’ τ_{X}^{𝔣} defined as follows. We call a set of inclusions {𝕋 ↪ 𝕋_{i} | i ∈ J} of topologizing subcategories a cocover if there exists a finite subset J_{0} of J such that ⋂_{i∈J_{0}} 𝕋_{i} = 𝕋.

Two of the three standard properties of cocovers follow immediately:

(a) 𝕋 ⟶^{id} 𝕋 is a cocover;

(b) the composition of cocovers is a cocover: if {𝕋 ↪ 𝕋_{i} | i ∈ J} is a cocover and {𝕋_{i} ↪ 𝕋_{ij} | j ∈ J_{i}} is a cocover for every i ∈ J, then {𝕋 ↪ 𝕋_{ij} | i ∈ J, j ∈ J_{i}} is a cocover.

The third standard property – the invariance under the base change, acquires the following form:

(c) If {𝕋 ↪ 𝕋_{i} | i ∈ J} is a cocover, then, for any 𝕊 ∈ 𝔗(X), both {𝕋•𝕊 ↪ 𝕋_{i}•𝕊 | i ∈ J} and {𝕊 • 𝕋 ↪ 𝕊 • 𝕋_{i} | i ∈ J} are cocovers.

The property (c) follows from [R4, 4.2.1]. Its proof is also contained in the argument of A1.2.3(b) below.

We call the triple (𝔗(X), •; τ_{X}^{𝔣}) the noncommutative finite cosite of topologizing subcategories of C_{X}.

A1.1.1. Note. One can define a finer topological structure on 𝔗(X) by taking as cocovers all sets of inclusions {𝕋 ↪ 𝕋_{i} | i ∈ J} such that ⋂_{i∈J} 𝕋_{i} = 𝕋. The family τ_{X} of

such cocovers satisfies the conditions (a) and (b) above, but fails, in general, the invariance with respect to a base change. The situation improves when one considers instead of all topologizing subcategories coreflective or reflective topologizing categories. This is made precise below.

A1.2. Coreflective and reflective topologizing categories. Let 𝔗_{𝔠}(X) (resp. 𝔗^{𝔠}(X)) denote the preorder of all coreflective (resp. reflective) topologizing subcategories of the category C_{X}. Recall that a subcategory B of C_{X} is called coreflective (resp. reflective) if the inclusion functor B ↪ C_{X} has a right (resp. left) adjoint. By [R, III.6.2.1], both 𝔗_{𝔠}(X) and 𝔗^{𝔠}(X) are monoidal subcategories of the monoidal category (preorder) (𝔗(X), •). We shall use the same notation – τ_{X}^{𝔣}, for the restrictions of the topological structure τ_{X}^{𝔣} defined in A1.1 to 𝔗_{𝔠}(X) and to 𝔗^{𝔠}(X).

Notice that (𝔗_{𝔠}(X), •; τ_{X}^{𝔣}) is naturally anti-isomorphic to (𝔗^{𝔠}(X^{o}), •; τ_{X^{o}}^{𝔣}).

The term ’anti-isomorphic’ refers to the monoidal structure (𝕊 • 𝕋)^{o} = 𝕋^{o} • 𝕊^{o}, where 𝕊^{o} denotes the subcategory of C_{X^{o}} = C_{X}^{op} corresponding to the subcategory 𝕊 of C_{X}.

A1.2.1. Lemma. Suppose that C_{X} is an abelian category with supremums of sets of subobjects (for instance, if C_{X} has infinite coproducts). Then

(a) The intersection of any set of reflective topologizing subcategories is a reflective topologizing subcategory.

(b) ( ⋂_{i∈J} 𝕋_{i}) • 𝕊 = ⋂_{i∈J}(𝕋_{i} • 𝕊) for any set {𝕋_{i} | i ∈ J} of topologizing subcategories

and any coreflective subcategory 𝕊.

Proof. (a) See [R, III.6.2.2].

(b) The inclusion ( ⋂_{i∈J} 𝕋_{i}) • 𝕊 ⊆ ⋂_{i∈J}(𝕋_{i} • 𝕊) is obvious. On the other hand, let M be

an object of the subcategory ⋂_{i∈J}(𝕋_{i} • 𝕊); that is, for every i ∈ J, there exists an exact

sequence 0 ⟶ M_{i} ⟶ M ⟶ L_{i} ⟶ 0 such that M_{i} ∈ Ob𝕊 and L_{i} ∈ Ob𝕋_{i}. Since 𝕊 is a coreflective subcategory, the supremum M_{J} of the set of subobjects {M_{i} | i ∈ J} is an object of 𝕊. The canonical epimorphism M ⟶ M/M_{J} factors through M ⟶ L_{i} for each i ∈ J. Therefore, the object M/M_{J}, being a quotient of the object L_{i}, belongs to the subcategory 𝕋_{i} for each i ∈ J, hence it belongs to ⋂_{i∈J} 𝕋_{i}.

A1.2.2. Corollary. Let C_{X} be an abelian category with infinums of sets of quotient objects (i.e. the dual category C_{X^{o}} = C_{X}^{op} has maximums of sets of subobjects). Then

(a) The intersection of any set of coreflective topologizing subcategories is a coreflective topologizing subcategory.

(b) 𝕊 • ( ⋂_{i∈J} 𝕋_{i}) = ⋂_{i∈J}(𝕊 • 𝕋_{i}) for any set {𝕋_{i} | i ∈ J} of topologizing subcategories

and any reflective subcategory 𝕊.

Proof. The assertion is dual to the assertion of A1.2.1.

A1.2.3. Corollary. Let C_{X} be an abelian category with supremums of sets of subobjects and infinums of sets of quotient objects. Then

(a) The intersection of any set of reflective (resp. coreflective) topologizing subcategories is a reflective (resp. coreflective) topologizing subcategory.

(b) If 𝕊 is a reflective and 𝕌 a coreflective topologizing subcategory of C_{X}, then

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

for any set {𝕋_{i} | i ∈ J} of topologizing subcategories.

A1.2.3.1. Note. The conditions on C_{X} in A1.2.3 hold if the category C_{X} has infinite products and coproducts. In particular, they hold for any Grothendieck category, or the category of quasi-coherent sheaves on an arbitrary scheme.

A1.2.4. Interpretations. Let τ_{X} denote the family of cocovers on 𝔗(X) in the sense of A1.1.1; that is {𝕋 ↪ 𝕋_{i} | i ∈ J} is a cocover iff the intersection of all 𝕋_{i} coincides with 𝕋. Corollary A1.2.3 can be spelled as follows:

If the category C_{X} has supremums of sets of subobjects and infinums of sets of quotient objects, then τ_{X} induces the structure of a right cosite on the monoid (𝔗_{𝔠}(X), •) of coreflective topologizing subcategories and the structure of a left cosite on the monoid (𝔗^{𝔠}(X), •) of reflective topologizing subcategories.

A1.2.4.1. Zariski cosite. We denote by 𝔗_{𝔷}(X) the intersection 𝔗_{𝔠}(X) ⋂ 𝔗^{𝔠}(X). Objects of 𝔗_{𝔷}(X) – bireflective subcategories of C_{X}, are interpreted as Zariski closed subspaces. By this reason, we shall call them sometimes Zariski topologizing subcategories. Under conditions of A1.2.3 (i.e. if C_{X} and C_{X^{o}} have supremums of sets of subobjects), (𝔗_{𝔷}(X), •; τ_{X}) is a two-sided cosite. The latter means that for any set {𝕊, 𝕋_{i} | i ∈ J} of Zariski topologizing subcategories, the intersection ⋂_{i∈J} 𝕋_{i} is a Zariski topologizing

subcategory and

𝕊 • ( ⋂_{i∈J} 𝕋_{i}) = ⋂_{i∈J}(𝕊 • 𝕋_{i}),  ( ⋂_{i∈J} 𝕋_{i}) • 𝕊 = ⋂_{i∈J}(𝕋_{i} • 𝕊).  (1)

We call (𝔗_{𝔷}(X), •; τ_{X}) the noncommutative Zariski finite cosite of the ’space’ X. One of the reasons for these interpretaions comes from the following example.

A1.2.5. Example. Let C_{X} = R − mod for an associative ring R. For every twosided ideal α in R, let 𝕋_{α} denote the full subcategory of R − mod whose objects are modules annihilated by the ideal α. By [R, III.6.4.1], the map α ⟼ 𝕋_{α} is an isomorphism of the preorder (I(R), ⊇) of two-sided ideals of the ring R onto (𝔗^{𝔠}(X), ⊆). Moreover, 𝕋_{α} • 𝕋_{β} = 𝕋_{αβ} for any pair of two-sided ideals α, β. This means that the map α ⟼ 𝕋_{α} is an isomorphism of monoidal categories (preorders), where the monoidal structure on I(R) is the multiplication of ideals.

It follows from this description that every reflective topologizing subcategory of C_{X} = R − mod is coreflective, that is 𝔗^{𝔠}(X) = 𝔗_{𝔷}(X).

One can see that ⋂_{i∈J} 𝕋_{α_{i}} = 𝕋_{α_{J}}, where α_{J} = sup(α_{i}|i ∈ J). Thus, the cotopology

τ_{X} on 𝔗^{𝔠}(X) = 𝔗_{𝔷}(X) induces a (noncommutative) Zariski topology on I(R): the set of inclusions of two-sided ideals {α_{i} ↪ α | i ∈ J} is a cover if α = sup(α_{i}|i ∈ J_{0}).

The invariance with respect to base change in I(R) is expressed by the equalities

β sup(α_{i}| i ∈ J) = sup(βα_{i}| i ∈ J) and sup(α_{i}| i ∈ J)β = sup(α_{i}β| i ∈ J)

for any set of two-sided ideals {β, α_{i} | i ∈ J}. One can deduce directly from these equalities the base change invariance on 𝔗_{𝔷}(X) (in the case when C_{X} = R − mod). In fact, we have

⋂_{i∈J}(𝕋_{α_{i}} • 𝕋_{β}) = ⋂_{i∈J} 𝕋_{α_{i}β} = 𝕋_{sup(α_{i}β| i∈J)} = 𝕋_{sup(α_{i}| i∈J)β} =

𝕋_{sup(α_{i}| i∈J)} • 𝕋_{β} = ( ⋂_{i∈J} 𝕋_{α_{i}}) • 𝕋_{β}.

Similar calculation shows that ⋂_{i∈J}(𝕋_{β} • 𝕋_{α_{i}}) = 𝕋_{β} • ( ⋂_{i∈J} 𝕋_{α_{i}}).

A1.2.6. Example: reflective topologizing subcategories of the category of quasi-coherent sheaves on a scheme. Let C_{X} be the category Qcoh_{X} of quasicoherent sheaves on a scheme X = (X , O_{X}). Then elements of 𝔗^{𝔠}(X) are in one to one correspondence with quasi-coherent ideals of the structure sheaf O_{X}, or, equivalently, with closed subschemes of the scheme X.

