Spectra related with localizations.

Alexander Rosenberg

Introduction

Categorical realization of non-affine varieties was started by J.-P. Serre in 1955, even before Grothendieck developed scheme theory, with a description of the category of quasicoherent sheaves on a projective variety [Fac]. About thirty years later, Yu. I. Manin proposed to use Serre’s description as a definition of the category of quasi-coherent sheaves on P roj(R), where R is a noncommutative ℤ_{+}-graded ring [M1]. The proposal triggered an active study of noncommutative P roj (starting with [A], [AB], [AZ], [V1], [V2]) and largerly contributed to general philosophy according to which noncommutative spaces are represented by categories.

This philosophy was supported, at least in the case of noetherian schemes, by a spectral theory of abelian categories due to P. Gabriel [Gab]. Points of the Gabriel’s spectrum are isomorphism classes of indecomposable injectives. If A is the category of modules over a commutative noetherian ring, then isomorphism classes of indecomposable injectives are in bijective correspondence with prime ideals of this ring. Injective objects are naturally related with localizations: there are localizations at points of Gabriel’s spectrum and at sets of points. Gabriel used the injective spectrum for reconstruction of an arbitrary noetherian scheme (in particular, the P roj of a noetherian commutative graded ring) from its category of quasi-coherent sheaves.

A different spectrum of an abelian category, which does not rely on noetherian hypothesis, was introduced in [R1] (see also [R, Ch.3]). This spectrum is, in general, much smaller than the Gabriel’s spectrum, at the same time powerful enough to reconstruct the prime spectrum of an arbitrary commutative ring from the category of modules over this ring, or, more generally, a quasi-compact (not necessarily noetherian) scheme from its category of quasi-coherent sheaves [R4]. If an abelian category has enough objects of finite type (like the category of modules over an associative ring), then closed points of this spectrum are precisely isomorphism classes of simple objects. The spectrum can be realized as a certain family of Serre subcategories; so that there are localizations at points and at sets of points.

In this work, we introduce spectra directly related with localizations. Remarkably, they make sense for non-abelian categories and triangulated categories. A considerable part of the paper, however, is dedicated to the studying these spectra in the abelian case, where they are also new, and to the comparison with the already known spectra. In particular, we sketch briefly facts on associated points and prime decomposition showing that our spectral theory is capable to support a meaningful categorical version of local algebra.

The first part of this work can be regarded as an introduction to basic spectra associated with exact localizations of general ’spaces’, i.e. ’spaces’ represented by arbitrary categories regarded as categories of quasi-coherent sheaves.

Section 1 contains preliminaries on localizations and multiplicative systems, and the beginning of the dictionary of categorical pseudo-geometry: ’spaces’ represented by categories and morphisms represented by (their inverse image) functors. This formal duality between ’spaces’ and categories is reflected in notaions: a category C_{X} represents the ’space’ X and a functor C_{Y} ⟶^{f^{∗}} C_{X} represents a morphism X ⟶^{f} Y.

In Section 2, we introduce the spectrum of exact localizations, or, shortly, the 𝔏spectrum, of a ’space’ and discuss its functorial properties.

In Section 3, we define 𝔏-local ’spaces’ and show that the localization at a ’point’ of the 𝔏-spectrum is an 𝔏-local ’space’.

In Section 4, we introduce the complete 𝔏-spectrum and show its functoriality with respect to exact localizations.

In Section 5, we define the closed spectrum (resp. the complete closed spectrum) and the flat spectrum (resp. the complete flat spectrum) of a ’space’.

In Section C1 (here ’C’ stands for ’complemetary facts’), we introduce cosubspaces and coimmertions. These notions are mainly used in the case of ’spaces’ represented by abelian categories.

The second part is dedicated to the spectra of ’spaces’ represented by abelian categories. In Section 6.1, we translate the content of Sections 2, 3, and 4 to the language of thick subcategories (using a natural isomorphism between the preorder of thick subcategories of an abelian category and the preorder of its saturated multiplicative systems). In Section 6.2, we notice that this isomorphism induces an isomorphism between the preorder of left closed multiplicative systems and coreflective thick subcategories which leads to the description of the closed spectrum (introduced in Section 4) in terms of coreflective thick subcategories. In Section 6.3, we remind the notion of a Serre subcategory (as it is defined in [R1]) and introduce the corresponding spectra, Spec_{𝔰}^{0}(X) and Spec_{𝔰}^{1}(X). If C_{X} is a Grothendieck category, then Serre subcategories are precisely coreflective thick subcategories and the spectrum Spec_{𝔰}^{0}(X) is isomorphic to the closed spectrum.

In Section 6.4, we establish, among other facts, the functoriality of spectra with respect to coimmersions.

In Section 6.5, we study 𝔏-local ’spaces’ represented by abelian categories. We show that if X is 𝔏-local, then all simple objects of the category C_{X} (if any) are isomorphic to each other. In the case when C_{X} does have simple objects, this fact allows to associate with the local ’space’ X a skew field, k(X), defined uniquely up to isomorphism, which we call the residue skew field of X.

In Sections 7, we study spectra associated with the preorder of topologizing subcategories of an abelian category C_{X} and recover the spectrum, Spec(X), defined in [R1] (see also [R, Ch.3]) and mentioned above.

In Section 8, we define local ’spaces’ as ’spaces’ X such that the category C_{X} has the smallest non-trivial topologizing subcategory. This notion appeares in a different disguise in [R1]. Thick subcategories of a category C_{X} such that the corresponding quotient categories are local are points of the so called complete spectrum of X introduced in [R, Ch.6]. Taking the intersection of the complete spectrum with Serre subcategories, we obtain the spectrum Spec^{−}(X) which, also, made its first appearance in [R, Ch.6].

In Section 8.7, we show that if C_{X} is an abelian category with a Gabriel-Krull dimension, then the spectrum Spec^{−}(X) coincides with the Serre spectrum Spec_{𝔰}^{1}(X) (defined in 6.3). In general, the latter is much larger.

In Section C2, the spectrum Spec_{𝔰}^{0}(X) is extended to general ’spaces’.

The third part contains a fragment of noncommutative (categorical) local algebra. The results obtained here are not used in the rest of the work.

Section 9 treats the case of general ’spaces’: we introduce and study supports and associated points of a class of arrows of an arbitrary category.

In Section 10, we apply and extend the notions and facts of Section 9 to studying supports and associated points of objects of an abelian category.

The main objective of the last part of the work is to define meaningful spectra of a (’space’ represented by a) triangulated category.

In Section 11, we extend the notions of the spectra to the case of categories with an action of a monoidal category. This material, important by itself, is used futher only in the simplest case of so called ℤ-categories, in order to give a background to spectral theory of triangulated categories.

Section 12 is dedicated to the spectra of triangulated categories. We follow the pattern of the corresponding constructions for abelian categories replacing thick subcategories by thick triangulated subcategories.

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

I. General ’spaces’ and their spectra.

1. Preliminaries on ’spaces’ and localizations.

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

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

The map which assigns to any functor C_{Y} ⟶^{F} C_{X} the morphism X ⟶^{[F]} Y is a functor Cat^{op} ⟶ |Cat|^{o} which turns Cat^{op} into a fibered category over |Cat|^{o}.

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

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

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

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

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

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

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

For a general morphism X ⟶^{f} Y, it follows from the universal property of localizations that f = p_{f} ∘ f_{𝔠} is a decomposition of f into a localization and a conservative morphism.

1.3. Multiplicative systems. A family of arrows Σ of a category C_{X} is called a left multiplicative system if it has the following properties:

(S1) Σ is closed under composition and contains all identical arrows of C_{X}.

(SL2) Every diagram M′ ⟵^{s} M ⟶^{f} L, where s ∈ Σ, can be completed to a commutative square

M ⟶^{f} L
s ↓  ↓ s′
M′ ⟶^{f′} L′

where s′ ∈ Σ.

(SL3) If M \overrightarrow{⟶}^{f}_{g} N is a pair of arrows such that f ∘ s = g ∘ s for some s ∈ Σ, then

there exists a morphism N ⟶^{t} N′ of Σ such that t ∘ f = t ∘ g.

A family Σ ⊆ HomC_{X} is a right multiplicative system if it has dual properties. Finally, Σ is called a multiplicative system if it is both right and left multiplicative.

We denote by SM_{ℓ}(X) (resp. by SM_{𝔯}(X)) the family of all left (resp. right) multiplicative systems in C_{X}. We denote by SM(X) the family SM_{ℓ}(X) ⋂ SM_{𝔯}(X) of all multiplicative systems in C_{X}.

We regard SM_{ℓ}(X), SM_{𝔯}(X), and SM(X) as preorders with respect to ⊆.

1.3.1. Saturation. Let Σ be a family of morphisms of the category C_{X}. Let q_{Σ}

be the localization morphism Σ^{−1}X ⟶ X and C_{X} ⟶^{q_{Σ}^{∗}} C_{Σ^{−1}X} = Σ^{−1}C_{X} its canonical inverse image functor.

The family Σ^{𝔰} = Σ_{q_{Σ}} of all arrows of C_{X} which q∗_{Σ} transfers into isomorphisms (cf. 1.2) is called the saturation of Σ. A family of arrows Σ is called saturated if it coincides with its saturation.

1.3.2. Generalities on saturated families of arrows. It follows from the universal property of localizations, that for any morphism Y ⟶^{f} X, the family Σ_{f} of all arrows of C_{X} which f^{∗} transforms to isomorphisms (see 1.2) is saturated. In particular, the saturation of any family of arrows is saturated.

Any set, {Y_{i} ⟶^{f_{i}} X | i ∈ J}, of morphisms of ’spaces’ defines uniquely a morphism Y = ∐_{i∈J} Y_{i} ⟶^{f} X with an inverse image

C_{X} ⟶^{f^{∗}} C_{Y} = ∏_{i∈J} C_{Y_{i}}

uniquely determined by a choice of inverse images, C_{X} ⟶^{f_{i}^{∗}} C_{Y_{i}}, of morphisms f_{i}, i ∈ J. Evidently, Σ_{f} = ⋂_{i∈J} Σ_{f_{i}}. This shows that the intersection of any set of saturated families

of morphisms is saturated.

1.3.3. Saturation of multiplicative systems. If Σ is a left multiplicative system, then its saturation, Σ^{𝔰}, consists of all morphisms L ⟶^{u} M which can be inserted in a commutative diagram of the form

L ⟶^{u} M

s ↓ ↙_{v} ↓ t

V ⟶^{u_{1}} W

where s, t ∈ Σ (see [GZ, 1.1.3.5]).

It follows from this description that the saturation of a (left and right) multiplicative system Σ coincides with all arrows s ∈ HomC_{X} such that there exist morphisms 𝔲 and 𝔳 such that 𝔲 ∘ s ∈ Σ ∋ s ∘ 𝔳.

1.3.3.1. Proposition. The saturation of a (left and right) multiplicative system is a multiplicative system.

Proof. Let Σ be a multiplicative system. It suffices to show that the saturation, Σ^{𝔰}, of Σ has the properties (SL2) and (SL3).

Let M ⟶^{s} M′ be an element of Σ^{𝔰}; i.e. there exist morphisms M′ ⟶^{𝔲} M′′ and N ⟶^{𝔳} M such that 𝔲 ∘ s ∈ Σ ∋ s ∘ 𝔳. And let M ⟶^{f} L be an arbitrary morphism.

By the property (SL3), the diagram M′′ ⟵^{𝔲∘s} M ⟶^{f} L can be inserted in a commutative diagram

\begin{matrix}M & ⟶^{f} & L \\ 𝔲 ∘ s ↓ & & ↓ s′ \\ M′′ & ⟶^{f′} & L′ \end{matrix}  (1)

where s′ ∈ Σ. The diagram (1) can be rewritten as

M ⟶^{f} L
s ↓  ↓ s′
M′ ⟶^{f′∘𝔲} L′

which proves (SL2).

Let M′ \overrightarrow{⟶}^{f}_{g} L is a pair of arrows such that f ∘ s = g ∘ s. In particular, f ∘ (s ∘ 𝔳) =

g ∘ (s ∘ 𝔳). Since s ∘ 𝔳 ∈ Σ, there exists (by the property (SL3)) a morphism L ⟶^{t} L′ of Σ such that t ∘ f = t ∘ g.

1.3.3.2. Note. The analogous assertion is not true, in general, for left (or right) multiplicative systems. It is true, however, if the category C_{X} has finite colimits (finite limits in the case of right multiplicative systems); see 1.4.1(b) and 1.4.2 below.

1.3.4. Notations. We denote by S^{𝔰}M_{ℓ}(X) (resp. by S^{𝔰}M_{𝔯}(X)) the family of all saturated left (resp. right) multiplicative systems in C_{X}.

We denote by S^{𝔰}M(X) the family of all saturated (left and right) multiplicative systems in C_{X}; that is S^{𝔰}M(X) = S^{𝔰}M_{ℓ}(X) ⋂ S^{𝔰}M_{𝔯}(X).

We regard S^{𝔰}M_{ℓ}(X), S^{𝔰}M_{𝔯}(X), and S^{𝔰}M(X) as preorders with respect to ⊆.

It follows from 1.3.3.1 that the saturation, Σ ⟼ Σ^{𝔰} induces a functor

SM(X) ⟶ S^{𝔰}M(X)

which is left adjoint to the inclusion functor S^{𝔰}M(X) ⟶ SM(X).

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

The following assertion is a reformulation of Propositions 1.3.1 and 1.3.4 in [GZ].

1.4.1. Proposition. (a) Let Σ be a left multiplicative system in C_{X}. Then the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is right exact.

(b) Let f = p_{f} ∘ f_{𝔠} be the canonical decomposition of a morphism X ⟶^{f} Y into a

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

In particular, if the category C_{Y} has limits and colimits of finite diagrams, then f is exact iff both the localization p_{f} and the conservative component f_{𝔠} are exact. The exactness of p_{f} is equivalent to that Σ_{f} ∈ S^{𝔰}M(X).

1.4.2. Corollary. Suppose the category C_{X} has finite colimits. Then the saturation map, Σ ⟼ Σ^{𝔰} induces a functor SM_{ℓ}(X) ⟶ S^{𝔰}M_{ℓ}(X) which is left adjoint to the corresponding inclusion functor S^{𝔰}M_{ℓ}(X) ⟶ SM_{ℓ}(X).

1.4.3. Corollary. Suppose C_{X} has finite colimits. Then the intersection of any set of saturated left multiplicative systems is a saturated left multiplicative system.

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

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

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

2. The 𝔏-spectrum.

Fix a ’space’ X. Recall that S^{𝔰}M(X) denote the preorder (with resp. to ⊆) of all saturated (left and right) multiplicative systems of the category C_{X}. The preorder S^{𝔰}M(X) has the initial object – the family Iso(C_{X}) of all isomorphisms of C_{X}. Let S^{𝔰}M^{⋆}(X) denote S^{𝔰}M(X) − {Iso(C_{X})}.

For any Σ ⊆ HomC_{X}, denote by Σ̂ the union of all saturated multiplicative systems of C_{X} which do not contain Σ. It follows that if Σ_{1} ⊆ Σ_{2}, then Σ̂_{2} ⊆ Σ̂_{1}. Notice that if Σ_{1} and Σ_{2} are saturated multiplicative systems, then the inverse implication holds, i.e. Σ_{1} ⊆ Σ_{2} iff Σ̂_{2} ⊆ Σ̂_{1}.

2.1. Definition. The 𝔏-spectrum, Spec_{𝔏}^{0}(X), of X consists of all saturated multiplicative systems Σ such that Σ̂ is a saturated multiplicative system.

In other words, elements of Spec_{𝔏}^{0}(X), are saturated multiplicative systems Σ such that there exists the biggest saturated multiplicative system, Σ̂, which does not contain Σ. In particular, Spec_{𝔏}^{0}(X) ⊆ S^{𝔰}M^{⋆}(X).

2.1.1. Note. If C_{X} is a groupoid, then S^{𝔰}M^{⋆}(X) is empty, hence Spec_{𝔏}^{0}(X) = ∅.

2.1.2. Specialization preorder. We call the preorder, ⊇, on S^{𝔰}M(X) the specialization preorder: Σ is a specialization of Σ′ if Σ ⊆ Σ′.

It follows that if Σ, Σ′ are elements of Spec_{𝔏}^{0}(X), then Σ is a specialication of Σ′ iff the saturated multiplicative system Σ̂ is a specialization of Σ̂′.

2.2. Functorial properties of the 𝔏-spectrum. Let 𝔏_{𝔢}𝔈𝔰𝔭 denote the subcategory of |Cat|^{o} formed by exact localizations (cf. 1.2). Since identical morphisms are exact localizations, Ob𝔏_{𝔢}𝔈𝔰𝔭 = Ob|Cat|^{o}. Let 𝔓𝔒𝔯𝔡_{⋆} denote the category of preorders with initial objects; its morphisms are morphisms of preorders mapping initial objects to initial objects.

2.2.1. Lemma. The map X ⟼ S^{𝔰}M(X) gives a rise to a contravariant functor

S^{𝔰}M_{𝔬} : 𝔏_{𝔢}𝔈𝔰𝔭^{op} ⟶ 𝔓𝔒𝔯𝔡_{⋆}

and to a covariant functor

S^{𝔰}M : 𝔏_{𝔢}𝔈𝔰𝔭 ⟶ 𝔓𝔒𝔯𝔡_{⋆}.

Proof. Let X ⟶^{u} Y be an exact localization and C_{Y} ⟶^{u^{∗}} C_{X} its inverse image functor. Set Σ_{u} = Σ_{u^{∗}} = {s ∈ HomC_{Y} | u^{∗}(s) ∈ IsoC_{X}}. The functor u^{∗} induces a map

S^{𝔰}M(Y) ⟶^{S^{𝔰}M_{𝔬}(u)} S^{𝔰}M(X)

which assigns to a family Σ ∈ S^{𝔰}M(Y) the minimal saturated multiplicative system containing u^{∗}(Σ), and a map

S^{𝔰}M(X) ⟶^{S^{𝔰}M(u)} S^{𝔰}M(Y)

which sends any saturated multiplicative system Σ′ to its preimage, u^{∗^{−1}}(Σ′). Notice that S^{𝔰}M_{𝔬}(u)∘S^{𝔰}M(u) is the identical map. This shows that S^{𝔰}M_{𝔬}(u) and S^{𝔰}M(u) induce an isomorphism between S^{𝔰}M(X) and the preorder S^{𝔰}M_{Σ_{u}}(Y) of saturated multiplicative systems of C_{Y} containing Σ_{u}. Notice that the map S^{𝔰}M_{𝔬}(u) can be represented as the composition of the map

S^{𝔰}M(Y) ⟶ S^{𝔰}M_{Σ_{u}}(Y), Σ ⟼ Σ ∨ Σ_{u},

and the restriction of S^{𝔰}M_{𝔬}(u) to S^{𝔰}M_{Σ_{u}}(Y) (the inverse to S^{𝔰}M(u)). It is easy to see that both maps, u ⟼ S^{𝔰}M_{𝔬}(u) and u ⟼ S^{𝔰}M(u) are functorial.

2.2.2. Extended 𝔏-spectrum. For any ’space’ X, set Spec_{𝔏⋆}^{0}(X) = Spec_{𝔏}^{0}(X) ∪ {⋆_{X}}, where ⋆_{X} = Iso(C_{X}). We call Spec_{𝔏⋆}^{0}(X) the extended spectrum of X. Notice that { \atop Iso(C_{X})} = ∅. Thus, the added trivial multiplicative system ⋆_{X} can be viewed as ∞ (with respect to the specialization preorder ⊇).

2.2.3. Proposition. Any exact localization X ⟶^{u} Y induces a morphism of extended spectra Spec_{𝔏⋆}^{0}(Y) ⟶ Spec_{𝔏⋆}^{0}(X). This correspondence defines a contravariant functor, Spec_{𝔏⋆}^{0}, from the category 𝔏_{𝔢}𝔈𝔰𝔭 to the category 𝔓𝔒𝔯𝔡_{⋆} of preorders with initial objects.

Proof. Fix an inverse image functor, C_{Y} ⟶^{u^{∗}} C_{X}, of the morphism u. The map

S^{𝔰}M_{𝔬}(u) : S^{𝔰}M(Y) ⟶ S^{𝔰}M(X)

(cf. 2.2.1) induces a morphism of spectra Spec_{𝔏⋆}^{0}(Y) ⟶ Spec_{𝔏⋆}^{0}(X).

In fact, let Σ_{P} ∈ Spec_{𝔏}^{0}Y and Σ_{P} ⊈ Σ_{u}. Then Σ_{u} ⊆ Σ̂_{P}. By (the argument of) 2.2.1, the map S^{𝔰}M_{𝔬}(u) induces an isomorphism between S^{𝔰}M(X) and the preorder S^{𝔰}M_{Σ_{u}}(Y) of saturated multiplicative systems of C_{Y} containing Σ_{u}. In particular, the image, Σ̂′_{P}, of Σ̂_{P} is a saturated multiplicative system. It follows that Σ̂′_{P} is the biggest saturated multiplicative system in C_{X} which does not contain the image, Σ′_{P}, of the saturated multiplicative system Σ_{u} ∨ Σ_{P}. In fact, if Σ′_{P} ⊈ Σ′ for some saturated multiplicative system Σ′, then Σ_{P} is not contained in the saturated multiplicative system Σ = u^{∗^{−1}}(Σ′). Therefore Σ ⊆ Σ̂_{P}, hence the assertion.

2.2.4. Remarks. (a) For any S ∈ S^{𝔰}M(X), let U_{𝔏}(S) denote {Σ ∈ Spec_{𝔏}^{0}(X) | Σ ⊈ S} It follows that U_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | S ⊆ Σ̂}.

The argument of 2.2.3 proves that any exact localization, X ⟶^{u} Y, induces an injective map U_{𝔏}(Σ_{u}) ⟶ Spec_{𝔏}^{0}Y.

(b) In general, the map S^{𝔰}M(X) ⟶^{S^{𝔰}M(u)} S^{𝔰}M(Y) corresponding to an exact localization X ⟶^{u} Y does not induce a map Spec_{𝔏⋆}^{0}(X) ⟶ Spec_{𝔏⋆}^{0}(Y).

For any exact localization U ⟶^{u} X, set Spec_{𝔏}^{0}(U; X) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ_{u} ⊆ Σ̂}.

2.2.5. Proposition. Let {U_{i} →^{u_{i}} X | i ∈ J} be a conservative set of exact localizations. Then Spec_{𝔏}^{0}(X) = ⋃_{i∈J} Spec_{𝔏}^{0}(U_{i}; X).

Proof. By hypothesis, the family of localization functors, {C_{X} ⟶^{u_{i}^{∗}} C_{U_{i}} | i ∈ J}, is conservative, i.e. ⋂_{i∈J} Σ_{u_{i}} = Iso(C_{X}). Therefore, for every Σ ∈ Spec_{𝔏}^{0}(X), there exists i ∈ J such that Σ ⊈ Σ_{u_{i}} which means precisely that Σ_{u_{i}} ⊆ Σ̂ and the image of Σ ∨ Σ_{u_{i}} in S^{𝔰}M(U_{i}) belongs to Spec_{𝔏}^{0}(U_{i}) (see 2.2.3 and 2.2.4(a)), hence the assertion.

3. 𝔏-Local ’spaces’ and the spectrum Spec_{𝔏}^{0}(X).

We call a ’space’ X 𝔏-local (here 𝔏- stands for ’localization’), if S^{𝔰}M^{⋆}(X) has the smallest element, or, equivalently, the intersection, Σ^{X}, of all non-trivial saturated multiplicative systems is a non-trivial saturated multiplicative system.

3.1. Proposition. The following conditions on a ’space’ X are equivalent:

(a) The ’space’ X is 𝔏-local.

(b) The family of arrows Σ^{X} = ⋂_{Σ∈S^{𝔰}M^{⋆}(X)} Σ belongs to Spec_{𝔏}^{0}(X).

(c) The spectrum Spec_{𝔏}^{0}(X) has an element, Σ′, such that Σ̂′ = Iso(C_{X}).

Proof. (a) ⇒ (b) & (a) ⇒ (c): If X is 𝔏-local and Σ is a saturated multiplicative system, then Σ^{X} ⊈ Σ iff Σ ∉ S^{𝔰}M^{⋆}(X), that is if Σ = Iso(C_{X}).

(b) ⇒ (a) follows from definitions: if Σ^{X} ∈ Spec_{𝔏}^{0}(X), then Σ^{X} ∈ S^{𝔰}M^{⋆}(X), hence X is 𝔏-local.

(c) ⇒ (a): Let Σ′ ∈ Spec_{𝔏}^{0}(X) be such that Σ̂′ = Iso(C_{X}). Then Σ′ ∈ S^{𝔰}M^{⋆}(X) and Σ′ is contained in any non-trivial saturated multiplicative system, i.e. Σ′ = Σ^{X}.

3.2. Proposition. For any Σ ∈ Spec_{𝔏}^{0}(X), the ’space’ Σ̂^{−1}X is 𝔏-local.

Proof. The localization functor C_{X} ⟶^{q_{P}^{∗}} Σ̂^{−1}C_{X} induces an isomorphism between the preorder of (non-trivial) saturated multiplicative systems of C_{_{Σ^{−1}}X} = Σ̂^{−1}C_{X} and saturated multiplicative systems of C_{X} which contain Σ̂ properly. Since every saturated multiplicative system which contains Σ̂ properly contains Σ as well, the preorder of saturated multiplicative systems properly containing Σ̂ coincides with the preorder of saturated multiplicative systems containing Σ̂ ∨ Σ. Therefore the image of Σ̂ ∨ Σ in S^{𝔰}M(Σ̂^{−1}X) is the smallest element of S^{𝔰}M^{⋆}(Σ̂^{−1}X).

4. The complete 𝔏-spectrum.

For any ’space’ X, we define its complete 𝔏-spectrum, Spec_{𝔏}^{1}(X), as follows. Elements of Spec_{𝔏}^{1}(X) are saturated multiplicative systems, Σ_{x}, of C_{X} such that the ’space’ of fractions Σ_{x}^{−1}X is 𝔏-local. In other words, elements of Spec_{𝔏}^{1}(X) are saturated multiplicative systems, Σ_{x}, such that the intersection of all saturated multiplicative systems properly containing Σ_{x} is a saturated multiplicative system which contains Σ_{x} properly too. We consider Spec_{𝔏}^{1}(X) together with the preorder ⊆. By 3.2, there is a morphism Spec_{𝔏}^{0}(X) ⟶ Spec_{𝔏}^{1}(X) defined by Σ ⟼ Σ̂.

4.1. Proposition. The map X ⟼ Spec_{𝔏}^{1}(X) extends to a functor, Spec_{𝔏}^{1}, from the category 𝔏_{𝔢}𝔈𝔰𝔭 to the category 𝔓𝔒𝔯𝔡 of preorders.

Proof. The map S^{𝔰}M(X) ⟶^{S^{𝔰}M(u)} S^{𝔰}M(Y), Σ ⟼ u^{∗^{−1}}(Σ), corresponding to an exact localization X ⟶^{u} Y induces a map Spec_{𝔏}^{1}(X) ⟶ Spec_{𝔏}^{1}(Y).

Indeed, Σ^{−1}X ≃ S^{𝔰}M(u)(Σ)^{−1}Y, so that S^{𝔰}M(u)(Σ)^{−1}Y is 𝔏-local if Σ^{−1}X is 𝔏local, hence the assertion.

4.2. Note. For any ’space’ X,

Spec_{𝔏}^{1}(X) = ⋃_{Σ∈S^{𝔰}M(X)} Spec_{𝔏}^{0}(Σ^{−1}X) = ⋃_{Σ∈Spec_{𝔏}^{1}(X)} Spec_{𝔏}^{0}(Σ^{−1}X).

Here Spec_{𝔏}^{1}(Σ^{−1}X) is identified with its image in Spec_{𝔏}^{1}(X).

4.3. The extended complete 𝔏-spectrum. For any ’space’ X, set Spec_{𝔏⋆}^{1}(X) = Spec_{𝔏}^{1}(X) ∪ {⋆_{X}}, where ⋆_{X} = Iso(C_{X}). We call Spec_{𝔏⋆}^{1}(X) the extended complete 𝔏-spectrum of X.

4.3.1. Proposition. The map X ⟼ Spec_{𝔏}^{1}(X) gives rise to a contravariant functor

aS𝔭 : 𝔏_{𝔢}𝔈𝔰𝔭^{op} ⟶ 𝔓𝔒𝔯𝔡_{⋆}

and to a covariant functor

aS𝔭 : 𝔏_{𝔢}𝔈𝔰𝔭 ⟶ 𝔓𝔒𝔯𝔡_{⋆}

to the category 𝔓𝔒𝔯𝔡_{⋆} of preorders with initial objects.

Proof. The functor _{a}S𝔭 : 𝔏_{𝔢}𝔈𝔰𝔭 ⟶ 𝔓𝔒𝔯𝔡_{⋆} is the unique extension of the functor Spec_{𝔏}^{1} : 𝔏_{𝔢}𝔈𝔰𝔭 ⟶ 𝔓𝔒𝔯𝔡 of 4.1.

Let X ⟶^{u} Y be an exact localization. We define the map

aS𝔭(u) : Spec_{𝔏}^{1}(Y) ⟶ Spec_{𝔏}^{1}(X)

as follows. Let Σ_{x} ∈ Spec_{𝔏}^{1}(Y). If Σ_{u} ⊆ Σ_{x}, then ^{a}S𝔭(u)(Σ_{x}) is the minimal saturated multiplicative system containing u^{∗}(Σ_{x}). By transitivity of localizations, ^{a}S𝔭(u)(Σ_{x}) ∈ Spec_{𝔏}^{1}(X). If Σ_{u} ⊈ Σ_{x}, then ^{a}S𝔭(u) maps Σ_{x} to the trivial family, ⋆_{X} = Iso(C_{X}). It is easy to check that the map u ⟼^{a} S𝔭(u) is functorial.

4.4. Remark. The dualization functor, X ⟼ X^{o}, establishes an isomorphism between the preorder of left (saturated) multiplicative systems on X and right (saturated) multiplicative systems on X^{o}. This isomorphism induces an isomorphism of preorders of (saturated) multiplicative systems:

SM(X) ⟶^{∼} SM(X^{o}) and S^{𝔰}M(X) ⟶^{∼} S^{𝔰}M(X^{o}).  (1)

In particular, a ’space’ X is 𝔏-local iff its dual, X^{o} is 𝔏-local.

Thus, the isomorphism S^{𝔰}M(X) ⟶^{∼} S^{𝔰}M(X^{o}) induces isomorphisms of spectra

Spec_{𝔏}^{1}(X) ⟶^{∼} Spec_{𝔏}^{1}(X^{o}) and Spec_{𝔏}^{0}(X) ⟶^{∼} Spec_{𝔏}^{0}(X^{o}).  (2)

as well as the extended versions of these spectra.

The spectra Spec_{𝔏}^{1}(X) and Spec_{𝔏}^{0}(X) are too large, which is one of the reasons why the duality (2) takes place. In the next section, we single out smaller spectra inside of Spec_{𝔏}^{1}(X) and Spec_{𝔏}^{0}(X).

5. Closed spectra and flat spectra.

5.1. Σ-Torsion free objects. Let Σ ⊆ HomC_{X}. We say that an object, M, of the category C_{X} is Σ-torsion free if every morphism M ⟶^{t} N which belongs to Σ is a monomorphism. We denote by C_{X_{Σ}} the full subcategory of the category C_{X} whose objects are Σ-torsion free.

5.1.1. Lemma. Let Σ ⊆ HomC_{X} be such that for every diagram L̃ ⟵^{s} L ⟶^{g} M, where s ∈ Σ and g is a monomorphism, there exists a commutative diagram

\begin{matrix}L & ⟶^{g} & M \\ s ↓ & & ↓ t \\ L̃ & \underrightarrow{_{g}} & M̃ \end{matrix}  (1)

where t ∈ Σ (e.g. Σ is a left multiplicative system).

Then any subobject of a Σ-free object is Σ-free.

Proof. Let L ⟶^{g} M be a monomorphism, and L ⟶^{s} L̃ a morphism of Σ. Then there exists a commutative diagram (1) in which t ∈ Σ. If M is Σ-torsion free, then M ⟶^{t} M̃ is a monomorphism. Thus t ∘ g is a monomorphism. It follows from the equality t ∘ g = g̃ ∘ s that s is a monomorphism, hence the assertion.

5.2. Closed families of morphisms and closed spectra. Let Σ ⊆ HomC_{X}. We say that Σ is closed, or right closed, if for every M ∈ ObC_{X}, there exists a morphism M ⟶ M̃ of Σ such that M̃ ∈ ObC_{X_{Σ}}.

5.2.1. Proposition. Let Σ ⊆ HomC_{X} be a left saturated multiplicative system such that the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is continuous; and let q∗_{Σ} and q_{_{Σ}∗} be resp. its inverse and direct image functors. Then the following conditions on an object M of C_{X} are equivalent:

(i) M is Σ-torsion free.

(ii) An adjunction morphism M ⟶^{η_{Σ} (M)} q_{_{Σ}∗}q∗_{Σ}(M) is a monomorphism.