A1.3. Example: coreflective topologizing subcategories of an affine ’space’. Let C_{X} be the category R − mod of left modules over an associative ring R. We denote by I_{ℓ}(R) the set of left ideals of R.

Recall that a set 𝔉 of left ideals of R is called a topologizing filter if it is closed under finite intersections, contains with every left ideal 𝔪 left ideals (𝔪 : r) = {a ∈ R | ar ∈ 𝔪} for all r ∈ R and all left ideals containing 𝔪.

Topologizing filters of left ideals form a monoidal category (a preorder) with respect to the Gabriel multiplication defined as follows.

𝔉 ∘ 𝔊 = ⋃_{𝔪∈𝔊} 𝔉 ∘ {𝔪}, where 𝔉 ∘ {𝔪} = {𝔫 ∈ I_{ℓ}(R) | (𝔫 : r) ∈ 𝔉 for all r ∈ 𝔪}.

There is a natural bijective correspondence between topologizing filters of left ideals and coreflective topologizing subcategories of the category R −mod. Namely, to each coreflective topologizing subcategory 𝕋 of R − mod, there corresponds the filter 𝔉_{𝕋} formed by annihilators of elements of modules from 𝕋. The inverse map assigns to each topologizing filter 𝔉 the full subcategory 𝕋_{𝔉} whose objects are all R-modules M such that each element of M is annihilated by some left ideal from 𝔉. These maps are mutually inverse isomorphisms between the monoidal preorder (𝔗𝔉_{ℓ}(R), ∘) of topologizing filters of left ideals of the ring R and the monoidal preorder (𝔗_{𝔠}(X), •) of coreflective topologizing subcategories of C_{X} = R − mod.

To every left ideal 𝔪 in R, one can assign the smallest topologizing filter [𝔪] containing 𝔪. It is easy to see that [𝔪] consists of all left ideals 𝔫 which contain (𝔪 : x) = {r ∈ R | rx ⊂ 𝔪} for some finite set x of elements of R. The corresponding coreflective topologizing subcategory is formed by all R-modules M such that every element of M is annihilated by the left ideal (𝔪 : x) for some finite set x of elements of R.

Notice that if 𝔪 is a two-sided ideal, then 𝔪 ⊆ (𝔪 : x) for any x ⊂ R. In this case the filter [𝔪] consists of all left ideals of R containing 𝔪 and the corresponding topologizing subcategory coincides with the subcategory 𝕋_{𝔪} whose objects are modules annihilated by 𝔪 (see A1.2.1). If α and β are two-sided ideals, then [α] ∘ [β] = [αβ]. This shows that the map I(R) ⟶ 𝔗𝔉_{ℓ}(R), α ⟼ [α] is an embedding of monoidal preorders.

Thus, we have a commutative diagram of morphisms of monoidal preorders

(I(R), ·) ⟶ (𝔗𝔉_{ℓ}(R), ∘)
≀↓ ↓≀
(𝔗^{𝔠}(X), •) ⟶ (𝔗_{𝔠}(X), •)

where the lower horizontal arrow is the inclusion functor and C_{X} = R − mod.

A1.4. The cosites of thick and Serre subcategories. Consider the preorder 𝔗𝔥(X) of thick subcategories and the preorder 𝔖𝔢(X) of Serre subcategories of the category C_{X} together with cocovers induced from 𝔗(X).

A1.4.1. Proposition. (a) 𝔗𝔥(X) and 𝔖𝔢(X) are Grothendieck precosites.

(b) The map 𝔗(X) ⟶ 𝔗𝔥(X) which assigns to every topologizing subcategory 𝕋 the thick subcategory 𝕋^{∞} generated by 𝕋 is a morphism from the noncommutative precosite (𝔗(X), •) of topologizing subcategories to the Grothendick precosite 𝔗𝔥(X) of thick subcategories of C_{X}.

(c) The map 𝔗(X) ⟶ 𝔖𝔢(X), 𝕋 ⟼ 𝕋^{−}, is a morphism from the noncommutative precosite (𝔗(X), •) to the Grothendick precosite 𝔖𝔢(X) of Serre subcategories.

Proof. For any pair of topologizing subcategories 𝕊 and 𝕋, the thick subcategory (𝕊•𝕋)^{∞} generated by 𝕊•𝕋 coincides with the coproduct 𝕊^{∞}⊔𝕋^{∞} of the thick subcategories 𝕊^{∞} and 𝕋^{∞} generated respectively by 𝕊 and 𝕋.

Similarly, (𝕊 • 𝕋)^{−} coincides with the coproduct 𝕊^{−} ∨ 𝕋^{−} of Serre subcategories generated respectively by 𝕊 and 𝕋.

In other words, the maps

𝔗(X) ⟶ 𝔗𝔥(X), 𝕋 ⟼ 𝕋^{∞}, and 𝔗(X) ⟶ 𝔖𝔢(X), 𝕋 ⟼ 𝕋^{−},  (1)

are morphisms of monoidal preorders resp.

(𝔗(X), •) ⟶ (𝔗𝔥(X), ⊔) and (𝔗(X), •) ⟶ (𝔖𝔢(X), ∨).

It remains to verify that the maps (1) transfer cocovers to cocovers, or, equivalently, for any finite set {𝕋_{i} | i ∈ J} of topologizing subcategories of C_{X}, there are equalities

( ⋂_{i∈J} 𝕋_{i})^{∞} = ⋂_{i∈J} 𝕋_{i}^{∞} and ( ⋂_{i∈J} 𝕋_{i})^{−} = ⋂_{i∈J} 𝕋_{i}^{−}.

The first equality is proven in [R4, 4.6.1] and the second equality is the assertion [R4, 4.1]. Altogether proves (b) and (c). The assertion (a) is a consequence of (b) and (c). In fact, for any finite set {𝕊, 𝕋_{i} | i ∈ J} of thick subcategories, we have by (b)

⋂_{i∈J}(T_{i} ⊔ S) = ⋂_{i∈J}(S • T_{i})^{∞} = ( ⋂_{i∈J} S • T_{i})^{∞} = (S • (⋂_{i∈J} T_{i}))^{∞} = S ⊔ ( ⋂_{i∈J} T_{i}),

Similarly, if {𝕊, 𝕋_{i} | i ∈ J} are Serre subcategories, then it follows from (c) that

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

hence the assertion.

A1.5. Monoidal subcategories of (𝔗(X), •) and topologies on spectra. Any full monoidal subcategory 𝔊 of (𝔗(X), •) closed under arbitrary intersections defines a topology τ_{𝔊} on Spec_{𝔱}^{0}(X) (hence on Spec(X)) by taking V_{𝔱}^{0}(𝕋) = {P ∈ Spec_{𝔱}^{0}(X) | P ⊆ 𝕋} (resp. V (𝕋) = {P ∈ Spec(X) | P ⊆ 𝕋}), 𝕋 ∈ 𝔊, as the set of closed subsets.

The map 𝔊 ⟼ τ_{𝔊} is a surjective map from the family of full monoidal subcategories of (𝔗(X), •) closed under arbitrary intersections onto the set of topologies on Spec_{𝔱}^{0}(X) which are coarser than the topology τ_{𝔱}^{0} corresponding to 𝔗(X).

A1.6. Zariski topology. Suppose that the category C_{X} has supremums of sets of subobjects (for instance, C_{X} has infinite coproducts). Then, by [R, III.6.2.2], the intersection of any set of reflective topologizing subcategories is a reflective topologizing subcategory. Taking as 𝔊 the subcategory 𝔗^{𝔠}(X) of reflective topologizing subcategories, we obtain the Zariski topology on Spec_{𝔱}^{0}(X) which we denote by τ_{𝔷}^{0}. Its restriction to Spec(X) will be denoted by τ_{𝔷}.

A1.6.1. Proposition. Suppose C_{X} has the property (sup) and a generator of finite type. Then the topological space (Spec(X), τ_{𝔷}) is quasi-compact.

Proof. See [R, III.6.5.2.1].

A1.6.2. Example. Example A1.2.6 shows that if C_{X} is the category of quasicoherent sheaves on a (commutative) scheme X, then the elements of 𝔗^{𝔠}(X) are categories of quasi-coherent sheaves on closed subschemes of X. Suppose that the scheme X is quasicompact and quasi-separated (more generally, quasi-compact and the embeddings of every point has a direct image functor). Then Spec(X) is the set of points of the underlying space of the scheme X and closed sets of the Zariski topology on Spec(X) are spectra of closed subschemes. So that the Zariski topology on Spec(X) coincides with the Zariski topology in the conventional sense.

A1.6.3. Example: Zariski topology on an affine noncommutative scheme. Let C_{X} = R − mod for an associative unital ring R. It follows from A1.6.1 that the topological space (Spec(X), τ_{𝔷}) is quasi-compact.

This fact is a special case of a much stronger assertion: the open subset U of the space (Spec(X), τ_{𝔷}) is quasi-compact iff U = U(𝕋_{α}) = Spec(X)−V (𝕋_{α}) for a finitely generated two-sided ideal α of the ring R (cf. A1.2.1).

Two different proofs of this theorem can be found in [R]: I.5.6 and III.6.5.3.1. One of its consequences is that quasi-compact open sets form a base of the Zariski topology on Spec(X). In fact, every two-sided ideal α is the supremum of a set {α_{i} | i ∈ J} of its two-sided subideals, so that U(𝕋_{α}) = U(sup(𝕋_{α_{i}} | i ∈ J)) = ⋃_{i∈J} U(𝕋_{α_{i}}) (see A1.2.1).

A1.6.4. Note. Unlike the commutative case, the Zariski topology is trivial or too coarse in many important examples of noncommutative affine schemes. It follows from the previous discussion that if C_{X} = R − mod, then the Zariski topology on Spec(X) is trivial iff R is a simple ring (i.e. it does not have non-trivial two-sided ideals). In particular, the Zariski topology on Spec(X) is trivial if C_{X} is the category of D-modules on the affine space 𝔸^{n}, because the algebra A_{n} of differential operators on 𝔸^{n} is simple. It is not sufficiently rich in the case when C_{X} is the category of representations of a semisimple Lie algebra over a field of characteristic zero.

A1.7. Some other canonical topologies. A way to define a topology on Spec_{𝔱}^{0}(X) (and on Spec(X)) is to single out a class of topologizing subcategories, Ξ, of C_{X}, take the smallest monoidal subcategory 𝔊_{Ξ} of (𝔗(X), •) which contains Ξ and is closed under arbitrary intersections (which are products in (𝔗(X), ⊆)) and obtain this way the topology τ_{𝔊_{Ξ}}^{0}. This is the same as taking the smallest topology on Spec_{𝔱}^{0}(X) for which the sets V_{𝔱}^{0}(𝕋), 𝕋 ∈ Ξ, are closed.