Proof. Let Σ be a saturated multiplicative system and C_{X} ⟶^{q_{Σ}^{∗}} C_{Σ^{−1}X} = Σ^{−1}C_{X} a localization functor at Σ. If the family Σ is flat, the functor q∗_{Σ} has a right adjoint, q_{_{Σ}∗}. For

every M ∈ ObC_{X}, the adjunction arrow, M ⟶^{η_{Σ} (M)} q_{_{Σ}∗}q∗_{Σ}(M), belongs to Σ. In particular, if M is a Σ-torsion free object, then the adjunction morphism η_{Σ}(M) is a monomorphism.

Let M ⟶^{s} N be a morphism from Σ. Then the upper horizontal arrow in the commutatative diagram

q_{_{Σ}∗}q∗_{Σ}(M) ⟶^{q_{_{Σ}∗}q_{Σ}^{∗} (s)} q_{_{Σ}∗}q∗_{Σ}(N)
η_{Σ}(M) ↑  ↑ η_{Σ}(N)
M  ⟶^{s}  N

is an isomorphism. If η_{Σ}(M) is a monomorphism, then η_{Σ}(N) ∘ s = q_{_{Σ}∗}q∗_{Σ}(s) ∘ η_{Σ}(M) is a monomorphism. Therefore s is a monomorphism. This shows that the object M is Σ-torsion free.

5.2.2. Corollary. Let Σ be a left saturated multiplicative system such that the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is continuous. Then Σ is closed.

Proof. For every M ∈ ObC_{X}, the adjunction arrow, M ⟶^{η_{Σ} (M)} q_{_{Σ}∗}q∗_{Σ}(M) belongs to Σ. If M̃ = q_{_{Σ}∗}q∗_{Σ}(M), then the adjunction arrow η_{Σ}(M̃) is an isomorphism, in particular it is a monomorphism.

5.2.3. Closed spectra. We denote by ℭS^{𝔰}M(X) the preorder of all closed saturated multiplicative systems on X. The complete closed spectrum, Spec_{ℭ}^{1}(X), is defined by

Spec_{ℭ}^{1}(X) = ℭS^{𝔰}M(X) ⋂ Spec_{𝔏}^{1}(X);

that is elements of Spec_{ℭ}^{1}(X) are closed saturated multiplicative systems, Σ, such that Σ^{−1}X is 𝔏-local.

We call Spec_{ℭ}^{0}(X) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ∈ Spec_{ℭ}^{1}(X)} the closed spectrum of X.

5.3. Continuous localizations and flat spectra. Let Σ ⊆ HomC_{X}. Recall that an object M of C_{X} is called left closed for Σ if C_{X}(s, M) is a bijection for each morphism s of Σ [GZ, I.4].

5.3.1. Lemma. (a) Let Σ ⊆ HomC_{X}, and let M be an object of C_{X} such that C_{X}(s, M) is a surjection for each morphism s of Σ. Then every morphism M ⟶^{t} N which belongs to Σ is a retraction (i.e. u ∘ t = id_{M} for some morphism u). In particular, M is Σ-torsion free.

(b) Suppose for any diagram L̃ ⟵^{s} L ⟶^{g} M such that s ∈ Σ, there exists a commutative diagram

\begin{matrix}L & ⟶^{f} & M \\ s ↓ & & ↓ t \\ L̃ & \underrightarrow{_{f}} & M̃ \end{matrix}  (1)

where t ∈ Σ (e.g. Σ is a left multiplicative system). Then C_{X}(s, M) is a surjection for each morphism s of Σ iff every morphism M ⟶^{t} N which belongs to Σ is a retraction.

Proof. (a) If M ⟶^{t} N is a morphism of Σ, then the map

C_{X}(N, M) ⟶ C_{X}(M, M), f ⟼ f ∘ t,

is surjective. In particular, there exists a morphism N ⟶^{u} M such that u ∘ t = id_{M}.

(b) Suppose that the object M is such that every morphism M ⟶ N which belongs to Σ is a retraction. Then C_{X}(s, M) is surjective for any morphism L ⟶^{s} L̃ of Σ.

In fact, let L ⟶^{f} M be an arbitrary morphism. By hypothesis, there is a commutative diagram (1), where t ∈ Σ. By condition, t is a retraction, i.e. there exists a morphism M̃ ⟶^{u} M such that u ∘ t = id_{M}. Then (u ∘ f̃) ∘ s = u ∘ (t ∘ f) = f. This shows the surjectivity of C_{X}(s, M).

5.3.2. Proposition. Suppose that Σ ⊆ HomC_{X} is a left multiplicative system. Then the following conditions on an object M of C_{X} are equivalent:

(a) M is left closed for Σ;

(b) C_{X}(s, M) is surjective for any s ∈ Σ;

(c) any morphism M ⟶ N which belongs to Σ is a retraction.

Proof. The implications (b) ⇔ (c) follow from 5.3.1(b). The implication (a) ⇒ (b) holds by definition. The implication (b) ⇒ (a) is proven in [GZ, 1.4.1.1].

5.3.3. Localizations and continuous localizations. Let X ⟶^{f} Y be a morphism with an inverse image functor C_{Y} ⟶^{f^{∗}} C_{X}. An object N of C_{Y} is called f-free over an object M of C_{X}, if there exists a morphism f^{∗}(N) ⟶^{u} M such that for any morphism

f^{∗}(L) ⟶^{v} M there exists a unique morphism L \underrightarrow{_{v}} N satisfying v = u ∘ f^{∗}(ṽ). In other words, (N, f^{∗}(N) ⟶^{u} M) is a final object of the category f^{∗}/M, or, what is the same, the object N represents the functor C_{X}(f^{∗}(−), M) : C_{Y}^{op} ⟶ Sets. We denote by C_{L(f)} the full subcategory of the category C_{Y} generated by f-free objects.

Let C_{𝔇(f_{∗})} denote the full subcategory of C_{X} generated by all M ∈ ObC_{X} such that the functor C_{X}(f^{∗}(−), M) : C_{Y}^{op} ⟶ Sets is representable. A choice for each M ∈ ObC_{𝔇(f_{∗})} of an object, f_{∗}(M), of the subcategory C_{L(f)} representing the functor C_{X}(f^{∗}(−), M) extends uniquely to a functor 𝔇(f_{∗}) ⟶ C_{Y} taking values in C_{L(f)}. Let C_{ℜ(f_{∗})} denote f^{∗^{−1}}(𝔇(f_{∗})). The morphism f is continuous iff 𝔇(f_{∗}) = X and, therefore, ℜ(f_{∗}) = Y.

5.3.3.1. Proposition. Suppose that Σ ⊆ HomC_{X} is a left multiplicative system. And let Σ^{−1}X ⟶^{q_{Σ}} X be the localization morphism. Then

(a) C_{L(q_{Σ} )} is the full subcategory of C_{X} generatated by all objects which are left closed for Σ.

(b) The subcategory C_{ℜ(q_{Σ} )} is generated by all M ∈ ObC_{X} such that there exists a morphism M ⟶^{s} N, where N is left closed for Σ and q_{Σ}(s) is invertible.

(c) The composition of the inclusion C_{L(q_{Σ} )} ⟶ C_{X} and the canonical localization

functor C_{X} ⟶^{q_{Σ}^{∗}} Σ^{−1}C_{X} is a fully faithful functor injective on objects. This functor induces an isomorphism 𝔇(q_{Σ}) ⟶^{∼} L(q_{Σ}).

(d) The inclusion functor C_{L(q_{Σ} )} \underrightarrow{_{q_{_{Σ}∗}}} C_{ℜ(q_{Σ} )} has a left adjoint, q∗_{Σ}.

Proof. (a) Let Y ⟶^{f} X be a morphism, M an object of C_{Y} such that the functor C_{(}f^{∗}(−), M) is representable. Then any object, N, representing C_{(}f^{∗}(−), M) is, obviously, left closed for Σ_{f} = {s ∈ HomC_{X} | f^{∗}(s) ∈ Iso(C_{Y})}.

If f = q_{Σ}, then the converse is true: if N ∈ ObC_{X} is left closed for Σ, then it follows from the universal property of the localization at Σ that the object N represents the functor

C_{Σ^{−1}X}(q∗_{Σ}(−), q∗_{Σ}(N)).

(b) By definition, the subcategory C_{ℜ(q_{Σ} )} is generated by all M ∈ ObC_{X} such that the functor C_{Σ^{−1}X}(q∗_{Σ}(−), q∗_{Σ}(M)) is representable by some object, N, of the category C_{X}. In particular, there exists a canonical morphism M ⟶^{t} N corresponding to the identical arrow q∗_{Σ}(M) ⟶ q∗_{Σ}(M). It follows that q∗_{Σ}(t) is an isomorphism.

(c) The canonical localization functor C_{X} ⟶^{q_{Σ}^{∗}} C_{Σ^{−1}X} is identical on objects, hence the composition of q∗_{Σ} with the inclusion functor C_{L(q_{Σ} )} ⟶ C_{X} is injective on objects. For any M ∈ ObC_{L(q_{Σ} )} and any L ∈ ObC_{X}, we have a functorial isomorphism C_{Σ^{−1}X}(q∗_{Σ}(L), q∗_{Σ}(M)) ≃ C_{X}(L, M). In particular, the composition of the embedding C_{L(q_{Σ} )} ⟶ C_{X} with q∗_{Σ} is a fully faithful functor.

(d) By (b), for any M ∈ ObC_{ℜ(q_{Σ} )}, there exists a morphism M ⟶^{t} N, where N is left closed for Σ. It follows from 5.3.2 that the object N here is defined uniquely up to isomorphism. A choice of N for every M ∈ ObC_{ℜ(q_{Σ} )} defines a functor, q∗_{Σ}, from C_{ℜ(q_{Σ} )} to C_{L(q_{Σ} )}. This functor is a left adjoint to the inclusion functor C_{L(q_{Σ} )} ⟶ C_{ℜ(q_{Σ} )}.

One of the corollaries of 5.3.3.1 is the following fact:

5.3.3.2. Proposition [GZ, 1.4.1]. Suppose that Σ ⊆ HomC_{X} is a left multiplicative system. Then the following conditions are equivalent:

(a) The canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is continuous.

(b) For every object M of the category C_{X}, there exists an object M̃ left closed for Σ and a morphism M ⟶^{s} M̃ such that q_{Σ}(s) is invertible.

5.3.4. Continuous and flat multiplicative systems. We call Σ ⊆ HomC_{X} continuous if it is a left multiplicative system and the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is continuous. It follows from [GZ, 1.1.3] that if Σ is continuous, then the saturation of Σ is a left multiplicative system, hence it is continuous. We denote by 𝔏𝔠_{ℓ}(X) the preorder of all continuous left saturated multiplicative systems and by 𝔏𝔠(X) the preorder of all continuous saturated multiplicative systems, i.e. 𝔏𝔠(X) = 𝔏𝔠_{ℓ}(X) ∩ S^{𝔰}M(X).

We will call continuous saturated multiplicative systems flat.

5.3.5. The flat spectra Spec_{𝔣𝔏}^{0}(X) and Spec_{𝔣𝔏}^{1}(X). The elements of the flat complete 𝔏-spectrum Spec_{𝔣𝔏}^{1}(X) are flat multiplicative systems Σ such that the ’space’ of fractions Σ^{−1}X is 𝔏-local.

We call Spec_{𝔣𝔏}^{1}(X) the complete flat 𝔏-spectrum of X.

The flat 𝔏-spectrum, Spec_{𝔣𝔏}^{0}(X), of the ’space’ X is defined by setting

Spec_{𝔣𝔏}^{0}(X) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ∈ Spec_{𝔣𝔏}^{1}(X)}.

It follows that

Spec_{𝔣𝔏}^{0}(X) ⊆ Spec_{ℭ}^{0}(X) and Spec_{𝔣𝔏}^{1}(X) ⊆ Spec_{ℭ}^{1}(X).

We leave to the reader the definition of the extended versions of these spectra.

C1. Cosubspaces and coimmersions.

Let X be a ’space’. We call Y a cosubspace of X if C_{Y} is a full subcategory of the category C_{X} which is closed under finite limits and colimits taken in C_{X} and, in addition, the following condition holds:

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

We call a morphism X ⟶^{u} V a coimmersion if it induces an equivalence between V and a cosubspace of X.

C1.1. Proposition. Let X be a ’space’ such that the category C_{X} has finite limits and colimits. Then any coimmersion X ⟶^{𝔳} Y induces a morphism S^{𝔰}M(X) ⟶^{𝔳_{𝔏}} S^{𝔰}M(Y) which maps every saturated multiplicative system Σ to its preimage 𝔳^{∗^{−1}}(Σ) by an inverse image functor 𝔳^{∗} of the coimmersion 𝔳.

The functor 𝔳^{∗} determines a functor 𝔳_{Σ}^{∗} : 𝔳_{𝔏}(Σ)^{−1}C_{Y} ⟶ Σ^{−1}C_{X} which is an inverse image functor of a coimmersion.

Proof. We can and will assume that Y is a cosubspace of X and 𝔳^{∗} is the inclusion functor C_{Y} ⟶ C_{X}.

(a) The morphism X ⟶^{𝔳} Y induces a morphism S^{𝔰}M(X) ⟶^{𝔳_{𝔏}} S^{𝔰}M(Y) which maps every saturated multiplicative system Σ of C_{X} to its restriction, Σ|_{Y}, to C_{Y}.

We need to verify that Σ|_{Y} belongs to S^{𝔰}M(Y). Let q_{Σ}^{∗} be the localization functor C_{X} ⟶ Σ^{−1}C_{X} = C_{Σ^{−1}X}. Since Σ is saturated, it coincides with the family {t ∈ HomC_{X} | q_{Σ}^{∗}(t) is invertible}. In particular, Σ|_{Y} = {t ∈ HomC_{Y} | q_{Σ}^{∗}(t) is invertible}. Therefore Σ|_{Y} is saturated too. Because C_{X} has finite limits and colimits, it follows from [GZ, 1.3.2 and 1.3.4] that Σ is a saturated multiplicative system iff the localization functor q_{Σ}^{∗} is exact, i.e. it preserves finite limits and colimits. Since the category C_{Y} is closed

under finite limits and colimits taken in C_{X}, that is the inclusion functor C_{Y} ⟶^{𝔳^{∗}} C_{X} is exact, the restriction of q_{Σ}^{∗} to C_{Y} (i.e. the composition q_{Σ}^{∗} ∘ 𝔳^{∗}) is exact. Therefore Σ|_{Y} is a multiplicative system.

(b) For any Σ ∈ S^{𝔰}M(X), there is a commutative diagram

\begin{matrix}C_{Y} & ⟶^{𝔳^{∗}} & C_{X} \\ q_{Σ|_{Y}}^{∗} ↓ & & ↓ q_{Σ}^{∗} \\ Σ|_{Y}^{−1}C_{Y} & ⟶^{𝔳_{Σ}^{∗}} & Σ^{−1}C_{X} \end{matrix}  (1)

in which the functor 𝔳_{Σ}^{∗} is uniquely determined by the other three. Notice that 𝔳_{Σ}^{∗} is an inverse image functor of a coimmersion.

In fact, by [GZ, 1.3.4], the functor 𝔳_{Σ}^{∗} is exact, because the functors q_{Σ|_{Y}}^{∗} and q_{Σ}^{∗} ∘ 𝔳^{∗} are exact (see the diagram (1)). It follows from the description of the categories of fractions Σ|_{Y}^{−1}C_{Y} and Σ^{−1}C_{X} (cf. [GZ, 1.3]) that the functor 𝔳_{Σ}^{∗} is fully faithful.

Moreover, if Y is a cosubspace of X and C_{Y} ⟶^{𝔳^{∗}} C_{X} is the inclusion functor, then

Σ|_{Y}^{−1}Y is a cosubspace of Σ^{−1}X and C_{Σ|_{Y}^{−1}Y} = Σ|_{Y}^{−1}C_{Y} ⟶^{𝔳_{Σ}^{∗}} Σ^{−1}C_{X} = C_{Σ^{−1}X} is the inclusion functor of a cosubspace. Here we take the standard realization of categories of fractions (see [GZ, 1.3]).

II. Spectra of ’spaces’ represented by abelian categories.

6. Spectra in terms of thick and Serre subcategories.

Fix a ’space’ X such that C_{X} is an abelian category. Let 𝔗𝔥(X) denote the preorder (with respect to ⊆) of all thick subcategories of the category C_{X}. Recall that a full subcategory, 𝕋, of the category C_{X} is thick if it is closed under subquotients and extensions. In other words, an object M in the exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 belongs to 𝕋 iff M′ and M′′ are objects of 𝕋.

6.1. Thick subcategories, saturated multiplicative systems, and Spec_{𝔏}^{0}. There is a bijective map from S^{𝔰}M(X) to the preorder 𝔗𝔥(X) which assigns to each saturated multiplicative system Σ the kernel of the localization functor C_{X} ⟶ Σ^{−1}C_{X}. The inverse map assigns to each thick subcategory 𝕋 of the category C_{X} the family, Σ_{𝕋}, of all morphisms s of C_{X} whose kernel and cokernel are objects of 𝕋. One can see that the map 𝔗𝔥(X) ⟶ S^{𝔰}M(X), 𝕋 ⟼ Σ_{𝕋}, (hence its inverse) is an isomorphism of preorders.

The trivial saturated multiplicative system, Iso(C_{X}), corresponds to the zero subcategory of the category C_{X}. Thus, the isomorphism 𝔗𝔥(X) ⟶^{∼} S^{𝔰}M(X) induces an isomorphism between the preorder 𝔗𝔥_{⋆}(X) of all nonzero thick subcategories of C_{X} and the preorder S^{𝔰}M_{⋆}(X) of all non-trivial saturated multiplicative systems of C_{X}.

Let Spec_{𝔗𝔥}^{0}(X) denote the preorder (with respect to ⊆) of all thick subcategories, P, of the category C_{X} such that there exists the biggest thick subcategory, P̂, of C_{X} which does not contain P. The map 𝕋 ⟼ Σ_{𝕋} induces an isomorphism of preorders

Spec_{𝔗𝔥}^{0}(X) ⟶^{∼} Spec_{𝔏}^{0}(X).

For any object M of the category C_{X}, let [M]_{•} denote the thick envelope of M which is, by definition, the smallest thick subcategory of C_{X} containing M.

6.1.1. Lemma. Let P ∈ Spec_{𝔗𝔥}^{0}(X). Then P = [M]_{•} for any M ∈ ObP − ObP̂.

Proof. (a) Let P, P′ ∈ Spec_{𝔗𝔥}^{0}(X). It follows from 2.1.2 that P′ ⊆ P iff P̂′ ⊆ P̂.

(b) For any M ∈ ObP − ObP̂, the category P̂ is the biggest thick subcategory which does not contain M, that is [M]_{•} ∈ Spec_{𝔗𝔥}^{0}(X) and { \atop [M]_{•}} = P̂. By (a) above, the latter equality implies that P = [M]_{•}.

6.1.2. Representatives of elements of Spec_{𝔗𝔥}^{0}(X). Let Spec_{𝔗𝔥}^{0}(X) denote the family of all objects M of the category C_{X} such that there exists the biggest thick subcategory, { \atop [M]_{•}}, which does not contain M. It follows from 6.1.1 that

Spec_{𝔗𝔥}^{0}(X) = {[M]_{•} | M ∈ Spec_{𝔗𝔥}^{0}(X)}.

6.1.3. The categories ⟨M⟩_{•} and another description of Spec_{𝔗𝔥}^{0}(X). For any object M of the category C_{X}, denote by ⟨M⟩_{•} the full subcategory of C_{X} whose objects are N ∈ ObC_{X} such that M ∉ [N]_{•}.

We define a relation ⋟ on ObC_{X} by M ⋟ L iff L ∈ [M]_{•}.

6.1.3.1. Lemma. The following conditions on objects L, M of C_{X} are equivalent:

(i) M ⋟ L; (ii) [L]_{•} ⊆ [M]_{•}; (iii) ⟨L⟩_{•} ⊆ ⟨M⟩_{•}.

Proof. The implications (i)⇔(ii) are evident. The implications (ii)⇒(iii) and (iii)⇒(i) are left to the reader.

The equivalence of (i) and (ii) in 6.1.3.1 shows, in particular, that ⋟ is a preorder.

6.1.3.2. Note. It follows from 6.1.3.1 that for any object M of the category C_{X}, the subcategory ⟨M⟩_{•} is closed with respect ⋟; i.e. if N ∈ Ob⟨M⟩_{•} and N ⋟ L, then L ∈ Ob⟨M⟩_{•}. In particular, the subcategory ⟨M⟩_{•} contains all subquotients of its objects. But, in general, ⟨M⟩_{•} is not closed under extensions.

6.1.3.3. Proposition. (a) For any object M of the category C_{X}, the subcategory ⟨M⟩_{•} is the union of all thick subcategories of C_{X} which do not contain the object M.

(b) The following conditions on an object M of the category C_{X} are equivalent: (i) M ∈ Spec_{𝔗𝔥}^{0}(X); (ii) the subcategory ⟨M⟩_{•} is thick.

For every M ∈ Spec_{𝔗𝔥}^{0}(X), the subcategory { \atop [M]_{•}} coincides with ⟨M⟩_{•}.

Proof. (a) If 𝕋 is a thick subcategory which does not contain the object M, then, obviously, M ∉ Ob[N]_{•} for any N ∈ Ob𝕋, that is 𝕋 ⊆ ⟨M⟩_{•}. This observation shows also that ⟨M⟩_{•} is the union of its thick subcategories [N]_{•}, N ∈ Ob⟨M⟩_{•}.

(b) (i)⇒(ii). Let M ∈ Spec_{𝔗𝔥}^{0}(X). If N ∈ Ob⟨M⟩_{•}, that is M does not belong to the

thick subcategory [N]_{•} spanned by N, then [N]_{•} ⊆ { \atop [M]_{•}}. Therefore ⟨M⟩_{•} is a subcategory of { \atop [M]_{•}}. The inverse inclusion, { \atop [M]_{•}} ⊆ ⟨M⟩_{•}, is obvious.

(ii)⇒(i). Notice that any thick subcategory 𝕋 of the category C_{X} which does not contain the object M is a subcategory of ⟨M⟩_{•}. This shows that if ⟨M⟩_{•} is a thick subcategory itself, the it is the biggest thick subcategory of C_{X} which does not contain M, hence the assertion.

6.2. Coreflective thick subcategories and the closed spectrum. Recall that a subcategory 𝕋 of the category C_{X} is called coreflective, or (right) closed, if the inclusion functor 𝕋 ↪ C_{X} has a right adjoint. In other words, the subcategory 𝕋 is coreflective iff every object of C_{X} has the biggest subobject which belongs to 𝕋.

6.2.1. Proposition. (a) Let 𝕋 be a full subcategory of the category C_{X} closed under coproducts and taking quotients (e.g. 𝕋 is topologizing). If the family Σ_{𝕋} = {s ∈ HomC_{X} | Ker(s) ∈ Ob𝕋 ∋ Cok(s)} is closed, then the subcategory 𝕋 is coreflective.

(b) Suppose the 𝕋 is a coreflective subcategory of C_{X} closed under extensions. Then the family Σ_{𝕋} is closed.

(c) Suppose 𝕋 is closed under taking quotients and extensions (e.g. 𝕋 is a thick subcategory). Then the family Σ_{𝕋} is closed iff the subcategory 𝕋 is coreflective.

Proof. Notice that Σ_{𝕋}-torsion free objects are precisely 𝕋-torsion free objects.

(a) Suppose, Σ_{𝕋} is closed; i.e. for every object M of the category C_{X}, there exists a morphism M ⟶^{s} N of Σ_{𝕋} (that is Ker(s) ∈ Ob𝕋 ∋ Cok(s)) such that N is 𝕋-torsion free. Then Ker(s) is the 𝕋-torsion of M. In fact, if L ↪ M is a subobject of M such that L ∈ Ob𝕋, then the image, L_{s}, of L ⊕ Ker(s) ⟶ M is a subobject of M which belongs to 𝕋 and Ker(s) ↪ L_{s}. The image of the composition L_{s} ↪ M →^{s} N is a subobject of N which belongs to 𝕋. Since N is 𝕋-torsion free, this image equals to zero. Therefore Ker(s) ↪ L_{s} is an isomorphism which means that L is a subobject of Ker(s).

(b) Let 𝕋 be a coreflective subcategory closed under extensions. For an arbitrary object M, let 𝔱_{𝕋}M denote the 𝕋-torsion of M and M_{𝕋} the quotient M/𝔱_{𝕋}M. By definition, the canonical epimorphism M ⟶ M_{𝕋} belongs to Σ_{𝕋}. Notice that the object M_{𝕋} is 𝕋torsion free.

In fact, let L ↪ M_{𝕋} be a subobject of M_{𝕋} such that L ∈ Ob𝕋. Then we have a short exact sequence

0 ⟶ 𝔱_{𝕋}M ⟶ L̃ ⟶ L ⟶ 0,

where L̃ = L ×_{M_{𝕋}} M. Since 𝕋 is closed under extensions, L̃ ∈ Ob𝕋. Because L̃ is a subobject of M, it factors through 𝔱_{𝕋}M ↪ M, hence 𝔱_{𝕋}M ⟶ L̃ is an isomorphism, or, equivalently, L = 0.

(c) The assertion follows from (a) and (b).

6.2.2. The closed spectrum. Let ℭ𝔗𝔥(X) denote the preorder of coreflective thick subcategories. We set

Spec_{ℭ}^{0}(X) = {P ∈ Spec_{𝔗𝔥}^{0}X | P̂ ∈ ℭ𝔗𝔥(X)}.

The map 𝕋 ⟼ Σ_{𝕋} induces an isomorphism of the preorder Spec_{ℭ}^{0}(X) onto the closed spectrum Spec_{ℭ}^{0}(X) of X.

Let Spec_{ℭ}^{0}(X) denote the family of all objects M such that ⟨M⟩_{•} is a coreflective thick subcategory of C_{X} and M is ⟨M⟩_{•}-torsion free. It follows from this definition that Spec_{ℭ}^{0}(X) ⊆ Spec_{𝔗𝔥}^{0}X.

6.2.3. Proposition. Spec_{ℭ}^{0}(X) = {[M]_{•} | M ∈ Spec_{ℭ}^{0}(X)}.

Proof. Let P ∈ Spec_{ℭ}^{0}(X), and let M ∈ ObP − ObP̂. By 6.1.1, P = [M]_{•} and, therefore, P̂ = ⟨M⟩_{•} (cf. 6.1.3). Since the category P is coreflective, the object M has the P̂-torsion, 𝔱_{P}M ↪ M, and (by 6.2.1(c)) the quotient object, M_{P} = M/𝔱_{P}M, is P̂-torsion free. Since M ∈ ObP − ObP̂, the object M_{P} is nonzero and belongs to the subcategory P. By 6.1.1 and 6.1.3.3, P = [M_{P}]_{•} and P̂ = ⟨M_{P}⟩_{•}.

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

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

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

(a) The subcategory 𝕋^{−} is thick (i.e. it is closed under taking subquotients and extensions).

(b) (𝕋^{−})^{−} = 𝕋^{−}.

(c) 𝕋 ⊆ 𝕋^{−} iff any nonzero subquotient of an object of 𝕋 has a nonzero subobject which is isomorphic to an object of 𝕋.

Proof. See [R, III.2.3.2.1].

6.3.2. Corollary. For any subcategory, 𝕋, of the category C_{X}, Ob𝕋^{−} consists of all N ∈ ObC_{X} such that any nonzero object of [N]_{•} contains a nonzero subobject which belongs to 𝕋.

Proof. By 6.3.1, 𝕋^{−} is a thick subcategory of C_{X}, hence N ∈ Ob𝕋^{−} iff [N]_{•} ⊆ 𝕋^{−}.

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

6.3.4. The spectrum Spec_{𝔰}^{0}(X). For any object L of the category C_{X}, denote by ⊥L the full subcategory of C_{X} whose objects are all N ∈ ObC_{X} such that C_{X}(N, L) = 0. Let Spec_{𝔰}^{0}(X) denote the family of all nonzero objects M of the category C_{X} such that M ∈ Ob[N]_{•} for any N ∈ ObC_{X} −Ob^{⊥}M. In other words, if N is an object of the category C_{X} such that if there exists a nonzero morphism N ⟶ M, then M ∈ Ob[N]_{•}.

Thus, if M ∈ Spec_{𝔰}^{0}(X), then Ob⟨M⟩_{•} = {N ∈ ObC_{X} | [N]_{•} ⊆^{⊥} M}.

6.3.5. Proposition. For every M ∈ Spec_{𝔰}^{0}(X), the subcategory ⟨M⟩_{•} is a Serre subcategory. In particular, Spec_{𝔰}^{0}(X) ⊆ Spec_{𝔗𝔥}^{0}(X).

Proof. By 6.1.3.2, ⟨M⟩_{•} is closed under taking subquotients (in C_{X}). Therefore, by 6.3.1(c), ⟨M⟩_{•} is a subcategory of its “Serre closure” ⟨M⟩_{•}^{−}. Notice that ⟨M⟩_{•} is properly contained in ⟨M⟩_{•}^{−} iff M ∈ Ob⟨M⟩_{•}^{−}.

In fact, if N ∈ Ob⟨M⟩_{•}^{−} − Ob⟨M⟩_{•}, then, by the definition of ⟨M⟩_{•}, the object M belongs to the minimal thick subcategory, [N]_{•}, spanned by N. By 6.3.1(a), the subcategory ⟨M⟩_{•}^{−} is thick, hence it contains the subcategory [N]_{•}.

If M is a nonzero object and M ∈ ⟨M⟩_{•}^{−}, there exists a nonzero morphism L ⟶ M with L ∈ Ob⟨M⟩_{•}. But, if M ∈ Spec_{𝔰}^{0}(X), then C_{X}(L, M) = 0 for all L ∈ Ob⟨M⟩_{•}. Thus, ⟨M⟩_{•}^{−} = ⟨M⟩_{•}.

6.3.6. Proposition. The following conditions on a nonzero object M of the category C_{X} are equivalent:

(i) M ∈ Spec_{𝔰}^{0}(X); (ii) M is ⟨M⟩_{•}-torsion free; (iii) ⟨M⟩_{•} is a Serre subcategory of C_{X} and M is ⟨M⟩_{•}-torsion free.

Proof. Clearly (iii)⇒(ii). If M ∈ Spec_{𝔰}^{0}(X), then M is ⟨M⟩_{•}-torsion free, because if there is a nonzero morphism

N ⟶ M, then M ∈ [N]_{•}, i.e. N ∉ Ob⟨M⟩_{•}. Together with 6.3.5, this proves the implication (i)⇒(iii).

(ii)⇒(i). By definition of Spec_{𝔰}^{0}(X), a nonzero object, M, belongs to Spec_{𝔰}^{0}(X) iff for any nonzero morphism N ⟶ M, the object M belongs to the thick subcategory [N]_{•} spanned by N.

Suppose the conditions of (ii) hold, and let N ⟶ M be a nonzero morphism. Since M is ⟨M⟩_{•}-torsion free, and ⟨M⟩_{•} is closed under taking subquotients in C_{X} (see 6.1.3.2), the object N does not belong to ⟨M⟩_{•} which means that M ∈ [N]_{•}.

We denote by Spec_{𝔰}^{0}(X) the preordered subset of Spec_{𝔗𝔥}^{0}(X) formed by all [M]_{•} such that M ∈ Spec_{𝔰}^{0}(X).

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

sup(N ∩ L | N ∈ Ω) ⟶ (supΩ) ∩ L

is an isomorphism.

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

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

Proof. See [R, III.2.4.4].

6.3.9. 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}).

6.3.10. Proposition. Suppose that X has the property (sup). Then

Spec_{𝔰}^{0}(X) = {P ∈ Spec_{𝔗𝔥}^{0}(X) | P̂ is a Serre subcategory of C_{X}}

= {[M]_{•} | M ≠ 0 and ⟨M⟩_{•} is a Serre subcategory of C_{X}} = Spec_{ℭ}^{0}(X).

Proof. It follows from 6.3.6 that

Spec_{ℭ}^{0}(X) = {M ∈ Spec_{𝔰}^{0}(X) | ⟨M⟩_{•} is coreflective}.

If X has the property (sup), then, by 6.3.8, every Serre subcategory is coreflective. The assertion follows now from 6.2.3.

6.4. Cosubspaces and coimmersions. Let C_{X} be an abelian category. Then Y is a cosubspace of X iff C_{Y} is a topologizing subcategory of C_{X}. Here ’topologizing’ means that C_{Y} is a full subcategory of C_{X} closed under finite coproducts and subquotients taken in C_{X}. In particular, C_{Y} is an abelian category.

For any cosubspace Y of X, we define cosubspaces Y^{−} and Y^{∞} by setting C_{Y^{−}} := C_{Y}^{−} and taking as C_{Y^{∞}} the thick envelope of C_{Y} in C_{X}.

We call a cosubspace Y thick if C_{Y} is a thick subcategory of C_{X}, i.e. Y = Y^{∞}. We call Y a Serre cosubspace if Y = Y^{−}.

A morphism X ⟶^{𝔳} Y is a coimmersion if it induces an isomorphism between a cosubspace of X and Y.