A1.7.1. The topology τ^{∗}. For instance, taking as Ξ the class of all topologizing subcategories [M], where M is an object of finite type, we obtain a topology τ^{∗} on Spec(X) which in the case when C_{X} is the category of modules over a commutative ring (more generally, the category of quasi-coherent sheaves on a quasi-compact quasi-separated scheme; see A1.2.6 above) coincides with the Zariski topology. It is drastically different in most of noncommutative cases. For any simple ring R (in particular, for any Weyl algebra A_{n}), the Zariski topology is trivial, while the topology τ^{∗} separates distinct points of the spectrum in Kolmogorov’s sense, i.e. (Spec(X), τ^{∗}) is a Kolmogorov’s space.

A1.7.2. The topology τ_{𝔰}. We take as Ξ the set Spec(X) and denote the corresponding topology on Spec(X) by τ_{𝔰}. This means that finite unions of sets V (P) form a base of the closed sets of the topology τ_{𝔰}.

Notice that if the category C_{X} has enough objects of finite type (i.e. every nonzero object of C_{X} has a nonzero subobject of finite type), then the topology τ_{𝔰} is coarser than the topology τ^{∗}. In fact, in this case every element P of Spec(X) is of the form [M] for some object M of finite type.

A1.8. Functorialities.

A1.8.1. Proposition. Let C_{X} and C_{Y} be abelian categories, and let X ⟶^{f} Y be a continuous morphism such that adjunction arrows f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{X}} and Id_{C_{Y}} ⟶^{η_{f}} f_{∗}f^{∗} are monomorphisms. Then the map 𝕋 ⟼ [f^{∗^{−1}}(𝕋)] defines a morphism of monoids

(𝔗(X), •) ⟶^{𝔗(f)} (𝔗(Y), •).

Proof. (a) Let X ⟶^{f} Y be a morphism such that f^{∗} is semi-exact; i.e. f^{∗} maps any exact sequence M′ ⟶ M ⟶ M′′ to an exact sequence (for instance, f^{∗} is right, or left exact). Then f^{∗^{−1}}(𝕋) • f^{∗^{−1}}(𝕊) ⊆ f^{∗^{−1}}(𝕋 • 𝕊) for any pair 𝕋, 𝕊 of subcategories of C_{X}. In particular, [f^{∗^{−1}}(𝕋)] • [f^{∗^{−1}}(𝕊)] ⊆ [f^{∗^{−1}}(𝕋 • 𝕊)].

In fact, if M′ ⟶ M ⟶ M′′ is an exact sequence with f^{∗}(M′) ∈ Ob𝕊 and f^{∗}(M′′) ∈ Ob𝕋, then f^{∗}(M) ∈ Ob𝕋 • 𝕊, because the sequence f^{∗}(M′) ⟶ f^{∗}(M) ⟶ f^{∗}(M′′) is exact, due to the semi-exactness of the functor f^{∗}.

(a1) Notice that the inverse image functor of a continuous morphism is right exact, hence semi-exact.

(b) In order to prove the inverse inclusion, [f^{∗^{−1}}(𝕋)] • [f^{∗^{−1}}(𝕊)] ⊇ [f^{∗^{−1}}(𝕋 • 𝕊)], it suffices to show that [f^{∗^{−1}}(𝕋)] • [f^{∗^{−1}}(𝕊)] ⊇ f^{∗^{−1}}(𝕋 • 𝕊).

Let f^{∗}(M) ∈ Ob𝕋 • 𝕊; i.e. there is an exact sequence L′ ⟶ f^{∗}(M) ⟶ L′′ with L′ ∈ Ob𝕊 and L′′ ∈ Ob𝕋. Consider the commutative diagram

f^{∗}f_{∗}(L′) ⟶ f^{∗}f_{∗}f^{∗}(M) ⟶ f^{∗}f_{∗}(L′)
ϵ_{f}(L′) ↓  ϵ_{f}f^{∗}(M) ↓  ↓ ϵ_{f}(L′′)
L′ ⟶  f^{∗}(M)  ⟶  L′′

Since ϵ_{f} is a monomorphism and ϵ_{f}f^{∗}(M) is a strict epimorphism (coretraction), ϵ_{f}f^{∗}(M) is an isomorphism. The monomorphness of ϵ_{f}(L′) and ϵ_{f}(L′′) imply that f^{∗}f_{∗}(L′) ∈ Ob𝕊 and f^{∗}f_{∗}(L′′) ∈ Ob𝕋. Thus, we have an exact sequence f_{∗}(L′) ⟶ f_{∗}f^{∗}(M) ⟶ f_{∗}(L′′) with f_{∗}(L′) ∈ Obf^{∗^{−1}}(𝕊) and f_{∗}(L′′) ∈ Obf^{∗^{−1}}(𝕋), hence f_{∗}f^{∗}(M) ∈ Obf^{∗^{−1}}(𝕋) • f^{∗^{−1}}(𝕊). If the adjunction morphism M ⟶ f_{∗}f^{∗}(M) is a monoarrow, the object M belongs to the subcategory [f^{∗^{−1}}(𝕋) • f^{∗^{−1}}(𝕊)] = [f^{∗^{−1}}(𝕋)] • [f^{∗^{−1}}(𝕊)].

A1.8.2. Note. The conditions of A1.8.1 hold if C_{Y} is a coreflective full subctegory of C_{X} and f^{∗} is the inclusion functor C_{Y} ↪ C_{X}. In this case, the adjunction arrow Id_{C_{Y}} ⟶^{η_{f}} f_{∗}f^{∗} is an isomorphism, and the second adjunction arrow, f^{∗}f_{∗} ⟶^{ϵ_{f}} Id_{C_{X}} , is a monomorphism.

Appendix 2. Complements on Spec_{𝔱}^{0}(X) and Spec(X).

A2.1. The difference between Spec_{𝔱}^{0}(X) and Spec(X). The following two assertions provide examples of elements of Spec_{𝔱}^{0}(X) which do not belong to Spec(X).

A2.1.1 Proposition. Let M be a semisimple object of the category C_{X}. Then the topologizing subcategory [M] spanned by M belongs to Spec_{𝔱}^{0,0}(X) = Spec_{𝔱}^{0}(X)−Spec(X) iff M is isomorphic to an infinite coproduct of copies of a simple object L.

The topologizing subcategory [M] belongs to Spec(X) iff M is isomorphic to a finite coproduct of copies of a simple object L.

Proof. (a) Let M be isomorphic to the coproduct L^{J} of J copies of a simple object L. (a1) If J is finite, then [M] = [L] is a closed point of Spec(X). Conversely, if [M] belongs to Spec(X), then [M] = [L], or, equivalently, L ≻ M. The

latter means that M is a subquotient of a finite coproduct of copies of the simple object L, hence M is isomorphic to a finite coproduct of copies of L.

(a2) Suppose that J is infinite and the object M belongs to the Gabriel product 𝕊_{2} • 𝕊_{1} of two topologizing subcategories of C_{X}; that is there exists an exact sequence 0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0 such that M_{i} ∈ Ob𝕊_{i}, i = 1, 2. The objects M_{1} and M_{2}, being subquotients of the semisimple object L^{J} are isomorphic to coproducts of resp. J_{1} and J_{2} copies of L. Since J is infinite, then either J_{1} or J_{2} should be isomorphic to J. If J_{i} ≃ J, then M_{i} ≃ M, hence M ∈ Ob𝕊_{i}.

(b) If M is a semisimple object which has non-isomorphic simple components, then M ≃ M_{1} ⊕ M_{2}, where M_{1}, M_{2} are nonzero objects and Hom(M_{1}, M_{2}) = 0. This implies that [M] = [M_{1}] • [M_{2}] and M does not belong to [M_{1}] ⋃[M_{2}]. By 4.3(a), [M] is not an element of Spec_{𝔱}^{0}(X).

A2.1.2. Corollary. If M is isomorphic to the product of infinite set of copies of a simple object of the category C_{X}, then [M] belongs to Spec_{𝔱}^{0,0}(X).

Proof. For every object N of C_{X}, let N^{o} denote the corresponding object of the dual category C_{X^{o}} = C_{X}^{op}. The object M^{o} is isomorphic to the coproduct of an infinite set of copies of a simple object L^{o}; hence, by A2.1.1, the topologizing subcategory [M^{o}] belongs to Spec_{𝔱}^{0,0}(X^{o}). But, as it was observed in 4.2, Spec_{𝔱}^{0}(X) is naturally isomorphic to Spec_{𝔱}^{0}(X^{o}); in particular, [M] belongs to Spec_{𝔱}^{0}(X). It remains to show that [M] does not belong to Spec(X).

Suppose that, on the contrary, [M] belongs to Spec(X), i.e. ˆ[M] = ⟨M⟩ is a Serre subcategory. Then M has a nonzero subquotient, M′, which is ⟨M⟩-torsion free. It follows that [M] = [M′]. Every subquotient of a product of a family of simple objects is isomorphic to the product of its subfamily. In particular, M′ is isomorphic to the product of J copies of the simple object L; hence the simple object L is a subobject of M′. Since M′ belongs to Spec(X), this implies that L ≻ M′, hence M′ is isomorphic to a finite product of copies of L. But then the equality [M] = [M′] implies that L ≻ M, hence M is isomorphic to a finite product of copies of L, which contradicts to the initial hypothesis.

A2.1.3. Note. Given an object L and a set J, let L^{⊕J} (resp. L^{ΠJ}) denote the direct sum, (resp. product) of J copies of the object L. Let L be a simple object of the category C_{X}, and let J and I be infinite sets such that Card(I) < Card(J). Then there are proper inclusions [L^{⊕J}] ⊊ [L^{ΠJ}] ⊊ [L^{ΠI}] ⊋ [L^{⊕I}] ⊋ [L^{⊕J}]. In particular, [L^{⊕J}], [L^{⊕I}], [L^{ΠJ}], and [L^{ΠI}] are four distinct elements of Spec_{𝔱}^{0,0}(X).

A2.2. Functorialities and topologies. For any topologizing subcategory 𝕋 of the category C_{X}, set

U_{𝔱}^{1}(𝕋) = {P ∈ Spec_{𝔱}^{1}(X) | 𝕋 ⊆ P} and U_{𝔱}^{0}(𝕋) = {Q ∈ Spec_{𝔱}^{0}(X) | Q ⊈ 𝕋}.

A2.2.1. Proposition. (a) If 𝕋, 𝕊 are topologizing categories, then

U_{𝔱}^{0}(𝕋 • 𝕊) = U_{𝔱}^{0}(𝕋) ⋂ U_{𝔱}^{0}(𝕊) and U_{𝔱}^{1}(𝕋 • 𝕊) = U_{𝔱}^{1}(𝕋) ⋂ U_{𝔱}^{1}(𝕊).