6.4.1. Gabriel multiplication. Recall that the Gabriel product, 𝕋 •_{X} 𝕊, of subcategories 𝕋 and 𝕊 of the category C_{X} is the full subcategory of C_{X} generated by all M ∈ ObC_{X} for which there exists an exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

with M′ ∈ Ob𝕊 and M′′ ∈ Ob𝕋.

We define infinitesimal neighborhoods of the subcategory 𝕋 by 𝕋^{•2} = 𝕋 •_{X} 𝕋 and 𝕋^{•n} = 𝕋^{•n−1} •_{X} 𝕋 for n ≥ 3. Finally, we set 𝕋^{∞} = ⋃_{n≥2} 𝕋^{•n}.

6.4.2. Gabriel product of topologizing subcategories. Let C_{X} be an abelian category and 𝕊, 𝕋 a pair of subcategories of C_{X}.

(i) If the subcategories 𝕊 and 𝕋 are topologizing, (i.e. they are closed under subquotients and coproducts taken in C_{X}), then 𝕋 •_{X} 𝕊 is topologizing too.

(ii) Gabriel product of topologizing subcategories is associative and has a unit element: 𝕋 •_{X} 𝕆 = 𝕆 •_{X} 𝕋 = 𝕋 for any topologizing subcategory 𝕋. Here 𝕆 denotes the zero subcategory of C_{X}.

(iii) If 𝕊 and 𝕋 are coreflective (resp. reflective) topologizing subcategories, then 𝕋•_{X} 𝕊 is coreflective (resp. reflective).

(iv) A topologizing subcategory 𝕋 of C_{X} is thick iff 𝕋 •_{X} 𝕋 = 𝕋.

(v) Let 𝕋 be a topologizing subcategory. Then 𝕋^{∞} = ⋃_{n≥2} 𝕋^{•n} is the thick envelope of 𝕋.

(vi) In particular, if 𝕋 and 𝕊 are topologizing subcategories of C_{X}, then (𝕋 •_{X} 𝕊)^{∞} is the thick envelope of 𝕋 ∪ 𝕊.

6.4.3. Gabriel product of coimmersions. If V ⟵^{𝔳} X ⟶^{𝔴} W are coimmersions, then we define their Gabriel coproduct, V •_{X} W by setting C_{V •_{X}W} = 𝔳^{∗}(C_{V}) •_{X} 𝔴^{∗}(C_{W}). Since 𝔳^{∗}(C_{V}) and 𝔴^{∗}(C_{X}) are equivalent to topologizing subcategories (resp. 𝔳^{∗}(C_{V}) •_{X} 0 and 𝔴^{∗}(C_{X}) •_{X} 0) of C_{X}, the subcategory C_{V •_{X}W} is topologizing, i.e. V •_{X} W is a cosubspace of X.

6.4.4. Proposition. (a) Let X ⟶^{𝔳} Y be a morphism having an exact inverse image functor, C_{Y} ⟶^{𝔳^{∗}} C_{X}. The the map 𝕋 ⟼ 𝔳^{∗^{−1}}(𝕋) sends topologizing (resp. thick) subcategories of the category C_{X} to topologizing (resp. thick) subcategories of C_{Y}.

(b) If X ⟶^{𝔳} Y is a coimmersion, then the map 𝕋 ⟼ 𝔳^{∗^{−1}}(𝕋) is compatible with Gabriel multiplication:

𝔳^{∗^{−1}}(𝕋 •_{X} 𝕊) = 𝔳^{∗^{−1}}(𝕋) •_{Y} 𝔳^{∗^{−1}}(𝕊)  (1)

for any pair 𝕊 and 𝕋 of topologizing subcategories of the category C_{X}.

Proof. The argument is left to the reader.

6.4.4.1. Note. If Y is a cosubspace of X and 𝔳^{∗} the inclusion functor, then we write 𝕋|_{Y} instead of 𝔳^{∗^{−1}}(𝕋). The equality (1) turns into (𝕋 •_{X} 𝕊)|_{Y} = 𝕋|_{Y} •_{Y} 𝕊|_{Y} .

6.4.5. Proposition. Every coimmersion, X ⟶^{𝔳} Y, induces morphisms of spectra which are vertical arrows of the commutative diagram

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

(2)

Proof. It suffices to show that an inverse image functor, 𝔳^{∗}, of the coimmersion 𝔳 induces embeddings Spec_{𝔰}^{0}(Y) ⟶ Spec_{𝔰}^{0}(X) and Spec_{𝔗𝔥}^{0}(Y) ⟶ Spec_{𝔗𝔥}^{0}(X). These embeddings determine the remaining two vertical arrows of the diagram (2).

For the rest of the argument, we can and will assume that Y is a cosubspace of X.

(a) Let M ∈ Spec_{𝔰}^{0}(Y). We claim that M ∈ Spec_{𝔰}^{0}(X), i.e. for any nonzero subobject, L, of the object M the thick subcategory, [L]_{•}, of C_{X} spanned by L contains M.

By hypothesis, the subcategory C_{Y} is closed under taking subquotients (in particular, subobjects) and coproducts in C_{X}, i.e. inclusion functor 𝔳^{∗} is exact. This means that every subobject, L, of M in C_{X} belongs to C_{Y}. Since M ∈ Spec_{𝔰}^{0}(Y), the thick subcategory [L]_{•}^{Y} of C_{Y} spanned by L contains M. But, [L]_{•}^{Y} is the subcategory of [L]_{•}, hence M ∈ Ob[L]_{•}.

(b) Let now M ∈ Spec_{𝔗𝔥}^{0}(Y). By 6.1.3.3, this means that the full subcategory ⟨M⟩_{•}^{Y} of C_{Y} whose objects are N ∈ ObC_{Y} such that M ∉ [N]_{•}^{Y} is a thick subcategory of C_{Y}. We claim that ⟨M⟩_{•} is a thick subcategory of the category C_{X}.

Let 𝕋 and 𝕊 be thick subcategories of the category C_{X} which do not contain the object M. Then their Gabriel product, 𝕋 • 𝕊, does not contain M.

Indeed, 𝕋|_{Y} and 𝕊|_{Y} do not contain M which means precisely that 𝕋|_{Y} and 𝕊|_{Y} are subcategories of the category ⟨M⟩_{•}. Since the subcategory ⟨M⟩_{•}^{Y} is thick, that is ⟨M⟩_{•}^{Y} •_{Y} ⟨M⟩_{•}^{Y} = ⟨M⟩_{•}^{Y} , the Gabriel product 𝕋|_{Y} •_{Y} 𝕊|_{Y} is a subcategory of ⟨M⟩_{•}, hence does not contain the object M. Since 𝕋|_{Y} •_{Y} 𝕊|_{Y} = (𝕋 •_{X} 𝕊)|_{Y} (see 6.4.3 and 6.4.3.1), M does not belong to 𝕋 •_{X} 𝕊. Same argument shows that the Gabriel product of any number of thick subcategories of C_{X} which do not contain M does not contain M. In particular, M ∉ ⋃_{n≥1}(𝕋 •_{X} 𝕊)^{•n} = (𝕋 •_{X} 𝕊)^{∞}. But, (𝕋 •_{X} 𝕊)^{∞} is the thick envelope of 𝕊 and 𝕋 (see 6.4.2(vi)).

This shows that ⟨M⟩_{•} •_{X} ⟨M⟩_{•} = ⟨M⟩_{•}, i.e. ⟨M⟩_{•} is a thick subcategory of C_{X}.

6.4.6. Corollary. (a) For any pair of coimmersions W ⟵^{𝔴} X ⟶^{𝔳} V, there are inclusions

Spec_{𝔰}^{0}(V •_{X} W) ⊇ Spec_{𝔰}^{0}(V) ⋃ Spec_{𝔰}^{0}(W)
Spec_{𝔗𝔥}^{0}(V •_{X} W) ⊇ Spec_{𝔗𝔥}^{0}(V) ⋃ Spec_{𝔗𝔥}^{0}(W)  (3)

(b) If the functors 𝔴^{∗} and 𝔳^{∗} induce equivalences between the categories C_{V} and C_{W} and thick subcategories of the category C_{X}, then the inclusions in (3) can be replaced by equalities. In particular, in this case

Spec_{𝔰}^{0}(V •_{X} W) = Spec_{𝔰}^{0}(V) ⋃ Spec_{𝔰}^{0}(W)

Spec_{𝔗𝔥}^{0}(V •_{X} W) = Spec_{𝔗𝔥}^{0}(V) ⋃ Spec_{𝔗𝔥}^{0}(W)  (4)

Proof. (a) The inclusions

Spec_{𝔰}^{0}(V •_{X} W) ⊇ Spec_{𝔰}^{0}(V) ⋃ Spec_{𝔰}^{0}(W)

and

Spec_{𝔗𝔥}^{0}(V •_{X} W) ⊇ Spec_{𝔗𝔥}^{0}(V) ⋃ Spec_{𝔗𝔥}^{0}(W)

follow from 6.4.5.

(b) For the rest of the argument, we can and will assume that V and W are cosubspaces of X, such that C_{V} and C_{W} are thick subcategories of C_{X}.

Let M ∈ Spec_{𝔰}^{0}(V •_{X} W), i.e. there exists an exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

such that M′ ∈ ObC_{W} and M′′ ∈ ObC_{V}. If M′ is nonzero, then M ∈ Ob[M′]_{•}. Since C_{W} is a thick subcategory, [M′]_{•} ⊆ C_{W}; in particular, M is an object of the subcategory C_{W}. Because C_{W} is a thick subcategory of C_{X}, the intersection ⟨M⟩_{•} ⋂ C_{W} coincides with ⟨M⟩_{•}^{W}. Therefore M ∈ Spec_{𝔰}^{0}(W).

If the object M′ is zero, then M ∈ ObC_{V} which implies that M ∈ Spec_{𝔰}^{0}(V).

The argument for Spec_{𝔗𝔥}^{0} is similar and is left to the reader.

6.4.7. Proposition. Let Y be a cosubspace of X such that C_{Y} is a thick subcategory of C_{X}. Then there are natural inclusions:

Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔰}^{0}(Y) ∐ Spec_{𝔰}^{0}(X/Y)  (5)

and

Spec_{𝔗𝔥}^{0}(X) ⟶ Spec_{𝔗𝔥}^{0}(Y) ∐ Spec_{𝔗𝔥}^{0}(X/Y).  (6)

Here X/Y is defined by C_{X/Y} = C_{X}/C_{Y}.

Proof. (i) It follows from 2.2.3 that the exact localization X/Y ⟶^{q_{Y}} X induces an embedding

Spec_{𝔗𝔥}^{0}(X) − Spec_{𝔗𝔥}^{0}(Y) ⟶ Spec_{𝔗𝔥}^{0}(X/Y).

This map, together the identical map Spec_{𝔗𝔥}^{0}(Y) onto itself, determines a morphism (6). (ii) The morphism (6) can be also described in terms of the map

Spec_{𝔗𝔥}^{0}(X) − ObC_{Y} ⟶ Spec_{𝔗𝔥}^{0}(X/Y)

induced by an inverse image (localization) functor C_{X} ⟶^{q_{Y}^{∗}} C_{X}/C_{Y} .

In fact, if M ∈ ObC_{X} − ObC_{Y}, then, C_{Y} = Ker(q_{Y}^{∗}) ⊆ ⟨M⟩_{•}. Therefore q_{Y}^{∗} (⟨M⟩_{•}) = ⟨q_{Y}^{∗} (M)⟩_{•}. Because if 𝕋 is a thick subcategory of C_{X}/C_{Y} which does not contain q_{Y}^{∗} (M), then q_{Y}^{∗^{−1}}(𝕋) is a thick subcategory of C_{X} which does not contain M, hence q_{Y}^{∗^{−1}}(𝕋) ⊆ ⟨M⟩_{•} which implies that 𝕋 ⊆ q_{Y}^{∗} (⟨M⟩_{•}). If M ∈ Spec_{𝔗𝔥}^{0}(X), i.e. the subcategory ⟨M⟩_{•} is thick, then q_{Y}^{∗} (⟨M⟩_{•}) = ⟨q_{Y}^{∗} (M)⟩_{•} is a thick subcategory of the category C_{X}/C_{Y}, that is q_{Y}^{∗} (M) ∈ Spec_{𝔗𝔥}^{0}(X/Y).

(iii) Let M ∈ Spec_{𝔰}^{0}(X) − ObC_{Y}, i.e. M is ⟨M⟩_{•}-torsion free. Since Ker(q_{Y}^{∗}) ⊆ ⟨M⟩_{•}, this implies that q_{Y}^{∗} (M) is ⟨q_{Y}^{∗} (M)⟩_{•}-torsion free, hence q_{Y}^{∗} (M) belongs to Spec_{𝔰}^{0}(X/Y).

6.5. 𝔏-local ’spaces’ and categories. Let C_{X} be an abelian category. We define the residue thick cosubspace, X_{⋆}, of the ’space’ X by setting C_{X_{⋆}} equal to the intersection of all nonzero thick subcategories of X. Then X is 𝔏-local iff the residue thick cosubspace of X is non-trivial, i.e. C_{X_{⋆}} is a nonzero subcategory of C_{X}. It follows that C_{X_{⋆}} = [M]_{•} and ⟨M⟩_{•} = 0 for any nonzero object M of C_{X_{⋆}}. In particular, every nonzero object of C_{X_{⋆}} belongs to Spec_{𝔰}^{0}(X).

6.5.1. Proposition. Suppose X is 𝔏-local and the category C_{X} has simple objects. Then all simple objects of C_{X} are isomorphic to each other, and C_{X_{⋆}} = [M]_{•}, where M is a simple object.

Proof. Let M ∈ ObC_{X} be a simple object. In particular, M is nonzero, hence C_{X_{⋆}} ⊆ [M]_{•}. But, if L is a nonzero object of the minimal thick subcategory, [M]_{•}, spanned by M, then M is a subobject of L. Therefore, M ∈ [L] ⊆ [L]_{•}. This shows the inverse inclusion, [M]_{•} ⊆ C_{X_{⋆}}, hence C_{X_{⋆}} = [M]_{•}.

If L is another simple object, then [M]_{•} = [L]_{•}, in particular, L ∈ Ob[M]_{•} which implies that M is a subobject of L (see above). Since the object L is simple, M ≃ L.

6.5.2. Residue (skew) field of an 𝔏-local category with simple objects. Suppose X is a 𝔏-local category such that the category C_{X} has a simpe object, M. We denote by k(X) the ring C_{X}(M, M)^{o} opposite to the ring of endomorphims of the object M. Since M is simple, k(X) is a skew field which we call the residue skew field of the 𝔏-local ’space’ X. By 6.5.1, the residue field of X is defined uniquely up to isomorphism.

6.5.3. The thick spectrum and the Serre spectrum. For any thick subcategory, 𝕋 of the category C_{X}. let 𝕋^{⋆} denote the residue category of 𝕋 defined as the intersection of all thick subcategories of C_{X} properly containing 𝕋. We denote by Spec_{𝔗𝔥}^{1}(X) the preorder of all thick subcategories 𝕋 such that 𝕋 ≠ 𝕋^{⋆}.

One can see that Spec_{𝔗𝔥}^{1}(X) can be identified with the family of all thick cosubspaces, P, of X (that is, C_{P} is a thick subcategory of C_{X}) such that the quotient ’space’ X/P (defined by C_{X/P} = C_{X}/C_{P}) is 𝔏-local. The canonical functor

𝔗𝔥(X) ⟶ S^{𝔰}M(X), 𝕋 ⟼ Σ_{𝕋},

(cf. 6.1) induces an isomorphism from Spec_{𝔗𝔥}^{1}(X) to the complete spectrum, Spec_{𝔏}^{1}(X), of X (defined in Section 4).

Let Spec_{𝔰}^{1}(X) denote the family of all Serre cosubspaces, P, of X (i.e. C_{P} is a Serre subcategory of C_{X}) such that X/P is 𝔏-local. By 3.2 and 6.3.6, we have inclusions

Spec_{𝔗𝔥}^{0}(X) ↪ Spec_{𝔗𝔥}^{1}(X) and Spec_{𝔰}^{0}(X) ↪ Spec_{𝔰}^{1}(X).

It follows from 4.2 that

Spec_{𝔗𝔥}^{1}(X) = ⋃_{C_{Y} ∈𝔗𝔥(X)} Spec_{𝔗𝔥}^{0}(X/Y) = ⋃_{P∈Spec_{𝔗𝔥}^{1}(X)} Spec_{𝔗𝔥}^{0}(X/P).

Similarly,

Spec_{𝔰}^{1}(X) = ⋃_{C_{Y} ∈𝔖𝔢(X)} Spec_{𝔰}^{0}(X/Y) = ⋃_{P∈Spec_{𝔰}^{1}(X)} Spec_{𝔰}^{0}(X/P).

Here 𝔖𝔢(X) is the family of all Serre subcategories of the category C_{X}.