In particular, U_{𝔱}^{i}(𝕋) = U_{𝔱}^{i}([𝕋]_{•}), where [𝕋]_{•} is the thick subcategory generated by 𝕋 and i = 0, 1.

(b) U_{𝔱}^{0}( ⋂_{j∈J} 𝕋_{j}) = ⋃_{j∈J} U_{𝔱}^{0}(𝕋_{j}) for any family {𝕋_{j} | j ∈ J} of topologizing categories.

(c) Suppose that 𝕋 is a thick subcategory of C_{X}. Then there is a commutative diagram

Spec_{𝔱}^{0}(X) ⟵ U_{𝔱}^{0}(𝕋) ⟶ Spec_{𝔱}^{0}(X/𝕋)
↓  ↓  ↓
Spec_{𝔱}^{1}(X) ⟵ U_{𝔱}^{1}(𝕋) ⟶ Spec_{𝔱}^{1}(X/𝕋)

(4)

of injective maps.

Proof. (a) The equality U_{𝔱}^{0}(𝕋 • 𝕊) = U_{𝔱}^{0}(𝕋) ⋂ U_{𝔱}^{0}(𝕊) follows from 4.3(a).

The inclusions 𝕊 ⊆ 𝕋 • 𝕊 ⊇ 𝕋 imply that U_{𝔱}^{1}(𝕋 • 𝕊) ⊆ U_{𝔱}^{1}(𝕋) ⋂ U_{𝔱}^{1}(𝕊). The inverse inclusion follows from the fact that elements of U_{𝔱}^{1}(−) are thick subcategories: if P is a thick subcategory containing 𝕊 and 𝕋, then P contains 𝕋 • 𝕊.

(b) The equality U_{𝔱}^{0}( ⋂_{j∈J} 𝕋_{j}) = ⋃_{j∈J} U_{𝔱}^{0}(𝕋_{j}) is straightforward.

(c) Let 𝕋 be a thick subcategory of C_{X}, and let P ∈ U_{𝔱}^{1}(𝕋), i.e. 𝕋 ⊆ P ≠ P^{𝔱}. Then the topologizing subcategory (𝕋 • P^{𝔱} • 𝕋)/𝕋 of C_{X}/𝕋 coincides with (P/𝕋)^{𝔱}. One can see that (𝕋 • P^{𝔱} • 𝕋)/𝕋 ≠ P/𝕋, because P^{𝔱} ≠ P; that is P/𝕋 is an element of Spec_{𝔱}^{1}(X/𝕋). The map U_{𝔱}^{1}(𝕋) ⟶ Spec_{𝔱}^{1}(X/𝕋) assigns to each P its quotient P/𝕋.

The map U_{𝔱}^{0}(𝕋) ⟶ Spec_{𝔱}^{0}(X/𝕋) assigns to each Q ∈ U_{𝔱}^{0}(𝕋) the topologizing subcategory Q_{𝕋} generated by the image of Q in C_{X}/𝕋, which coincides with (𝕋 • Q • 𝕋)/𝕋 (see 5.1.1). Since Q ⊈ 𝕋, the topologizing subcategory 𝕋 is contained in the thick subcategory Q̂; hence Q̂/𝕋 is a thick subcategory of the quotient category C_{X}/𝕋. One can see that Q̂/𝕋 = Q̂_{𝕋}; so that Q_{𝕋} belongs to Spec_{𝔱}^{0}(X/𝕋).

A2.2.1.1. Corollary. Suppose that 𝕋 is a thick subcategory of C_{X}. Then there is a commutative diagram

Spec(X) ⟵ U_{𝔱}^{0,1}(𝕋) ⟶ Spec(X/𝕋)
≀↓  ↓  ↓≀
Spec_{𝔱}^{1,1}(X) ⟵ U_{𝔱}^{1,1}(𝕋) ⟶ Spec_{𝔱}^{1,1}(X/𝕋),

(5)

where U_{𝔱}^{1,1}(𝕋) = {P ∈ U_{𝔱}^{1}(𝕋) | P = P^{−}}, U_{𝔱}^{0,1}(𝕋) = {Q ∈ Spec(X) | Q ⊈ 𝕋}, and all horizontal arrows are injective maps.

Proof. The assertion is a consequence of A2.2.1(c).

A2.2.2. The ’specialization’ topology on Spec_{𝔱}^{0}(X). It follows from the assertions A2.2.1 (a) and (b) that {U_{𝔱}^{0}(𝕋) | 𝕋 ∈ 𝔗(X)} is the set of open sets of a topology on Spec_{𝔱}^{0}(X) which we denote by τ_{𝔱}^{0}. Let V_{𝔱}^{0}(𝕋) be the complement to U_{𝔱}^{0}(𝕋) in (Spec_{𝔱}^{0}(X), i.e.

V_{𝔱}^{0}(𝕋) = {Q ∈ Spec_{𝔱}^{0}(X) | Q ⊆ 𝕋}.

Recall that a topological space 𝔛 is called sober if every irreducible closed subset of 𝔛 has a unique generic point.

A topological space is called a Kolmogorov’s space if for every pair of its distinct points, one of the points has an open neighborhood which does not contain the other point.

A2.2.3. Proposition. (Spec_{𝔱}^{0}(X), τ_{𝔱}^{0}) is a sober Kolmogorov’s topological space.

Proof. (a) Let Q_{1} and Q_{2} be distinct elements of (Spec_{𝔱}^{0}(X), τ_{𝔱}^{0}). If Q_{1} ⊈ Q_{2}, then Q_{1} ∈ U_{𝔱}^{0}(Q_{2}) ∉ Q_{2}. This shows that (Spec_{𝔱}^{0}(X), τ_{𝔱}^{0}) is a Kolmogorov’s space.

(b) For every topologizing subcategory 𝕋, let 𝕋_{𝔯𝔢𝔡,0} denote the supremum of all Q ∈ V_{𝔱}^{0}. Here 𝔯𝔢𝔡 stands for ’reduced’. It follows that 𝕋_{𝔯𝔢𝔡,0} is the smallest topologizing subcategory of C_{X} such that V_{𝔱}^{0}(𝕋_{𝔯𝔢𝔡,0}) = V_{𝔱}^{0}(𝕋).

The closed set V_{𝔱}^{0}(𝕋) is irreducible iff 𝕋_{𝔯𝔢𝔡,0} is an element of Spec_{𝔱}^{0}(X).

In fact, suppose that V_{𝔱}^{0}(𝕋) is irreducible, and let 𝕊_{1}, 𝕊_{2} be a pair of topologizing subcategories such that 𝕋_{𝔯𝔢𝔡,0} ⊆ 𝕊_{1} • 𝕊_{2}. Then

V_{𝔱}^{0}(𝕋) = V_{𝔱}^{0}(𝕋_{𝔯𝔢𝔡,0}) ⊆ V_{𝔱}^{0}(𝕊_{1} • 𝕊_{2}) = V_{𝔱}^{0}(𝕊_{1}) ⋃ V_{𝔱}^{0}(𝕊_{2}).

Since V_{𝔱}^{0}(𝕋) is irreducible, it is contained in V_{𝔱}^{0}(𝕊_{i}) for i = 1, or i = 2. But then 𝕋_{𝔯𝔢𝔡,0} ⊆ 𝕊_{i} for the same i. This shows that 𝕋_{𝔯𝔢𝔡,0} belongs to Spec_{𝔱}^{0}(X).

Conversely, if Q belongs to Spec_{𝔱}^{0}(X) and 𝕊_{1}, 𝕊_{2} are topologizing subcategories such that V_{𝔱}^{0}(Q) ⊆ V_{𝔱}^{0}(𝕊_{1}) ⋃ V_{𝔱}^{0}(𝕊_{2}) = V_{𝔱}^{0}(𝕊_{1} • 𝕊_{2}). Then Q ⊆ 𝕊_{1} • 𝕊_{2}, hence Q ⊆ 𝕊_{i} for i = 1 or 2, which, in turn, means that V_{𝔱}^{0}(Q) ⊆ V_{𝔱}^{0}(𝕊_{i}).

Altogether proves that the topological space (Spec_{𝔱}^{0}(X), τ_{𝔱}^{0}) is sober.

A2.3. The spectrum Spec(X) as a subspace of (Spec_{𝔱}^{0}(X), τ_{𝔱}^{0}). The topology τ_{𝔱}^{0} on Spec_{𝔱}^{0}(X) (cf. A2.2.2) induces the topology τ^{≻} on Spec(X) (see 1.6). Its closed sets are V (𝕋) = {Q ∈ Spec(X) | Q ⊆ 𝕋}, where 𝕋 runs though the family 𝔗(X) of topologizing subcategories of C_{X}.

For every topologizing subcategory 𝕋, let 𝕋_{𝔯𝔢𝔡} denote the supremum of all Q ∈ V (𝕋). It follows that 𝕋_{𝔯𝔢𝔡} is the smallest topologizing subcategory of C_{X} such that V (𝕋_{𝔯𝔢𝔡}) = V (𝕋). We call a topologizing subcategory 𝕋 reduced if 𝕋 = 𝕋_{𝔯𝔢𝔡}.

One can see that the map 𝕋 ⟼ V (𝕋) is a bijection between the preorder 𝔗_{𝔯𝔢𝔡}(X) of reduced topologizing subcategories and the preorder of closed subsets of (Spec(X), τ^{≻}) such that V (sup(𝕋, 𝕊)) = V (𝕋) ⋃ V (𝕊).

A2.3.1. Proposition. (a) The subspace V (𝕋) is irreducible iff 𝕋_{𝔯𝔢𝔡} belongs to Spec(X).

Proof. (a) Let 𝕋_{𝔯𝔢𝔡} ⊆ 𝕊_{1} • 𝕊_{2}. Then V (𝕋_{𝔯𝔢𝔡}) ⊆ V (𝕊_{1} • 𝕊_{2}) = V (𝕊_{1}) ⋃ V (𝕊_{2}). Since V (𝕋_{𝔯𝔢𝔡}) is irreducible, V (𝕋_{𝔯𝔢𝔡}) ⊆ V (𝕊_{i}) for i = 1, or 2. Therefore 𝕋_{𝔯𝔢𝔡} ⊆ (𝕊_{i})_{𝔯𝔢𝔡} ⊆ 𝕊_{i}. By 4.3(a), this means that 𝕋_{𝔯𝔢𝔡} is an element of Spec_{𝔱}^{0}(X).

(b) If 𝕋 is reduced, i.e. 𝕋 = 𝕋_{𝔯𝔢𝔡}, then 𝕋 ∈ Spec_{𝔱}^{0}(X) iff 𝕋 ∈ Spec(X). In fact, since 𝕋 = 𝕋_{𝔯𝔢𝔡}, there exists an object Q ∈ V (𝕋) such that Q ⊈ 𝕋̂. This means

that 𝕋 = Q.