7. Spectra associated with the category of cosubspaces.

Let C_{X} be an abelian category. We denote by 𝔗(X) the preorder (with respect to ⊆) of all topologizing subcategories of the category C_{X}. The Gabriel multiplication, •_{X}, is a structure of a monoidal category on 𝔗(X) with the unit object 𝕆 = {0}.

7.1. The subcategories ⟨P⟩. Let 𝔗_{⋆}(X) denote 𝔗(X) − {𝕆}. For any P ∈ 𝔗_{⋆}(X), let P^{⊢} denote the family of all 𝕋 ∈ 𝔗(X) such that P ⊈ 𝕋. Let ⟨P⟩ denote the union of all 𝕋 ∈ P^{⊢}.

If P, P′ are nonzero topologizing subcategories of C_{X}, then P ⊆ P′ iff ⟨P⟩ ⊆ ⟨P′⟩.

7.2. Subcategories [M] and the preorder ≻. For any M ∈ ObC_{X}, let [M] denote the topologizing subcategory spanned by M; i.e. [M] is the intersection of all topologizing subcategories containing M.

For any nonzero object M of the category C_{X}, let ⟨M⟩ denote the union of all topologizing subcategories of C_{X} which do not contain the object M. It follows that ⟨M⟩ = ⟨[M]⟩.

For two objects, M and N, of the category C_{X}, we write M ≻ N if N is a subquotient of a coproduct of finite number of copies of M.

7.2.1. Lemma. For any two objects M, N of the category C_{X}, the following conditions are equivalent:

(i) M ≻ N, (ii) [N] ⊆ [M], (iii) ⟨N⟩ ⊆ ⟨M⟩.

Proof. (i)⇒(ii). If M ≻ N, then N ∈ Ob[M], hence [N] ⊆ [M]. (ii)⇒(i). The relation ≻ is transitive and for any object, N, a finite coproduct, N′,

of copies of N N ≻ N′. This implies that the full subcategory of C_{X} whose objects are L ∈ ObC_{X} such that M ≻ L is topologizing, hence it coincides with [M]. Thus, we have the implications: ([N] ⊆ [M]) ⇒ (N ∈ [M]) ⇒ (M ≻ N).

The equivalence of (ii) and (iii) is observed in 7.1.

7.3. Proposition. The following conditions on a nonzero topologizing subcategory P of C_{X} are equivalent:

(i) ⟨P⟩ is a thick subcategory of the category C_{X};

(ii) ⟨P⟩ is a topologizing subcategory of C_{X} and P^{⊢} is a monoidal subcategory of the monoidal category (𝔗(X), •_{X});

(iii) there exists an object M ∈ P such that P = [M] and ⟨M⟩ is a thick subcategory of the category C_{X}.

If the equivalent conditions above hold, then P = [M] for every M ∈ ObP − Ob⟨P⟩.

Proof. (i)⇒(ii). Suppose ⟨P⟩ is a thick subcategory of the category C_{X}. This means precisely the following:

(a) ⟨P⟩ is topologizing, hence it is a final object of the category P^{⊢};

(b) ⟨P⟩ = ⟨P⟩ •_{X} ⟨P⟩. If 𝕋 and 𝕊 are objects to P^{⊢}, then 𝕊 ⊆ ⟨P⟩ ⊇ 𝕋, whence 𝕋 •_{X} 𝕊 ⊆ ⟨P⟩ •_{X} ⟨P⟩ = ⟨P⟩. (ii)⇒(iii). Suppose that P ∈ 𝔗_{⋆}(X) is such that ⟨P⟩ is a topologizing subcategory of

C_{X}. Then P = [M] for every M ∈ ObP − Ob⟨P⟩.

In fact, [M] ⊆ P, or, equivalently, ⟨M⟩ ⊆ ⟨P⟩, for every M ∈ ObP. If M ∈ ObP − Ob⟨P⟩, then, since ⟨P⟩ ∈ 𝔗(X), the inverse inclusion holds: ⟨P⟩ ⊆ ⟨M⟩. Therefore ⟨M⟩ = ⟨P⟩, or, equivalently, P = [M].

The implication (iii)⇒(i) is obvious.

7.4. The spectrum Spec_{𝔱}^{0}(X). The spectrum, Spec_{𝔱}^{0}(X), of X consists of all P ∈ 𝔗_{⋆}(X) such that ⟨P⟩ is a thick subcategory of C_{X}. Here _{𝔱} stands for ’topologizing’.

Let Spec_{𝔱}^{0}(X) the family of all objects M of C_{X} such that [M] ∈ Spec_{𝔱}^{0}(X). In other words, M ∈ Spec_{𝔱}^{0}(X) iff ⟨M⟩ is a thick subcategory of C_{X}.

7.5. Proposition. Spec_{𝔱}^{0}(X) ⊆ Spec_{𝔗𝔥}^{0}X.

Proof. For every M ∈ ObC_{X}, the subcategory ⟨M⟩_{•} is contained in ⟨M⟩. In fact, Ob⟨M⟩_{•} = {N ∈ ObC_{X} | M ∉ [N]_{•}} ⊆ {N ∈ ObC_{X} | M ∉ [N]} = Ob⟨M⟩.

If M ∈ Spec_{𝔱}^{0}(X), i.e. ⟨M⟩ is a thick subcategory of the category C_{X}, then we have the inverse inclusion: ⟨M⟩ ⊆ ⟨M⟩_{•}. Therefore ⟨M⟩ = ⟨M⟩_{•} and ⟨M⟩_{•} is a thick subcategory of the category C_{X}, i.e. M ∈ Spec_{𝔗𝔥}^{0}X.

7.6. The spectra Spec(X) and Spec(X). Let Spec(X) denote the family of all nonzero objects M such that ⟨M⟩ is contained in ^{⊥}M. Recall that ^{⊥}M is the full subcategory of C_{X} whose objects are left orthogonal to M; i.e. Ob^{⊥}M = {L ∈ ObC_{X} | C_{X}(L, M) = 0}. In other words, M ∈ Spec(X) iff M ∈ [N] for every object N such that there exists a nonzero morphism N ⟶ M.

We set Spec(X) = {[M] | M ∈ Spec(X)}.

7.6.1. Proposition. (a) If M ∈ Spec(X), then ⟨M⟩ is a Serre subcategory of the category C_{X}. In particular, Spec(X) is contained in Spec_{𝔱}^{0}(X).

(b) Spec(X) = Spec_{𝔱}^{0}(X) ⋂ Spec_{𝔰}^{0}(X).

(c) If X has the property (sup), then Spec(X) = {P ∈ 𝔗_{⋆}(X) | ⟨P⟩ is a Serre subcategory of C_{X}}.

Proof. (a) Since ⟨M⟩ is closed under taking subquotients, ⟨M⟩ ⊆ ⟨M⟩^{−} for any nonzero object M of the category C_{X}. Since ⟨M⟩^{−} is a Serre (in particular topologizing) category, ⟨M⟩ ≠ ⟨M⟩^{−} iff M ∈ Ob⟨M⟩^{−}. The latter implies the existence of a nonzero morphism N ⟶ M with N ∈ ⟨M⟩. If M were an object of Spec(X), then M ∈ [N] ⊆ ⟨M⟩ which is impossible by the definition of ⟨M⟩. Therefore ⟨M⟩ = ⟨M⟩^{−}.

(b) If M ∈ Spec(X), then ⟨M⟩ is a Serre (in particular, thick) subcategory of the category C_{X}. By (the argument of) 7.5, ⟨M⟩_{•} = ⟨M⟩. In particular, ⟨M⟩_{•} is a Serre subcategory and M is ⟨M⟩_{•}-torsion free, i.e. M ∈ Spec_{𝔰}^{0}(X). Together with (a) above, this gives the inclusion Spec(X) ⊆ Spec_{𝔱}^{0}(X) ⋂ Spec_{𝔰}^{0}(X).

Conversely, let M ∈ Spec_{𝔱}^{0}(X) ⋂ Spec_{𝔰}^{0}(X). This means that ⟨M⟩ is a thick subcategory and ⟨M⟩_{•} is a Serre subcategory of the category C_{X}, and M is ⟨M⟩_{•}-torsion free. But, by (the argument of) 7.5, the subcategories ⟨M⟩ and ⟨M⟩_{•} coincide. Therefore M ∈ Spec(X).

(c) If X has a property (sup), then by 6.3.10, Spec_{𝔰}^{0}(X) = {P ∈ 𝔗_{⋆}(X) | ⟨P⟩_{•} is a Serre subcategory of C_{X}}.

It follows from (the argument of) (b) above that Spec(X) = Spec_{𝔱}^{0}(X) ⋂ Spec_{𝔰}^{0}(X) = {P ∈ 𝔗_{⋆}(X) | ⟨P⟩_{•} = ⟨P⟩ is a Serre subcategory of C_{X}}.

7.6.2. Proposition. The following conditions are equivalent:

(a) Every object of Spec_{𝔰}^{0}(X) contains a subobject which belongs to Spec(X).

(b) The map which assigns to every topologizing subcategory, 𝕋, of the category C_{X} the minimal thick subcategory, 𝕋_{•}, containing 𝕋 induces an isomorphism

Spec(X) ⟶^{∼} Spec_{𝔰}^{0}(X).

Proof. (a)⇒(b). It follows from 7.6.1 that the map 𝕋 ⟼ 𝕋_{•} induces an embedding Spec(X) ↪ Spec_{𝔰}^{0}(X). By definition, Spec_{𝔰}^{0}(X) = {[P]_{•} | P ∈ Spec_{𝔰}^{0}(X)}. If P′ is a nonzero subobject of an object P ∈ Spec_{𝔰}^{0}(X), then [P′]_{•} = [P]_{•}. If P′ ∈ Spec(X), then [P]_{•} is the image of an object of Spec(X).

(b)⇒(a). By (b), for every P′ ∈ Spec_{𝔰}^{0}(X), there exists a P ∈ Spec(X) such that [P]_{•} = [P′]_{•}. In particular, P′ ∈ Ob[P]_{•}. Since [P]_{•} ⊆ [P]^{−} and P′ is a nonzero object, P′ has a nonzero subobject, P′′, which belongs to the subcategory [P]. Because P′ ∈ Spec_{𝔰}^{0}(X), and P′′ is a nonzero subobject of P′, the objects P′ and P′′ are equivalent, i.e. [P′′]_{•} = [P′]_{•}, whence [P′′]_{•} = [P]_{•}. The equality [P′′]_{•} = [P]_{•} is equivalent to the equality ⟨P′′⟩_{•} = ⟨P ⟩_{•}, and ⟨P′′⟩_{•} = ⟨P ⟩. In particular, P′′ ∉ ⟨P ⟩. Since P ∈ Spec(X), the latter means that P′′ ≻ P, or, what is the same, P ∈ Ob[P′′]. Thus, P′′ ∈ Ob[P] and P ∈ Ob[P′′] which means that [P′′] = [P] ∈ Spec(X).

8. Local ’spaces’ and the related spectra.

Let C_{X} be an abelian category. We define the residue cosubspace, X_{𝔱}, of X by setting C_{X_{𝔱}} equal to the intersection of all nonzero topologizing subcategories of C_{X}. We call X and the category C_{X} local if the residue subcategory C_{X_{𝔱}} is nonzero.

Thus, if X is local, then [M] = C_{X_{𝔱}} and ⟨M⟩ = 𝕆 for every nonzero object of the residue subcategory C_{X_{𝔱}}. In particular, every nonzero object of C_{X_{𝔱}} belongs to Spec(X), and all nonzero objects of C_{X_{𝔱}} are equivalent to each other, i.e. they define a unique smallest element of Spec(X).

8.1. Note. Local categories are defined in [R, Ch.3] as categories with quasi-final objects. An object P is called quasi-final if N ≻ P for every nonzero object N. One can see that quasi-final objects are precisely nonzero objects of the residue subcategory C_{X_{𝔱}}.

8.2. Proposition. (a) Every local ’space’ is 𝔏-local.

(b) Let X be an 𝔏-local ’space’ with the residue thick cosubspace X_{⋆} such that Spec_{𝔱}^{0}(X)_{⋆} is non-empty. Then X is local.

Proof. (a) If X is local, then the residue thick subcategory (the intersection of all nonzero thick subcategories of C_{X}) coincides with the thick envelope of the residue subcategory C_{X_{𝔱}}.

(b) Let X be a 𝔏-local space with non-empty Spec_{𝔱}^{0}(X)_{⋆}. Let P ∈ Spec_{𝔱}^{0}(X)_{⋆}, that is P is a nonzero object of the residue thick subcategory C_{X_{⋆}} and ⟨P ⟩ is a thick subcategory. Since P ∉ Ob⟨P ⟩ and C_{X_{⋆}} = [P]_{•} is the smallest nonzero thick subcategory of the category C_{X}, the subcategory ⟨P ⟩ is zero. This means that P is a quasi-final object (see 8.1), or, equivalently, P is contained in any nonzero topologizing subcategory of the category C_{X}. Therefore X is local with the residue subcategory [P].

8.3. Corollary. If C_{X} has simple objects, then X is local iff it is 𝔏-local.

Proof. Suppose X is 𝔏-local and it has a simple object, M. By 6.5.1, the residue thick subcategory of C_{X} coincides with the minimal thick subcategory, [M]_{•}, containing M. In particular, M ∈ Spec(X). The assertion now follows from 8.2(b).

8.4. Proposition. For every P ∈ Spec(X), the category C_{X}/⟨P ⟩ is local.

Proof. The localization functor C_{X} ⟶^{q_{⟨P ⟩}^{∗}} C_{X}/⟨P ⟩ is exact and its kernel coincides with the Serre subcategory ⟨P ⟩. Every nonzero object of the quotient category C_{X}/⟨P ⟩ is isomorphic to an object q∗_{⟨P ⟩}(N), where N ∉ Ob⟨P ⟩, i.e. N ≻ P. Being exact, the functor q∗_{⟨P ⟩} respects the relation ≻; in particular, q∗_{⟨P ⟩}(N) ≻ q∗_{⟨P ⟩}(P). Since q∗_{⟨P ⟩}(P) ≠ 0, this shows that q∗_{⟨P ⟩}(P) is a quasi-final object of the category C_{X}/⟨P ⟩, that is [q∗_{⟨P ⟩}(P)] is the residue subcategory of C_{X}/⟨P ⟩.

8.5. The complete spectrum and the S-spectrum. The elements of the complete spectrum, Spec_{𝔱}^{1}(X), of the ’space’ X is the family of all thick subcategories C_{Y} such that the quotient ’space’ X/Y (defined by C_{X/Y} = C_{X}/C_{Y}) is local.

The S-spectrum, Spec^{−}(X), is formed by Serre subcategories, C_{V} of C_{X} such that X/V is local. Thus, Spec^{−}(X) ⊆ Spec_{𝔱}^{1}(X).

The notion of the complete spectrum is introduced in [R, Ch.6], details on the Sspectrum can be found in [R, Ch.3 and Ch.6], where it is called the flat spectrum.

The map P ⟼ ⟨P⟩ induces injective morphisms

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

Since ’local’ implies ’𝔏-local’, we have inclusions

Spec_{𝔱}^{1}(X) ⊆ Spec_{𝔗𝔥}^{1}(X) and Spec^{−}(X) ⊆ Spec_{𝔰}^{1}(X).  (2)

The image of the map Spec(X) ⟶ Spec^{−}(X) is described as follows.

8.5.1. Proposition. An element P of Spec^{−}(X) belongs to the image of Spec(X) iff there exists an element, M, of Spec(X) which belongs to ObP^{⋆} − ObP.

Proof. The image of Spec(X) in Spec^{−}(X) consists of Serre subcategories of the form ⟨M⟩, where M ∈ Spec(X). By definition of ⟨M⟩, the object M does not belong to ⟨M⟩ and, obviously, does belong to ⟨M⟩^{⋆} = ([M] ∪ ⟨M⟩)^{−}.

Conversely, let P ∈ Spec^{−}(X) and let M ∈ Spec(X) does not belong to the subcategory P. The claim is that P = ⟨M⟩.