A2.3.2. Proposition. (Spec(X), τ^{≻}) is a sober Kolmogorov’s space.

Proof. (a) It follows from A2.3.1 that every irreducible closed subset of Spec(X) is of the form V (Q) for some Q ∈ Spec(X), i.e. (Spec(X), τ^{≻}) is a sober topological space.

(b) Let Q and Q′ be elements of Spec(X) such that Q ⊈ Q′. This means that the closure of Q′ in the topology τ^{≻} does not contain Q; or, equivalently, there is an open neighborhood of Q which does not contain the point Q′.

Appendix 3. Supports and specializations. Krull filtrations.

A3.1. Support in Spec(X). Let M be an object of an abelian category C_{X}. The support of M in Spec(X) is the set Supp(M) of all [P] ∈ Spec(X) such that M ≻ P, or, equivalently, [P] ⊆ [M].

A3.2. Supports in Spec^{1}(X) and in Spec^{−}(X). The support of an object M of C_{X} in Spec^{1}(X) is the set Supp^{1}(X) of all P ∈ Spec^{1}(X) such that M ∉ ObP.

The support of M in the S-spectrum is the set

Supp^{−}(M) = Supp^{1}(M) ⋂ Spec^{−}(X) = Supp^{1}(M) ⋂ 𝔖𝔢(X).

A3.3. Lemma. Let M be an object of C_{X}.

(a) The following conditions are equivalent:

(a1) P ∈ Supp^{1}(M);

(a2) M ≻ L for some nonzero object L of P^{⊛} − P.

(b) The following conditions are equivalent:

(b1) P ∈ Supp^{−}(M);

(b2) M ≻ L for some nonzero object L of P_{⊛} = P^{⊛} ∩ P^{⊥}.
Proof. Let C_{X} ⟶^{q_{P}^{∗}} C_{X/P} be the localization functor at P ∈ Spec^{1}(X).

(a1)⇒(a2). The condition P ∈ Supp^{−}(M) means precisely that q_{P}^{∗}(M) ≠ 0. On the other hand, q_{P}^{∗}(L′) is a quasi-final object of C_{X/P} for every nonzero object L′ of P^{⊛} − P. Therefore, q_{P}^{∗}(M) ≻ q_{P}^{∗}(L′). The latter means that there exists a diagram

M^{⊕n} ⟵^{𝔧′} K′ ⟶^{𝔢′′} L′′ ⟶^{𝔰} L′  (9)

such that Ker(𝔧′) and Cok(𝔰) are objects of P, 𝔢′′ is an epimorphism and 𝔰 is a monomorphism. Replacing K′ by K = K/Ker(𝔧′) and L′ by the cokernel of the composition

Ker(𝔧′) ⟶ K′ ⟶^{𝔢′′} L′′, we obtain the diagram M^{⊕n} ⟵^{𝔧} K ⟶^{𝔢} L in which 𝔧 is a monomorphism and 𝔢 is an epimorphism; i.e. M ≻ L. Since q_{P}^{∗}(L) is isomorphic to q_{P}^{∗}(L′) and L′ is an object of P^{⊛} − P, the object L belongs to P^{⊛} − P too.

(a2)⇒(a1) & (b2)⇒(b1). If M ≻ L and L ∉ ObP, then M ∉ ObP, i.e. P ∈ Supp^{1}(X). (b1)⇒(b2). If P ∈ Supp^{−}(M) and L′ is a nonzero object of P^{⊛} ∩ P^{⊥}, then q_{P}^{∗}(M) ≻

q_{P}^{∗}(L′) which is expressed by the diagram (9). Since this time L′ is a P-torsion object, the

composition of Ker(𝔧′) ⟶ K′ ⟶^{𝔢′′} L′′ is zero. Therefore, replacing K′ by K = K′/Ker(𝔧′),

we obtain a diargram M^{⊕n} ⟵^{𝔧} K ⟶^{𝔢′′} L′′ ↪ L′ in which 𝔧 is a monomorphism and 𝔢′′ is an epimorphism. So that M ≻ L′′, where L′′ is a subobject of an object of P^{⊛} ∩ P^{⊥}, hence L′′ belongs to P^{⊛} ∩ P^{⊥}.

A3.4. Proposition. Let P_{1}, P_{2} be elements of Spec^{−}(X). Then the following conditions are equivalent:

(a) P_{2} ⊆ P_{1};

(b) for every nonzero object M_{1} of P_{1}^{⊛} ∩ P_{1}^{⊥}, there exists a nonzero object M_{2} of P_{2}^{⊛} ∩ P_{2}^{⊥} such that M_{1} ≻ M_{2}.

(c) There exists a nonzero object M_{1} of P_{1}^{⊛} ∩ P_{1}^{⊥} with the following property: for any nonzero subobject L_{1} of M_{1}, there is an object M_{2} of P_{2}^{⊛} ∩ P_{2}^{⊥} such that L_{1} ≻ M_{2}.

Proof. (a)⇒(b). If P_{2} ⊆ P_{1} and M_{1} is a nonzero object of P_{1}^{⊛} ∩ P_{1}^{⊥}), then P_{2} ∈ Supp^{−}(M_{1}). By A3.3(b), there exists an object M_{2} of P_{2}^{⊛} ∩ P_{2}^{⊥} such that M_{1} ≻ M_{2}.

(b)⇒(a). Suppose that P_{2} ⊈ P_{1}; and let N be an object of P_{2} − P_{1}. In particular, P_{1} ∈ Supp^{−}(N). By A3.3(b), there exists a nonzero object M_{1} of P_{1}^{⊛} ∩ P_{1}^{⊥} such that N ≻ M_{1}. By condition (b), M_{1} ≻ M_{2} for some nonzero object M_{2} of P_{2}^{⊛} ∩ P_{2}^{⊥} which implies that N ≻ M_{2}. The latter is impossible, because N ∈ ObP_{2} and M_{2} ∉ ObP_{2}. Therefore P_{2} ⊆ P_{1}.

Obviously, (b)⇒(c).

(c)⇒(a). Replacing X by X/(P_{1} ∩ P_{2}) and the objects M_{1} and M_{2} by their images in C_{X/(P_{1}∩P_{2})}, we can assume that P_{1} ∩ P_{2} = 0. Suppose that P_{2} ≠ 0. Then P_{2} is a local subcategory. If P_{1} = 0, then C_{X} is local too, and nonzero objects of P_{1}^{⊛} ∩ P_{1}^{⊥} = 0^{𝔱} are precisely quasi-final objects of C_{X}. Since P_{2} ≠ 0, it contains 0^{𝔱}; in particular, M_{1} ∈ ObP_{2}. This contradicts to the condition (c) according to which M_{1} ≻ M_{2} for some M_{2} ∈ P_{2}^{⊛}∩P_{2}^{⊥}.

Suppose now that both P_{1} and P_{2} are nonzero, hence both of them are local. There exists a quasi-final object L_{2} of P_{2} and a monomorphism L_{2} ⟶ M_{1} such that M_{1}/L_{2} ∈ ObP_{1}. By condition (c), there exists a nonzero object M_{2} of P_{2}^{⊛} ∩ P_{2}^{⊥} such that L_{2} ≻ M_{2}. Since L_{2} ∈ P_{2}, we run into a contradiction again. Altogether shows that P_{2} = 0.

A3.5. The Krull filtration of Spec^{−}(X) and the associated filtration of X. Fix an abelian category C_{X}. For every cardinal α, we define a subset 𝔖_{α}^{−}(X) of Spec^{−}(X) as follows.

𝔖_{0}^{−}(X) = ∅;

if α is not a limit cardinal, then 𝔖_{α}^{−}(X) consists of all P ∈ Spec^{−}(X) such that any P′ ∈ Spec^{−}(X) properly contained in P belongs to 𝔖_{α−1}^{−}(X);

if α is a limit cardinal, then 𝔖_{α}^{−}(X) = ⋃_{β<α} 𝔖_{β}^{−}(X).

It follows from this definition (borrowed from [R, VI.6.3]) that 𝔖_{1}^{−}(X) consists of all closed points of Spec^{−}(X).

We denote by 𝔖_{ω}^{−}(X) the union of all 𝔖_{α}^{−}(X). The filtration {𝔖_{α}^{−}(X)} determines a filtration

C_{X_{0}} ↪ C_{X_{1}} ↪ … C_{X_{α}} ↪ …  (5)

of the category C_{X} (or the ’space’ X) by taking as C_{X_{α}} the full subcategory of C_{X} generated by objects M such that Supp^{−}(M) ⊆ 𝔖_{α}^{−}(X). Recall that Supp^{−}(M) = {P ∈ Spec^{−}(X) | M ∉ ObP}. In particular, C_{X_{ω}} is the full subcategory of C_{X} generated by all M ∈ ObC_{X} such that Supp^{−}(M) ⊆ 𝔖_{ω}^{−}(X).

It follows from the general properties of supports that C_{X_{α}} is a Serre subcategory of C_{X} and Spec^{−}(X_{α}) is naturally identified with 𝔖_{α}^{−}(X); in particular, Spec^{−}(X_{ω}) is identified with 𝔖_{ω}^{−}(X).

A3.6. The Krull dimension. For every element P of Spec^{−}(X_{ω}), there is the biggest cardinal, 𝔥𝔱^{−}(P), among all the cardinals α such that P ∉ 𝔖_{α}^{−}(X). The cardinal 𝔥𝔱^{−}(P) is called the hight of P ([R, VI.6.3]).

The Krull dimension of X is the supremum of all 𝔥𝔱^{−}(P), where P runs through Spec^{−}(X_{ω}) (in [R] it is called the flat dimension).

An object M of C_{X} is said to have a Krull dimension if it belongs to the subcategory C_{X_{ω}}. Finally, the ’space’ X (or the category C_{X}) has a Krull dimension if X = X_{ω} (that is C_{X} = C_{X_{ω}}) and every nonzero object of C_{X} has a nonempty support, i.e. C_{X_{0}} = 𝕆.

A3.7. The Krull dimension and the Gabriel-Krull dimension. We recall the notion of the Gabriel filtration of an abelian category as it is defined in [R, 6.6]. Let C_{X} be an abelian category. The Gabriel filtration of X assigns to every cardinal α a Serre subcategory C_{X_{α}^{−}} of C_{X} which is constructed as folows:

Set C_{X_{0}^{−}} = 𝕆.

If α is not a limit cardinal, then C_{X_{α}^{−}} is the smallest Serre subcategory of C_{X} containing all objects M such that the localization q_{α−1}^{∗}(M) of M at C_{X_{α−1}^{−}} has a finite length.