Since M ∉ ObP, we have the inclusion P ⊆ ⟨M⟩. Let q^{∗} denote a localization functor C_{X} ⟶ C_{X}/P. By [R, 3.2.2], since M belongs to Spec(X) and does not belong to P, the object q^{∗}(M) belongs to Spec(X/|P|). Here C_{X/|P|} = C_{X}/P. Because M ∈ ObP^{⋆}, the object q^{∗}(M) is quasi-final. The latter means that q^{∗}(N) ≻ q^{∗}(M) for every N ∈ ObC_{X} − ObP; i.e., there exists a diagram

⨁_{m times} q^{∗}(N) \underleftarrow{_{𝔧}} q^{∗}(K̃) \underrightarrow{_{𝔢}} q^{∗}(M),  (3)

where 𝔧̃is a monomorphism and 𝔢̃ is an epimorphism. To the diagram (3), there corresponds a diagram

⨁_{m times} N ⟵^{𝔧′} K′ ⟶^{s} K̃ ⟵^{t} K′′ ⟶^{𝔢′′} M  (4)

such that q^{∗}(𝔧′) is a monomorphism, q^{∗}(𝔢′′) is an epimorphism, and q^{∗}(s), q^{∗}(t) are isomorphisms. The diagram (4) can be replaced by the diagram

⨁_{m times} N ⟵^{𝔧_{1}} K_{1} ⟶^{𝔢_{1}} M  (5)

where K_{1} = K′ ×_{K} K′′ and 𝔧_{1} (resp. 𝔢_{1}) is the composition of 𝔧′ (resp. 𝔢′′) with the projection K_{1} ⟶ K′ (resp. K_{1} ⟶ K′′). Since q^{∗}(𝔧_{1}) is a monomorphism, Ker(𝔧_{1}) is an object of P. Notice that the composition of K_{1} ⟶^{𝔢_{1}} M with the canonical monomorphism Ker(𝔧_{1}) ↪ K_{1} is zero, because M is P-torsion free. Therefore 𝔢_{1} factors through a morphism K = K_{1}/Ker(𝔧_{1}) ⟶^{𝔢} M. Thus, we have obtained a diagram

⨁_{m times} N ⟵^{𝔧} K ⟶^{𝔢} M  (4)

in which 𝔧 is a monomorphism and 𝔢 is nonzero. Let M_{1} denote the image of 𝔢. The diagram (4) means, by definition, that N ≻ M_{1}. Since M ∈ Spec(X) and M_{1} is a nonzero subobject of M, M_{1} ≻ M. Therefore, N ≻ M for every N ∈ ObC_{X} −ObP, or, equivalently, ⟨M⟩ ⊆ P.

8.6. The flat spectra. Recall that the flat complete 𝔏-spectrum, Spec_{𝔣𝔏}^{1}(X) is formed by all saturated multiplicative systems Σ such that the ’space’ of fractions Σ^{−1}X is 𝔏-local and the localization, Σ^{−1}X ⟶^{q_{Σ}} X, is continuous (i.e. it has a direct image functor) (see 5.3.5).

The flat 𝔏-spectrum, Spec_{𝔣𝔏}^{0}(X) is defined as {Σ ∈ S^{𝔰}M^{⋆}(X) | Σ̂ ∈ Spec_{𝔣𝔏}^{1}(X)}. The map 𝕋 ⟼ Σ_{𝕋} identifies Spec_{𝔣𝔏}^{1}(X) and Spec_{𝔣𝔏}^{0}(X) with corresponding preorders

of coreflective thick (hence Serre) subcategories.

8.6.1. Proposition. (a) There are inclusions:

Spec_{𝔣𝔏}^{1}(X) ⊆ Spec_{ℭ}^{1}(X) ⊆ Spec_{𝔰}^{1}(X) and Spec_{𝔣𝔏}^{0}(X) ⊆ Spec_{ℭ}^{0}(X) ⊆ Spec_{𝔰}^{0}(X).

(b) If C_{X} is a Grothendieck category, then

Spec_{𝔣𝔏}^{1}(X) = Spec_{ℭ}^{1}(X) = Spec_{𝔰}^{1}(X) and Spec_{𝔣𝔏}^{0}(X) = Spec_{ℭ}^{0}(X) = Spec_{𝔰}^{0}(X).

Proof. (a) Let q^{∗} denote the localization functor C_{X} ⟶ C_{X}/P, where P is a thick subcategory. If q^{∗} has a right adjoint, q_{∗}, then the subcategory P is coreflective, i.e. every object of C_{X} has a P-torsion. This follows from that the P-torsion of an object M is the kernel of the canonical (adjunction) morphism M ⟶ q_{∗}q^{∗}(M). The assertion (a) follows now from the fact that every coreflective topologizing (in particular, thick) subcategory is a Serre subcategory of the category C_{X}.

(b) If C_{X} is a Grothendieck category, then every Serre subcategory of the category C_{X} is coreflective, and any exact localization whose kernel is a Serre subcategory has a right adjoint.

8.7. The spectra of abelian categories with Gabriel-Krull dimension. We recall the notion of the Gabriel filtration of an abelian category as it is presented in [R, 6.6]. Let C_{X} be an abelian category. The Gabriel filtration of X assigns to every ordinal 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 ordinal, 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 ordinal, 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}}. Clearly the quotient category C_{X}/C_{X_{ω}} has no simple objects.

An object M is said to have the Gabriel-Krull dimension β, if β is the smallest ordinal 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.

8.7.1. Proposition. Suppose C_{X} is an abelian category with Gabriel-Krull dimension. Then Spec^{−}(X) = Spec_{𝔰}^{1}(X) .

Proof. By [R, 6.6.0.3], if 𝕊 is a proper Serre subcategory of C_{X}, then the quotient category, C_{X}/𝕊, has simple objects. In particular, for any P ∈ Spec_{𝔰}^{1}(X), the 𝔏-local quotient category C_{X}/P has simple objects. By 8.3, any 𝔏-local category with simple objects is local, that is P ∈ Spec^{−}(X). This shows the inclusion Spec_{𝔰}^{1}(X) ⊆ Spec^{−}(X) which together with 8.5(2) above implies the equality Spec^{−}(X) = Spec_{𝔰}^{1}(X).

8.7.2. Proposition. Suppose C_{X} is an abelian category with Gabriel-Krull dimension and the property (sup). Then the following conditions are equivalent:

(a) Any nonzero object of C_{X} has a subobject which belongs to Spec(X).

(b) Spec^{−}(X) = Spec_{𝔰}^{1}(X) = Spec_{𝔰}^{0}(X) = Spec(X).

Proof. (a)⇒(b). The equality Spec^{−}(X) = Spec_{𝔰}^{1}(X) holds by 8.7.1. Since there are inclusions Spec(X) ⊆ Spec_{𝔰}^{0}(X) ⊆ Spec_{𝔰}^{1}(X), and Spec(X) ⊆ Spec^{−}(X), it suffices to show that Spec^{−}(X) ⊆ Spec(X).

Let P ∈ Spec^{−}(X), and let M be an object of C_{X} such that the image of M in C_{X}/P is a simple object. Since C_{X} has the property (sup), every Serre subcategory of C_{X} in particular, P, is coreflective. Therefore, every object of C_{X} has a P-torsion. Replacing the object M by its quotient by the P-torsion, we will assume that M is P-torsion free. By hypothesis (a), M has a subobject, M′ which belongs to Spec(X). Since M′ is P-torsion free, P ⊆ ⟨M′⟩. On the other hand, the Serre subcategory ⟨M′⟩/P of the category C_{X}/P does not contain the image of M, hence ⟨M′⟩/P = 𝕆, that is ⟨M′⟩ = P.

(b)⇒(a). Let N be a nonzero object of the category C_{X}, and let the Gabriel-Krull dimension of N be α + 1. Then N is C_{X_{α}}-torsion free and the image of N in C_{X}/C_{X_{α}} = C_{X/X_{α}} has a simple subobject, M. Let q_{α}^{∗} denote the localization functor C_{X} ⟶ C_{X/X_{α}}. We can and will assume that M is the image of a subobject, M′, of N. We claim that M′ ∈ Spec(X).

In fact, the preimage, P, of the Serre subcategory ⟨M⟩ ⊆ C_{X/X_{α}} in C_{X} belongs to Spec^{−}(X). Since, by (b), Spec^{−}(X) = Spec(X), i.e. P = ⟨P ⟩ for some P ∈ Spec(X). The object P can be chosen in such a way that q_{α}^{∗}(P) is a simple object of the category C_{X/X_{α}}. This implies that q_{α}^{∗}(P) is isomorphic to the object M. In particular, there exists a

monomorphism q_{α}^{∗}(P) ⟶ q_{α}^{∗}(N), or, equivalently, there exists a diagram P ⟵^{s} P′ ⟶^{h} N such that q_{α}^{∗}(s) is an isomorphism and q_{α}^{∗}(h) is a monomorphism. Taking quotient with respect to C_{X_{α}}-torsion, we can assume that both s and h are monomorphisms. Since P ∈ Spec(X), every nonzero subobject of P is equivalent to P. Thus P′ ∈ Spec(X) and [P′] = [P].

8.7.3. Corollary. Let C_{X} be the category R − mod of left modules over a left noetherian PI ring R. Then

Spec^{−}(X) = Spec_{𝔰}^{1}(X) = Spec_{𝔰}^{0}(X) = Spec(X) ≃ SpecR.

Here SpecR denotes the prime spectrum of the ring R.

Proof. (a) If C_{X} = R − mod for an arbitrary associative ring, then every object of Spec(X) is equivalent to any of its cyclic submodules, i.e. modules of the form R/p for some left ideal p. The set Spec_{ℓ}R = {p ∈ I_{ℓ}R | R/p ∈ Spec(X)} is called in [R] the left spectrum of the ring R. For any pair of left ideals, 𝔪 and 𝔫, in R, the relation R/𝔪 ≻ R/𝔫 (same as R/𝔪 ∈ [R/𝔫]) means exactly that there exists a finite set, x, of elements of R such that (𝔪 : x) ⊆ 𝔫 (cf. [R, 3.4]), where (𝔪 : x) = {r ∈ R | rx ⊂ 𝔪}. If 𝔪 ∈ Spec_{ℓ}R, and a finite set x ⊂ R is not contained in 𝔪, then (𝔪 : x) is equivalent to 𝔪, that is [R/𝔪] = [R/(𝔪 : x) (cf. [R, Ch.1]. For any 𝔪 ∈ Spec_{ℓ}R, the set (𝔪 : R) = {a ∈ R | aR ⊆ 𝔪} (which is the biggest two-sided ideal contained in 𝔪) is a prime ideal in R. If R is a PI ring, then for any left ideal, 𝔪, there exists a finite subset, x, of elements of the ring R such that (𝔪 : R) = (𝔪 : x). In particular, every object of Spec(X) is equivalent to the quotient module R/p, where p is a prime ideal. It remains to observe that the restriction of the map 𝔪 ⟼ [R/𝔪] to two-sided ideals is injective (see [R, Ch.1]). All together shows that the map p ⟼ [R/p] is an isomorphism of SpecR onto Spec(X).

(b) Let A be an associative unital ring, m a left ideal in A. Consider the set of ideals (m : x), where x runs through A − m. Suppose this set has a maximal element, (m : y), with respect to ≤. Then (m : y) ∈ Spec_{ℓ}(A).

In fact, for any r ∈ A − (m : y), we have (m : y) ≤ ((m : y) : r) = (m : ry) ≠ A. Thanks to the maximality of (m : y), ((m : y) : r) = (m : ry) ≤ (m : y), hence the assertion.

(c) Every nonzero R-module has a submodule which belongs to Spec(X).

It suffices to prove this fact for an arbitrary quotient module, R/m. For every x ∈ R, let x̄ denote the image of x in R/m. Then Ann(x̄) = (m : x). Thus, {(m : x) | x ∈ R − m} is the set of annihilators of nonzero elements of the module R/m. By (a), each ideal (m : x) is equivalent to the maximal two-sided ideal contained in (m : x) (the annihilator of the module R/(m : x), which is ((m : x) : R) = (m : Rx)

Since the ring R is noetherian, the set of two-sided ideals (m : Rx) | x ∈ R − m} has a maximal element, (m : Rz). Therefore, (m : z) is a maximal element of the set {(m : x) | x ∈ R − m}. By (b), (m : z) belongs to Spec_{ℓ}(R), hence the cyclic submodule Rz̄ ≃ R/(m : z) belongs to Spec(X).

The assertion follows now from 8.7.2.

8.7.4. Corollary. Let C_{X} be the category Qcoh_{X} of quasi-coherent sheaves on a scheme X = (X , O). If the scheme X is noetherian, then

Spec^{−}(X) = Spec_{𝔰}^{1}(X) = Spec_{𝔰}^{0}(X) = Spec(X) ≃ X .

Proof. Since the category of quasi-coherent sheaves on a noetherian scheme is locally noetherian, the equalities Spec^{−}(X) = Spec_{𝔰}^{1}(X) = Spec_{𝔰}^{0}(X) = Spec(X) follow from 8.7.2. By [R, 6.1.1 and 6.1.2], the spectrum Spec^{−}(X) is naturally isomorphic to the Gabriel’s spectrum of C_{X} formed by isomorphism classes of indecomposable injective objects. Gabriel proved (in [Gab]) that the underlying space of a noetherian scheme is isomorphic to the Gabriel’s spectrum of the category of quasi-coherent sheaves on that scheme.

C2. The spectrum Spec_{𝔖}^{0}(X).

Fix a triple (X; 𝔐, E), where X is a ’space’, 𝔐 (resp. E) is a family of monomorphisms (resp. epimorphisms) closed under the composition and such that 𝔐∩E contains the class Iso(C_{X}) of all isomorphisms of the category C_{X}.

A choice of 𝔐 to keep in mind is the class 𝔐_{s}(X) of all strict monomorphisms of the category C_{X}. Recall that an arrow L ⟶^{g} M is called a strict monomorphism if it is determined uniquely up to isomorphism by the family of all pairs of arrows M ⇉ N equalizing g. If fibred coproducts of pairs of monomorphisms M ⟵^{g} L ⟶^{g} M exist in C_{X}, then 𝔐_{s}(X) consists of all arrows L ⟶ M such that the canonical diagram L ⟶ M \overrightarrow{⟶} M ∐_{L} M is exact.

Dually, a standard choice of E is the family E_{s}(X) of all strict epimorphisms of C_{X}. If C_{X} has fibred products of pairs of epimorphisms M ⟶^{h} N ⟵^{h} M, then E_{s}(X) consists of all arrows L ⟶ M such that the canonical diagram M ∏_{N} M \overrightarrow{⟶} M ⟶ N is exact.

Thus, if C_{X} is an abelian category, then every monomorphism (resp. epimorphism) of C_{X} is strict.

C2.1. Definition of Spec_{𝔖}^{0}(X). Let Spec_{𝔖}^{0}(X) denote the preorder (with respect to ⊆) of all Σ ⊂ Hom(C_{X}) satisfying the following conditions:

(i) The family Σ contains all identical arrows, and Σ ⊈ Iso(C_{X}).

(ii) If ts ∈ Σ, and either s ∈ E, or t ∈ 𝔐, then s ∈ Σ.

(iii) If ts ∈ Σ and s ∈ Iso(C_{X}), then t ∈ Σ. (iv) Σ ∩ Σ̂ = Iso(C_{X}), and Σ̂ ∈ S^{𝔰}M(X).

Note that already the properties (i) and (iii) imply that, Iso(C_{X}) ⊊ Σ.

For any family Σ ⊆ Hom(C_{X}), we denote by Σ_{•} the smallest saturated multiplicative system containing Σ. It follows that Σ̂_{•} = Σ̂. We denote by Spec_{𝔖}^{0}(X) the preorder ({Σ_{•} | Σ ∈ Spec_{𝔖}^{0}(X)}, ⊆). It follows from (iv) above that Spec_{𝔖}^{0}(X) ⊆ Spec_{𝔏}^{0}(X).

C2.2. Another realization of Spec_{𝔖}^{0}(X). Consider the following equivalence relation on Spec_{𝔖}^{0}(X): Σ ∼ Σ′ iff Σ_{•} = Σ′_{•}; that is Σ and Σ′ are equivalent iff they have the same image in Spec_{𝔖}^{0}(X). Every equivalence class has the biggest element which is the union of all Σ′ ∈ Spec_{𝔖}^{0}(X) such that Σ′_{•} = Σ_{•}, or, what is the same, Σ̂′ = Σ̂.

In fact, the properties (i), (ii), and (iii) are preserved by the union of every family of classes having these properties. The property (iv) is preserved by the union of a family of equivalent classes.

We denote by 𝔖𝔭𝔢𝔠_{𝔖}^{0}(X) the preorder (with respect to ⊆) of the biggest elements of Spec_{𝔖}^{0}(X). The map Σ ⟼ Σ_{•} induces an isomorphism of preorders

𝔖𝔭𝔢𝔠_{𝔖}^{0}(X) ⟶^{∼} Spec_{𝔖}^{0}(X).

C2.3. Subcategories S_{P} and associated classes of morphisms. Let C_{X} be an abelian category; and let 𝔐 and E be the families of resp. all monomorphisms and all epimorphisms of C_{X}. For any subcategory 𝕋 of the category C_{X}, we set Σ_{𝕋} = {s ∈ HomC_{X} | Ker(s) ∈ Ob𝕋 ∋ Cok(s)}.