If β is a limit cardinal, then C_{X_{β}^{−}} is the smallest Serre subcategory containing all subcategories C_{X_{α}^{−}} for α < β.

Let C_{X_{ω}^{−}} denote the smallest Serre subcategory containing all the subcategories C_{X_{α}^{−}}. It follows that the quotient category C_{X/X_{ω}^{−}} has no simple objects.

An object M is said to have the Gabriel-Krull dimension β, if β is the smallest cardinal such that M belongs to C_{X_{β}^{−}}.

The ’space’ X has a Gabriel-Krull dimension if X = X_{ω}^{−}.

Every locally noetherian abelian category (e.g. the category of quasi-coherent sheaves on a noetherian scheme, or the category of left modules over a left noetherian associative algebra) has a Gabriel-Krull dimension.

It is argued in [R, VI.6] that if X has a Gabriel-Krull dimension, then the filtration (5) coincides with the Gabriel filtration of the category C_{X}. In particular, X has a Krull dimension: X = X_{ω} = X_{ω}^{−}. Thus, the Krull dimension is an extension of the Gabriel-Krull dimension to a wider class of ’spaces’.

A3.8. A description of Spec_{⋊}(X_{ω}). The filtration {𝔖_{α}^{−}(X)} of Spec^{−}(X) induces, via the isomorphism Spec_{⋊}(X) ⟶^{∼} Spec^{−}(X) (defined in 6.4), a filtration {𝔖_{α}^{⋊}(X)} of the spectrum Spec_{⋊}(X). We call it the Krull filtration of Spec_{⋊}(X).

A3.8.1. Proposition. The spectrum Spec_{⋊}(X_{ω}) of X_{ω} is naturally isomorphic to ⋃_{α} Spec_{⋊}(X_{α}/X_{α−1}), and Spec^{−}(X_{ω}) is isomorphic to ⋃_{α} Spec^{−}(X_{α}/X_{α−1}), where α

runs through non-limit cardinals. These isomorphisms are compatible with the isomorphisms Spec_{⋊}(X_{ω}) ⟶^{∼} Spec^{−}(X_{ω}) and Spec_{⋊}(X_{α}/X_{α−1}) ⟶^{∼} Spec^{−}(X_{α}/X_{α−1}).

Proof. More precisely,

Spec_{⋊}(X_{ω}) = ⋃_{α}(𝔖_{α}^{⋊}(X) − 𝔖_{α−1}^{⋊}(X)),

where α runs though non-limit cardinals, and for every non-limit cardinal α, there is a natural isomorphism

𝔖_{α}^{⋊}(X) − 𝔖_{α−1}^{⋊}(X) ⟶^{∼} Spec_{⋊}(X_{α}/X_{α−1}).  (6)

The isomorphism (6) is given by the map 𝔖_{α}^{⋊}(X) ⟶ 𝔗(X/X_{α−1}) which assigns to every element P_{⊛} of 𝔖_{α}^{⋊}(X) the smallest topologizing subcategory [q_{α−1}^{∗}(P_{⊛})] of C_{X/X_{α−1}} spanned by the image of P_{⊛}.

Let P ∈ Spec^{−}(X_{ω}), i.e. P ∈ 𝔖_{α}(X) for some α. Consider all cardinals β such that C_{X_{β}} ⊆ P. Since P is a Serre subcategory, the smallest Serre subcategory spanned by all C_{X_{β}} coincides with C_{X_{α−1}} for a non-limit cardinal α. The image P_{⊛} = P^{⊛} ∩ P^{⊥} of P in Spec_{⋊}(X) is an element of 𝔖_{α}^{⋊}(X) − 𝔖_{α−1}^{⋊}(X).

A3.9. The Krull filtrations and equivalences of categories.

A3.9.1. Proposition. Let C_{X} and C_{Y} be abelian categories. Any category equivalence C_{X} ⟶^{Θ} C_{Y} induces equivalences C_{X_{α}} ⟶^{Θ_{α}} C_{Y_{α}} for all cardinals α. In particular, Θ induces a category equivalence C_{X_{ω}} ⟶^{Θ_{ω}} C_{Y_{ω}}

Proof. The argument is by (transfinite) induction. The assertion is, obviously, true for α = 0. It is also true for α = 1: if P is a closed point of Spec^{−}(X), then [Θ(P)] is a closed point of Spec^{−}(Y).

Suppose now that Θ induces equivalences C_{X_{α}} ⟶^{Θ_{α}} C_{Y_{α}} for all cardinals α < β. We

claim that then it induces a category equivalence C_{X_{β}} ⟶^{Θ_{β}} C_{Y_{β}}.

(a) If β is a limit cardinal, then it follows from the definition of the filtration (cf. A3.5), that C_{X_{β}} = ( ⋃_{α<β} C_{X_{α}})^{−}. By the induction hypothesis, Θ induces a category

equivalence ⋃_{α<β} C_{X_{α}} ⟶ ⋃_{α<β} C_{Y_{α}}. It is easy to show that if Θ induces an equivalence

between a subcategory 𝕋 of C_{X} and a subcategory 𝕊 of C_{Y}, then Θ induces an equivalence 𝕋^{−} ⟶ 𝕊^{−}. In particular, Θ induces a category equivalence from C_{X_{β}} = ( ⋃_{α<β} C_{X_{α}})^{−} to

C_{Y_{β}} = ( ⋃_{α<β} C_{Y_{α}})^{−}.

(b) Suppose now that β is not a limit cardinal. By the induction hypothesis, Θ induces a category equivalence C_{X_{β−1}} ⟶ C_{Y_{β−1}}; hence Θ induces an equivalence between

quotient categories C_{X/X_{β−1}} \underrightarrow{_{Θ_{β−1}}} C_{Y/Y_{β−1}}. The equivalence Θ̂_{β−1} induces an equivalence C_{(X/X_{β−1})_{1}} ⟶ C_{(Y/Y_{β−1})_{1}}. Notice that C_{X_{β}} is the preimage of C_{(X/X_{β−1})_{1}} in C_{X}.

Similarly for C_{Y_{β}}. Therefore Θ induces a functor C_{X_{β}} ⟶^{Θ_{β}} C_{Y_{β}} and its quasi-inverse,

Θ^{∗}, induces a functor C_{Y_{β}} ⟶^{Θ_{β}^{∗}} C_{X_{β}}. Since Θ is an equivalence, Θ_{β} is an equivalence with a quasi-inverse Θ_{β}^{∗}.

A3.9.2. Proposition. Any category equivalence C_{X} ⟶^{Θ} C_{Y} induces isomorphisms

𝔖_{α}^{−}(X) ⟶^{∼} 𝔖_{α}^{−}(Y) and 𝔖_{α}^{⋊}(X) ⟶^{∼} 𝔖_{α}^{⋊}(Y)

for all cardinals α. In particular, Θ induces an isomorphisms

𝔖_{ω}^{−}(X) ⟶^{∼} 𝔖_{ω}^{−}(Y) and 𝔖_{ω}^{⋊}(X) ⟶^{∼} 𝔖_{ω}^{⋊}(Y).

Proof. The assertion follows from A3.9.1 and from the fact that the natural isomorphisms

𝔖_{α}^{−}(X) ≃ Spec^{−}(X_{α}), 𝔖_{α}^{⋊}(X) ≃ Spec_{⋊}(X_{α})

are compatible with category equivalences for all cardinals α. In particular, we have commutative diagrams

𝔖_{ω}^{−}(X)  ⟶^{∼} 𝔖_{ω}^{−}(Y)  𝔖_{ω}^{⋊}(X)  ⟶^{∼} 𝔖_{ω}^{⋊}(Y)
≀↓  ↓≀  and  ≀↓  ↓≀
Spec^{−}(X_{ω}) ⟶^{∼} Spec^{−}(Y_{ω})  Spec_{⋊}(X_{ω}) ⟶^{∼} Spec_{⋊}(Y_{ω})

Details are left to the reader.

A3.9.3. Corollary. Let C_{X} be an abelian category and C_{X} ⟶^{Θ} C_{X} an autoequivalence.

(a) If P ∈ Spec^{−}(X_{ω}), then Θ(P) ⊆ P ⇔ [Θ(P)] = P ⇔ P ⊆ [Θ(P)].

(b) If P_{⊛} ∈ Spec_{⋊}(X_{ω}), then Θ(P_{⊛}) ⊆ P_{⊛} ⇔ [Θ(P_{⊛})) = P_{⊛} ⇔ P_{⊛} ⊆ [Θ(P_{⊛})).

(a) If P ∈ Spec(X_{ω}), then Θ(P) ⊆ P ⇔ [Θ(P)] = P ⇔ P ⊆ [Θ(P)].

Here [Θ(P)] and [Θ(P_{⊛})) coincide with strictly full subcategories of C_{X} generated by resp. Θ(P) and Θ(P_{⊛}).

Proof. (a) (i) Let Θ(P) ⊆ P. If P ⊈ [Θ(P)], then 𝔥𝔱^{−}([Θ(P)]) < 𝔥𝔱^{−}(P). By A3.9.1, this implies that 𝔥𝔱^{−}([Θ^{∗}Θ(P)]) ≤ 𝔥𝔱^{−}([Θ(P)]) < 𝔥𝔱^{−}(P). But, since Θ^{∗} is a quasi-inverse to Θ, [Θ^{∗}Θ(P)] = P. Therefore P = [Θ(P)].

(ii) The implication P ⊆ [Θ(P)] ⇒ [Θ(P)] = P follows from (i), because the inclusion P ⊆ [Θ(P)] is equivalent to the inclusion Θ^{∗}(P) ⊆ P.

(b) The assertion (b) follows from (a) and the observation that the isomorphism Spec^{−}(X) ⟶^{∼} Spec_{⋊}(X) (cf. 6.4) is compatible with the actions of autoequivalences on resp. Spec^{−}(X) and Spec_{⋊}(X).

(c) The assertion (c) follows from (b) and an observation that the canonical embedding

Spec(X) ⟶ Spec_{⋊}(X), P ⟼ P ∩ ⟨P⟩^{⊥}

is compatible with the actions of autoequivalences on resp. Spec(X) and Spec_{⋊}(X). Details are left to the reader.

A4. Local properties of spectra and closed points.

A4.1. Closed points od spectra and Gabriel-Krull dimension. If X has a Gabriel-Krull dimension, then the set Spec_{𝔱}^{1,1}(X)_{1} of the closed points of Spec_{𝔱}^{1,1}(X) coincides with the set Spec^{−}(X)_{1} of the closed points of Spec^{−}(X). Since in this case Spec^{−}(X) = Spec_{𝔖𝔢}^{1}(X), the spectra Spec_{𝔖𝔢}^{1}(X), Spec^{−}(X), and Spec_{𝔱}^{1,1}(X) have the same sets of closed points.

A4.2. Lemma. Suppose that every nonzero object of C_{X} has a non-empty support in Spec(X) (for instance, C_{X} has enough objects of finite type; cf. A3.1). Then for any closed point P in Spec_{𝔱}^{1,1}(X) and for any thick subcategory 𝕋 of C_{X} such that 𝕋 ⊆ P, the subcategory P/𝕋 of C_{X}/𝕋 is a closed point in Spec_{𝔱}^{1,1}(X/𝕋).

Proof. By 3.2(ii), P = Q̂ for a uniquely determined by this equality element Q of Spec(X), which is a closed point in Spec(X), since P is a closed point in Spec_{𝔱}^{1,1}(X). We claim that image [q^{∗}(Q)] of Q is T (X/𝕋) is a closed point of Spec(X/𝕋).

In fact, let Q′ be a nonzero topologizing subcategory of C_{X}/𝕋 contained in [q^{∗}(Q)]. This means that the preimage Q′′ = q^{∗^{−1}}(Q′) of Q′ in C_{X} is a topologizing subcategory of C_{X} which does not contain 𝕋 and is contained in q^{∗^{−1}}([q^{∗}(Q)]). By 5.1.1, q^{∗^{−1}}([q^{∗}(Q)]) = 𝕋 • Q • 𝕋. Any object M of the subcategory 𝕋 • Q • 𝕋 can be described by the diagram

M′_{1} ⟵ 0 ; ↓ ; 0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0 ; ↓ ; M′′_{1} ⟶ 0  (1)

which incorporates two short exact sequences such that the objects M′_{1} and M_{2}, belong to 𝕋, and M′′_{1} ∈ ObQ. One can see from this description that M is an object of 𝕋 • Q • 𝕋 − 𝕋 iff M′′_{1} is an object of Q − 𝕋. It follows from the diagram (1) that M′′_{1} ∈ Ob[M]. Since 𝕋 ⊉ Q′′ ⊆ 𝕋 • Q • 𝕋, the topologizing subcategory Q′′ ∩ Q is not contained in 𝕋. In particular, it is nonzero. Let M be a nonzero element of Q′′ ∩ Q. By hypothesis, Supp(M) is non-empty; i.e. there exists an element Q̃ of Spec(X) such that Q̃ ⊆ [M]. Thus, we have inclusions Q̃ ⊆ [M] ⊆ Q′′ ∩ Q ⊆ Q. Since Q is a closed point of Spec(X), the inclusion Q̃ ⊆ Q implies that Q̃ = Q. Therefore the inclusions above can be replaced by equalities. In particular, Q′′ ∩ Q = Q, that is Q ⊆ Q′′ which means that Q′ coincides with [q^{∗}(Q)] = (𝕋 • Q • 𝕋)/𝕋.

A4.2.1. Corollary. Suppose that every nonzero object of C_{X} has a non-empty support in Spec(X). Then every closed point of Spec_{𝔱}^{1,1}(X) is a closed point of Spec^{1}(X).

Proof. Let P be a closed point of Spec_{𝔱}^{1,1}(X); and let P_{1} be an element of Spec^{1}(X) such that P_{1} ⊆ P. By A4.2, P/P_{1} is a closed point of Spec_{𝔱}^{1,1}(X/P_{1}). But, X/P_{1} is a local ’space’, hence it has a unique closed point – 0. This shows that P/P_{1} = 0, i.e. P = P_{1}.

A4.3. Proposition. Suppose that C_{X} is an abelian category with the property (sup). Let {T_{i} | i ∈ J} be a finite set of Serre subcategories of C_{X} such that ⋂_{i∈J} T_{i} = 0. Then

(a) A point P of Spec^{−}(X) is closed iff P/T_{i} is a closed point of Spec^{−}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P.

(b) Suppose that every nonzero object of C_{X} has a nonzero support in Spec(X). Then a point P of Spec_{𝔱}^{1,1}(X) is closed iff P/T_{i} is a closed point of Spec_{𝔱}^{1,1}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P.

Proof. (a) If P ∈ Spec^{−}(X), then P/T_{i} ∈ Spec^{−}(X/T_{i}) for all i such that T_{i} ⊆ P. And if P is a closed point, then P/T_{i} is a closed point.

In fact, if T_{i} ⊆ P, then P/T_{i} is an element of Spec^{1}(X/T_{i}); and P/T_{i} is a Serre subcategory of C_{X}/T_{i} (due to the reflectivity of the Serre subcategory T_{i} which is a consequence of the property (sup)). Therefore, it belongs to Spec^{−}(X/T_{i}). If P′ ∈ Spec^{−}(X/T_{i}) and P′ ⊆ P, then the preimage P′′ of P′ in C_{X} is a Serre subcategory which belongs to Spec^{−}(X) and is contained in P. Thus, if P is a closed point of Spec^{−}(X/T_{i}), then P′′ = P, hence P′ = P/T_{i}.

(a1) Conversely, let P/T_{i} be closed for all i ∈ J such that T_{i} ⊆ P. Then we claim that P is closed. If the number Card(J) = 1, then the statement is true by a trivial reason. In the general case, let P′ be an element of Spec^{−}(X) such that P′ ⊆ P. And let J^{P′} denote the set {i ∈ J | T_{i} ⊈ P′}. Since J is finite, by 9.3, there exists i ∈ J such that T_{i} ⊆ P′. Therefore Card(J^{P′}) < Card(J). By (the end of the argument of) A1.4.1(c) (or [R4, 4.2]),

P′ = (⋂_{i∈J} T_{i}) ∨ P′ = ⋂_{i∈J}(T_{i} ∨ P′) = ⋂_{i∈J^{P′}}(T_{i} ∨ P′).

So that {T′_{i} = (T_{i} ∨ P′)/P′, i ∈ J^{P′}} is a set of Serre subcategories of C_{X}/P′ whose intersection is zero. The point P̃ = P/P′ of Spec^{−}(X/P′) is such that that P̃/T′_{i} is a closed point of Spec^{−}(X/(T_{i} ∨P′) for all i ∈ J^{P′} such that T′_{i} ⊆ P̃. Since Card(J^{P′}) < Card(J), by induction hypothesis, P̃ is a closed point of X/P′. The latter ’space’ being local, this means that P̃ = 0, or, equivalently, P = P′.

(b) If P is a closed point of Spec_{𝔱}^{1,1}(X), then, by A4.2, P/T_{i} is a closed point of Spec_{𝔱}^{1,1}(X/T_{i}) for every i ∈ J such that T_{i} ⊆ P.

Conversely, suppose that P ∈ Spec_{𝔱}^{1,1}(X) is such that P/T_{i} is a closed point of the spectrum Spec_{𝔱}^{1,1}(X/T_{i}) if T_{i} ⊆ P. Let P′ be an element of Spec_{𝔱}^{1,1}(X) such that P′ ⊆ P. By 9.3, there exists i ∈ J such that T_{i} ⊆ P′; in particular, T_{i} ⊆ P. Since P′/T_{i} is a point of Spec_{𝔱}^{1,1}(X/T_{i}) and P/T_{i} is a closed point, the inclusion P′/T_{i} ⊆ P/T_{i} implies that P′/T_{i} and P/T_{i} coincide. Therefore, P′ = P.

A4.3.1. Corollary. Suppose that C_{X} is an abelian category with the property (sup). Let {T_{i} | i ∈ J} be a finite set of Serre subcategories such that ⋂_{i∈J} T_{i} = 0 and for every

i ∈ J, any element of Spec^{−}(X/T_{i}) contains a closed point of Spec^{−}(X/T_{i}). Then every element of Spec^{−}(X) contains a closed point of Spec^{−}(X).

Proof. Let P ∈ Spec^{−}(X). Since ⋂_{i∈J} T_{i} = 0, there exists J_{P} = {i ∈ J | T_{i} ⊆ P}

is non-empty. Fix an i ∈ J_{P}. By hypothesis, P_{i} ⊆ P, where P_{i}/T_{i} is a closed point of Spec^{−}(X/T_{i}). If J_{P_{i}} = {i}, then, by A4.3(a), P_{i} is a closed point of Spec^{−}(X). If J_{P_{i}} contains more than one element, we take j ∈ J_{P_{i}} − {i} and repeat the argument replacing P by P_{i}; and so on. Since J is finite, the process stabilizes. As a result, we find an element P′ of Spec^{−}(X) such that P′ ⊆ P and P′/T_{j} is a closed point of Spec^{−}(X/T_{j} for every j ∈ J_{P′}. By A4.3(a), the latter means that P′ is a closed point of Spec^{−}(X).

A4.3.2. Corollary. Suppose that C_{X} is an abelian category with the property (sup). Let {T_{i} | i ∈ J} be a finite set of Serre subcategories such that ⋂_{i∈J} T_{i} = 0 and for every

i ∈ J, the set Spec_{𝔱}^{1,1}(X/T_{i})_{1} of the closed points of Spec_{𝔱}^{1,1}(X/T_{i}) contains the set Spec^{−}(X/T_{i})_{1} of the closed points of Spec^{−}(X/T_{i}). Then Spec^{−}(X)_{1} ⊆ Spec_{𝔱}^{1,1}(X)_{1}.

Suppose that, in addition, one of the following conditions holds:

(a) For all i ∈ J, every element of Spec^{−}(X/T_{i}) contains a closed point.

(b) Every nonzero object of C_{X} has a non-empty support in Spec(X). Then Spec^{−}(X)_{1} and Spec_{𝔱}^{1,1}(X)_{1} coincide.

Proof. Let P be a closed point of Spec^{−}(X). By A4.3, P/T_{i} is a closed point of Spec^{−}(X/T_{i}) for all i ∈ J_{P} = {j ∈ J | T_{j} ⊆ P}. By hypothesis, P/T_{i} is a closed point of Spec_{𝔱}^{1,1}(X/T_{i}) for all i ∈ J_{P}. By 9.6.1, P ∈ Spec_{𝔱}^{1,1}(X). Since P is a closed point of the space Spec^{−}(X), it is, definitely, a closed point of its subspace Spec_{𝔱}^{1,1}(X). This shows the inclusion Spec^{−}(X)_{1} ⊆ Spec_{𝔱}^{1,1}(X)_{1}.

(a) Let P ∈ Spec_{𝔱}^{1,1}(X). Since P is an element of Spec^{−}(X), by A4.3.1, P ⊇ P′, where P′ is a closed point of Spec^{−}(X). By A4.3(a), for every i ∈ J_{P′} = {j ∈ J | T_{j} ⊆ P′}, the quotient subcategory P′/T_{i} is a closed point of Spec^{−}(X/T_{i}), hence, by hypothesis, it belongs to Spec_{𝔱}^{1,1}(X/T_{i}). By 9.6.1, the latter implies that P′ belongs to Spec_{𝔱}^{1,1}(X). Since it P′ is a closed point of Spec^{−}(X), it is a closed point of Spec_{𝔱}^{1,1}(X).

(b) If every nonzero object of C_{X} has a non-empty support, then, by A4.2.1, we have the inverse inclusion: Spec_{𝔱}^{1,1}(X)_{1} ⊆ Spec^{−}(X)_{1}.

A4.4. Proposition. Let C_{X} be an abelian category and {T_{i} | i ∈ J} a finite set of thick subcategories such that ⋂_{i∈J} T_{i} = 0. Suppose that each category C_{X/T_{i}} has enough ob-

jects of finite type. Then closed points of Spec(X) are in a natural bijective correspondence with the isomophism classes of simple objects of C_{X}.

Proof. Let u_{i}^{∗} denote the localization functor C_{X} ⟶ C_{X}/T_{i}. Let M be an object of Spec(X) such that [M] be a closed point of Spec(X). Since ⋂_{i∈J} T_{i} = 0, there is an i ∈ J

such that M ∉ T_{i}. Therefore, [u_{i}^{∗}(M)] is a closed point of Spec(X/T_{i}). Since the category C_{X/T_{i}} = C_{X}/T_{i} has enough objects of finite type, all closed points of Spec(X/T_{i}) correspond to simple objects. In particular, u_{i}^{∗}(M) is the direct sum of a finite number of copies of a simple object u_{i}^{∗}(L), and there is a monomorphism u_{i}^{∗}(L) ⟶ u_{i}^{∗}(M). This monomor-

phism is described by a diagram L ⟵^{s} L′ ⟶^{𝔧′} M such that u_{i}^{∗}(s) is an isomorphism and Ker(𝔧′) belongs to T_{i}. Since the object M is T_{i}-torsion free, the object L_{i} = L′/Ker(𝔧′) is T_{i}-torsion free too. It follows that u_{i}^{∗}(L_{i}) is isomorphic to u_{i}^{∗}(L). In particular, L_{i} is a nonzero subobject of M. Since M ∈ Spec(X), the object L_{i} also belongs to Spec(X) and [L_{i}] = [M]. So, we replace M by L_{i}. Repeating this procedure consecutively for all j ∈ J such that M does not belong to T_{j}, we replace M by its subobject, N such that for any j ∈ J, the object u_{j}^{∗}(N) is either zero, or simple. Since N belongs to Spec(X), it follows

that for every nonzero monomorphism N′ ⟶^{h} N, its image u_{i}^{∗}(h) is an isomorphism for every i ∈ J. The condition ⋂_{i∈J} T_{i} = 0 means that the family of localization functors

{C_{X} ⟶^{u_{i}^{∗}} C_{X}/T_{i} | i ∈ J} is conservative; hence h is an isomorphism. This shows that N is a simple object. Therefore, M is isomorphic to the coproduct of a finite number of copies of N.

The following proposition is a refinement of 1.6.2.

A4.5. Proposition. Suppose that C_{X} is an abelian category with the property (sup). Let {T_{i} | i ∈ J} be a finite set of Serre subcategories of C_{X} such that ⋂_{i∈J} T_{i} = 0, and for

every i ∈ J, the category C_{X/T_{i}} has enough objects of finite type. Then

(a) The intersection Spec(X) ⋂ Spec(X^{o}) coincides with the set Spec(X)_{1} of closed points of Spec(X), and closed points of Spec(X) are of the form [M], where M runs through simple objects of C_{X}.

(b) Closed points of Spec^{−}(X)) are in bijective correspondence with the isomorphism classes of simple objects of C_{X}.

Proof. (a) By A4.4, closed points of Spec(X) are of the form [M], where M runs through simple objects of C_{X}. Since simple objects of C_{X} and C_{X}^{op} = C_{X^{o}} are the same, the set Spec(X)_{1} of closed points of Spec(X) is contained in Spec(X) ∩ Spec(X^{o}).

(a1) Let 𝔘 denote the finite cover {U_{i} = X/T_{i} ⟶^{u_{i}} X | i ∈ J} associated with {T_{i} | i ∈ J}. And let Spec_{℘}^{1,1}(𝔘) = {P ∈ 𝔗𝔥(X) | P/T_{i} ∈ Spec_{𝔱}^{1,1}(U_{i}) if T_{i} ⊆ P}. By 9.6.1, the natural map Spec_{𝔱}^{1,1}(X) ⟶ Spec_{℘}^{1,1}(𝔘) is an isomorphism. This isomorphism and the embedding Spec_{𝔱}^{1,1}(X^{o}) ⟶ Spec_{℘}^{1,1}(𝔘^{o}) induce an injective map

Spec_{𝔱}^{1,1}(X) ⋂ Spec_{𝔱}^{1,1}(X^{o}) ⟶ Spec_{℘}^{1,1}(𝔘) ⋂ Spec_{℘}^{1,1}(𝔘^{o}),  (2)

where

Spec_{℘}^{1,1}(𝔘) ⋂ Spec_{℘}^{1,1}(𝔘^{o}) =
{P ∈ Spec^{−}(X) | P/T_{i} ∈ Spec_{𝔱}^{1,1}(U_{i}) ⋂ Spec_{𝔱}^{1,1}(U_{i}^{o}) if T_{i} ⊆ P_{i}}.

Since each category C_{U_{i}} = C_{X}/T_{i}, i ∈ J, has enough objects of finite type, it follows from 1.6.2 and the isomorphism Spec(U_{i}) ⟶^{∼} Spec_{𝔱}^{1,1}(U_{i}) (see 3.2(ii)) that the intersection Spec_{𝔱}^{1,1}(U_{i}) ⋂ Spec_{𝔱}^{1,1}(U_{i}^{o}) coincides with the set of closed points of Spec_{𝔱}^{1,1}(U_{i}) and these closed points are in bijective correspondence with isomorphism classes of simple objects of the category C_{U_{i}}. It follows now from (the argument of) A4.4 and the isomorphism Spec(X) ⟶^{∼} Spec_{𝔱}^{1,1}(X) (see 3.2(ii)) that the map (2) above is bijective.

(b) Notice that the conditions of this proposition imply the conditions (a) and (b) of A4.3.2. In particular, by A4.3.2, the spectra Spec^{−}(X) and Spec_{𝔱}^{1,1}(X) have the same closed points. The assertion follows this fact and from (a) above.

A4.6. Semilocal ’spaces’.

A4.6.1. Proposition. Suppose that there is a finite subset {P_{i} | i ∈ J} of Spec^{−}(X) such that ⋂_{i∈J} P_{i} = 0. Then P ∈ Spec^{−}(X) is a closed point iff it is a closed point of

Spec_{𝔱}^{1,1}(X), i.e. it is of the form P = ⟨L⟩ for an object L of Spec(X).

The set of closed points of Spec^{−}(X) coincides with the set of minimal elements of {P_{i} | i ∈ J}.

Proof. Let P be a closed point of Spec^{−}(X)). By 9.3, the set J_{P} = {i ∈ J | P_{i} ⊆ P} is not empty. Since P is a minimal element of Spec^{−}(X), the set J_{P} consists of all i ∈ J such that P_{i} = P. Thus, P/P_{i} is the zero subcategory of C_{X}/P_{i} which is the only closed point of the local space X/P_{i} = X/P. By 9.6.1, P is an element of Spec_{𝔱}^{1,1}(X). Since Spec_{𝔱}^{1,1}(X) is a subset of the spectrum Spec^{−}(X) and P is a closed point of the latter, it is a closed point of Spec_{𝔱}^{1,1}(X).

This argument shows that the set of closed points of Spec^{−}(X) is a subset of the set of minimal elements of {P_{i} | i ∈ J}, and that it is a subset of closed point of Spec_{𝔱}^{1,1}(X).

Notice that every minimal element of {P_{i} | i ∈ J} is a closed point of Spec^{−}(X).

In fact, let P_{j} be a minimal element of the set {P_{i} | i ∈ J}, and let P′ ∈ Spec^{−}(X) be a subcategory of P_{j}. The set J_{P′} = {i ∈ J | P_{i} ⊆ P′} is non-empty, and P_{m} ⊆ P′ ⊆ P_{j} for every m ∈ J_{P′}. Since P_{j} is a minimal element, this implies that P_{m} = P′ = P_{j}.

Let P ∈ Spec_{𝔱}^{1,1}(X). The set J_{P} = {i ∈ J | P_{i} ⊆ P} is not empty. Let P_{i} be a minimal element of {P_{j} | j ∈ J_{P}}. Then, by the argument above, P_{i} is a closed point of Spec^{−}(X), hence it is a closed point of Spec_{𝔱}^{1,1}(X) which is contained in P. Thus, if P is a closed point of Spec_{𝔱}^{1,1}(X), then P_{i} = P.

A4.6.2. Corollary. Let C_{X} be an abelian category. The following conditions are equivalent:

(a) There is a finite subset {P_{i} | i ∈ J} of Spec^{−}(X) such that ⋂_{i∈J} P_{i} = 0.

(b) The set Spec^{−}(X)_{1} of closed points of Spec^{−}(X) is finite, and the intersection ⋂_{P∈Spec^{−}(X)_{1}} P is zero.

(c) The set Spec^{−}(X)_{1} is finite, and the support in Spec^{−}(X) of any nonzero object of C_{X} contains a closed point.

(d) The set Spec_{𝔱}^{1,1}(X)_{1} of closed points of Spec_{𝔱}^{1,1}(X) is finite, and the support in Spec(X) of every nonzero object of C_{X} contains a closed point.

Proof. Obviously, (b) ⇒ (a). The implication (a) ⇒ (b) follows from A4.6.1.

(b) ⇔ (c). If ⋂_{P∈Spec^{−}(X)_{1}} P = 0, then for every nonzero object M of C_{X}, there

exists a closed point P of Spec^{−}(X) such that M ∉ ObP, which means precisely that P ∈ Supp^{−}(M). Conversely, if every nonzero object of C_{X} has an element of Spec^{−}(X)_{1} in its support, then ⋂_{P∈Spec^{−}(X)_{1}} P = 0.

(d) ⇒ (a). The support in Spec(X) of a nonzero object M contains a point Q, that is Q ⊆ [M], means precisely that [M] ⊈ Q̂, or, equivalently, M ∉ ObQ̂. By 3.2(ii), Q is a closed point of Spec(X) iff Q̂ ∈ Spec_{𝔱}^{1,1}(X)_{1}. Therefore, the condition (d) implies that

⋂_{P∈Spec_{𝔱}^{1,1}(X)_{1}} P = 0.

The implication (a) ⇒ (b) follows from A4.6.1.

A4.6.3. Definition. Let C_{X} be an abelian category. We call the ’space’ X semilocal if the equivalent conditions of A4.6.2 hold.

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