For any object P ∈ C_{X}, we denote by S_{P} the full subcategory of the category C_{X} whose objects are all subobjects of P.

C2.3.1. Proposition. For any nonzero object P, the family of arrows Σ_{S_{P}} has properties (i), (ii), (iii) of C2.1.

The family Σ_{S_{P}} belongs to Spec_{𝔖}^{0}(X) (i.e. it satisfies the property C2.1(iv)) iff P is an object of Spec_{𝔰}^{0}(X).

Proof. (i) For any P ∈ ObC_{X} the family of arrows Σ_{S_{P}} contains all isomorphisms of the category C_{X}. If P = 0, then Σ_{S_{P}} = Iso(C_{X}). If P ≠ 0, then Iso(C_{X}) ⊊ Σ_{S_{P}}, because, for example, the morphism 0 ⟶ P belongs to Σ_{S_{P}}.

(ii) Suppose that the composition of morphisms L ⟶^{s} M ⟶^{t} N belongs to Σ_{S_{P}}; i.e. Ker(ts) ∈ ObS_{P} ∋ Cok(ts). Since Ker(s) is a subobject of Ker(ts) the subcategory S_{P} is closed under taking subobjects, Ker(s) ∈ ObS_{P}. If s ∈ E, this means that s ∈ Σ_{S_{P}}.

If s ∉ E, but t ∈ 𝔐, then the sequence

0 ⟶ Cok(s) ⟶ Cok(ts) ⟶ Cok(t) ⟶ 0

is exact. In particular, Cok(s) ∈ ObS_{P}, because Cok(ts) ∈ ObS_{P}. Therefore s ∈ Σ_{S_{P}}. (iii) Evidently, if s is an isomorphism, then ts ∈ Σ_{S_{P}} iff t ∈ Σ_{S_{P}}. (iv) Let Σ ∈ S^{𝔰}M(X), and let Σ_{S_{P}} ⊈ Σ. If P ∈ Spec_{𝔰}^{0}(X), then Σ_{S_{P}} ∩ Σ = Iso(C_{X}). In fact, let 𝕋 be the kernel of the localization C_{X} ⟶ Σ^{−1}C_{X}. Clearly Σ_{S_{P}} ⊈ Σ iff

P ∉ Ob𝕋; and Σ_{S_{P}} ∩ Σ = Iso(C_{X}) iff S_{P} ∩ 𝕋 = 𝕆. Thus, the property (iv) is equivalent to the following property:

(iv’) For any thick subcategory 𝕋 of the category C_{X}, P ∉ Ob𝕋 iff S_{P} ∩ 𝕋 = 𝕆. Notice that S_{P} ∩ 𝕋 = 𝕆 iff 𝕋 ∈^{⊥} P. Thus, the condition (iv’) means precisely that

P is an object of Spec_{𝔰}^{0}(X).

C2.3.2. Proposition. Let C_{X} be an abelian category. The map P ⟼ Σ_{S_{P}} induces an isomorphism Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖}^{0}(X).

Proof. By C2.3.1, the map P ⟼ Σ_{S_{P}} induces a morphism Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖}^{0}(X). The claim is that this morphism induces an isomorphism Spec_{𝔰}^{0}(X) ⟶ Spec_{𝔖}^{0}(X).

Let Σ ∈ Spec_{𝔖}^{0}(X), i.e. Σ is a family of arrows of the category C_{X} satisfying the conditions (i)–(iv) in C2.1. In particular, Σ ∩ Σ̂ = Iso(C_{X}), or, what is the same, Σ − Σ̂ = Σ − Iso(C_{X}). Therefore, s ∈ Σ − Σ̂ iff s ∈ Σ and either Ker(s) ≠ 0, or Cok(s) ≠ 0.

(a) Suppose Ker(s) ≠ 0. We claim that Ker(s) is an object of Spec_{𝔗𝔥}^{0}X.

The morphism M ⟶^{s} N can be represented as a composition, ts′, of an epimorphism M ⟶^{s′} N′ and a monomorphism N′ ⟶^{t} N. By the property C2.1(ii), s′ ∈ Σ.

Let 𝕋 be a thick subcategory of the category C_{X}. Suppose Ker(s) has a nonzero subobject, M′, which belongs to 𝕋. Then s′ can be presented as a composition of two epimorphisms, M ⟶^{s′′} N′′ ⟶^{t′} N′ such that Ker(s′′) = M′. By C2.1(ii), s′′ ∈ Σ, hence s′′ ∈ Σ ∩ Σ_{𝕋}, where Σ_{𝕋} = {h ∈ HomC_{X} | Ker(h) ∈ Ob𝕋 ∋ Cok(h)}. Since s′′ is not an isomorphism, this implies that Σ ⊆ Σ_{𝕋}, in particular, Ker(s) ∈ Ob𝕋.

(b) Suppose now that s ∈ Σ is such that Ker(s) = 0 and Cok(s) ≠ 0. We claim that Cok(s) ∈ Spec_{𝔗𝔥}^{0}X.

Suppose Cok(s) has a nonzero subobject, M′, which belongs to a thick subcategory 𝕋 of the category C_{X}. Then we have a commutative diagram

0 ⟶  M ⟶^{s′} N′ ⟶ M′ ⟶ 0
id_{M} ↓  t ↓  ↓
0 ⟶  M ⟶^{s} N ⟶ Cok(s) ⟶ 0

whose vertical arrows are monomorphisms and rows are exact sequences. Here N′ = M′ ×_{Cok(s)} N. Since s = ts′ ∈ Σ, s′ ∈ Σ ∩ Σ_{𝕋}. Because s′ is not an isomorphism, this implies that Σ ⊆ Σ_{𝕋}, in particular, Cok(s) ∈ Ob𝕋.

(c) It follows from (a) and (b) that every Σ ∈ Spec_{𝔖}^{0}(X) contains either an epimorphism, s, such that Ker(s) ∈ Spec_{𝔰}^{0}(X), or a monomorphism, t, such that Cok(t) ∈ Spec_{𝔰}^{0}(X).

(c’) Consider the first case, i.e. suppose there exists an s ∈ Σ such that P_{s} = Ker(s) ∈ Spec_{𝔰}^{0}(X). It follows from the property C2.1(ii) that Σ_{[P_{s})} ⊆ Σ, hence { \atop Σ_{[P_{s})}} ⊆ Σ̂. On the other hand, Σ_{[P_{s})} contains the morphism s which does not belong to Σ̂. Therefore, the inverse inclusion, Σ̂ ⊆ { \atop Σ_{[P_{s})}}, holds, i.e. Σ̂ = { \atop Σ_{[P_{s})}} = Σ_{⟨P_{s}⟩}. The latter equalities are equivalent to the equalities Σ_{•} = Σ_{[P_{s})} = Σ_{[P_{s}]_{•}}.

(c”) Suppose now that s ∈ Σ is a monomorphism such that Cok(s) = P^{s} belongs to Spec_{𝔰}^{0}(X). It follows from the argument in (b) above that Σ_{[P^{s})} ⊆ Σ, hence { \atop Σ_{[P^{s})}} ⊆ Σ̂. Since s ∈ Σ_{[P^{s})} − Σ̂, the inverse inclusion holds, i.e. Σ_{•} = Σ_{[P^{s})} = Σ_{[P^{s}]_{•}}, or, equivalently, Σ_{•} = Σ_{[P^{s})} = Σ_{[P^{s}]_{•}}. This concludes the argument.

C2.4. A description of 𝔖𝔭𝔢𝔠_{𝔖}^{0}(X). Let C_{X} be an abelian category. For any M ∈ ObC_{X}, we denote by [M) the smallest full subcategory of the category C_{X} which contains all objects equivalent to M with respect to the preorder ≻ and is closed under taking subobjects. Recall that N ≻ L iff L is a subquotient of a finite coproduct of copies of N. Objects N and L are equivalent with respect to ≻ iff [N] = [L].

One can show that for any object M, the subcategory [M) is closed under finite coproducts. In particular, [M) is an additive (but, in general, not abelian) subcategory of the category C_{X}.

C2.4.1. Proposition. Let C_{X} be an abelian category. Then

𝔖𝔭𝔢𝔠_{𝔖}^{0}(X) = {Σ_{[P)} | P ∈ Spec_{𝔰}^{0}(X)}.

Proof. If P is an object of Spec_{𝔰}^{0}(X), then every nonzero object of the subcategory [P) is equivalent to P with respect to ≻. Thus, [P) is the union of the full subcategories S_{M} (whose objects are subobjects of M), where M runs through the class of objects equivalent to P, i.e. such that M ≻ P ≻ M, or, equivalently, [M] = [P]. The assertion follows now from (the arguments of) C2.3.1 and C2.3.2. Details are left to the reader.

III. Supports, associated points, and primary decomposition.

9. Supports and weakly associated points.

Fix a ’space’ X. For any S ⊆ HomC_{X}, let Supp_{𝔏}(S) denote {Σ ∈ Spec_{𝔏}^{0}(X) | S ⊈ Σ̂}. We call Supp_{𝔏}(S) the support of S in Spec_{𝔏}^{0}(X), or simply the support of S.

It follows that Supp_{𝔏}(S) is closed with respect to the specialization preorder: if Σ ∈ Supp_{𝔏}(S) and Σ′ ∈ Spec_{𝔏}^{0}(X) is contained in Σ, then Σ′ ∈ Supp_{𝔏}(S).

9.1. Proposition. For any set {S_{i} ⊆ HomC_{X} | i ∈ J},

Supp_{𝔏}(⋃_{i∈J} S_{i}) = ⋃_{i∈J} Supp_{𝔏}(S_{i}).

Proof. In fact, by definition, Supp_{𝔏}(⋃_{i∈J} S_{i}) = {Σ ∈ Spec_{𝔏}^{0}(X) | ⋃_{i∈J} S_{i} ⊈ Σ̂}, and {Σ ∈ Spec_{𝔏}^{0}(X) | ⋃_{i∈J} S_{i} ⊈ Σ̂} = ⋃_{i∈J}{Σ ∈ Spec_{𝔏}^{0}(X) | S_{i} ⊈ Σ̂} = ⋃_{i∈J} Supp_{𝔏}(S_{i}).

9.2. Corollary. Let S_{1} ⊆ HomC_{X} ⊇ S_{2}. Suppose that Iso(C_{X}) is contained either in S_{1}, or in S_{2}. Then

Supp_{𝔏}(S_{1}S_{2}) = Supp_{𝔏}(S_{1}) ⋃ Supp_{𝔏}(S_{2}).

Proof. It follows from the hypothesis that S_{1} ∪ S_{2} ⊆ S_{1}S_{2}, hence, by 9.1,

Supp_{𝔏}(S_{1}) ⋃ Supp_{𝔏}(S_{2}) ⊆ Supp_{𝔏}(S_{1} ∪ S_{2}) ⊆ Supp_{𝔏}(S_{1}S_{2}).

Let [S]_{•} denote the intersection of all saturated multiplicative systems containing S. Then Supp_{𝔏}(S) = Supp_{𝔏}([S]_{•}). Therefore, since S_{1}S_{2} ⊆ [S_{1} ∪ S_{2}]_{•},

Supp_{𝔏}(S_{1}S_{2}) ⊆ Supp_{𝔏}([S_{1} ∪ S_{2}]_{•}) = Supp_{𝔏}(S_{1} ∪ S_{2}) = Supp_{𝔏}(S_{1}) ⋃ Supp_{𝔏}(S_{2}),

whence the assertion.

9.3. Weakly associated points in Spec_{𝔏}^{0}(X). For any S ⊆ HomC_{X}, set

Ass_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | (Σ ∨ Σ̂) ∩ S ⊈ Σ̂}.

We call the elements of Ass_{𝔏}(S) weakly associated points of S in Spec_{𝔏}^{0}(X), or simply weakly associated points of S.

9.3.1. Proposition. (i) For any S ⊆ HomC_{X}, Ass_{𝔏}(S) ⊆ Supp_{𝔏}(S). (ii) If S is a saturated multiplicative system, then Ass_{𝔏}(S) = Supp_{𝔏}(S) = {Σ ∈

Spec_{𝔏}^{0}(X) | Σ ⊆ S}.

(iii) For any set {S_{i} ⊆ HomC_{X} | i ∈ J},

Ass_{𝔏}( ⋃_{i∈J} S_{i}) = ⋃_{i∈J} Ass_{𝔏}(S_{i}).

Proof. (i) The inclusion Ass_{𝔏}(S) ⊆ Supp_{𝔏}(S) follows from definitions.

(ii) Suppose S ∈ S^{𝔰}M(X). Then

Supp_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | S ⊈ Σ̂} = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ⊆ Ŝ}

and

{Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ⊆ Ŝ} = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ ⊆ S},

because Σ̂ ⊆ Ŝ iff Σ ⊆ S (see 2). Obviously, {Σ ∈ Spec_{𝔏}^{0}(X) | Σ ⊆ S} ⊆ Ass_{𝔏}(S). By (i), Ass_{𝔏}(S) ⊆ Supp_{𝔏}(S), hence the assertion.

(iii) We have:

Ass_{𝔏}( ⋃_{i∈J} S_{i}) = {Σ ∈ Spec_{𝔏}^{0}(X) | (Σ ∨ Σ̂) ⋂(⋃_{i∈J} S_{i}) ⊈ Σ̂}

= {Σ ∈ Spec_{𝔏}^{0}(X) | ⋃_{i∈J}((Σ ∨ Σ̂) ⋂ S_{i}) ⊈ Σ̂}

= ⋃_{i∈J}{Σ ∈ Spec_{𝔏}^{0}(X) | (Σ ∨ Σ̂) ⋂ S_{i} ⊈ Σ̂} = ⋃_{i∈J} Ass_{𝔏}(S_{i})

for any set {S_{i} ⊆ HomC_{X} | i ∈ J}.

9.3.2. Proposition. (i) Let Σ ∈ Spec_{𝔏}^{0}(X), and let Σ ⊇ S ⊈ Σ ∩ Σ̂. Then Supp_{𝔏}(S) = {Σ′ ∈ Spec_{𝔏}^{0}(X) | Σ′ ⊆ Σ}.

(ii) Suppose that the intersection, S_{•}, of all saturated multiplicative systems containing S belongs to Spec_{𝔏}^{0}(X) and S ∩ Ŝ = Iso(C_{X}) (e.g. S ∈ Spec_{𝔖}^{0}(X); cf. C2.1). Then Ass_{𝔏}(S) = {S_{•}}.

Proof. (i) Clearly Σ ∈ Supp_{𝔏}(S). Since Supp_{𝔏}(S) is closed under specializations, {Σ′ ∈ Spec_{𝔏}^{0}(X) | Σ′ ⊆ Σ} ⊆ Supp_{𝔏}(S). Suppose Σ′ ∈ Supp_{𝔏}(S), that is S ⊈ Σ̂′. Since S ⊆ Σ, this implies that Σ ⊈ Σ̂′, that is Σ̂′ ⊆ Σ̂, or, equivalently, Σ′ ⊆ Σ.

(ii) Suppose that the conditions of (ii) hold. Let Σ ∈ Ass_{𝔏}(S). Set S′ = S ∩ (Σ ∨ Σ̂). By definition, Σ ∈ Ass_{𝔏}(S), iff S′ ⊈ Σ̂; in particular, S′ ⊈ Iso(C_{X}). Because S′ ∩ Ŝ ⊆ Iso(C_{X}), the latter implies that S′ ⊈ Ŝ. Since Σ ∈ Supp_{𝔏}(S), it follows from (i) that Σ ⊆ S_{•} and, equivalently, Σ̂ ⊆ Ŝ.

Suppose Σ̂ ≠ Ŝ, i.e. Ŝ contains Σ̂ properly. Then, since Σ∨Σ̂ is the smallest saturated multiplicative system containing properly Σ̂, we have the inclusion Σ∨Σ̂ ⊆ Ŝ. In particular, S′ ⊆ Ŝ, which contradicts to the fact that S′ ⊈ Ŝ.

9.4. Support and weakly associated points in the complete 𝔏-spectrum. Fix a ’space’ X. Let S ⊆ HomC_{X}. We define the support of S in S^{𝔰}M(X) by

Supp_{𝔏}(S) = {Σ ∈ S^{𝔰}M(X) | S ⊈ Σ}

and the support, Supp_{𝔏}^{1}(S), of S in Spec_{𝔏}^{1}(X) by

Supp_{𝔏}^{1}(S) = Supp_{𝔏}(S) ⋂ Spec_{𝔏}^{1}(X) = {Σ ∈ Spec_{𝔏}^{1}(X) | S ⊈ Σ}.

These supports have the same general properties as the support in Spec_{𝔏}^{0}(X), as the following proposition shows.

9.4.1. Proposition. Let Supp(S) denote either Supp_{𝔏}(S), or Supp_{𝔏}^{1}(S).

(a) Let S ⊆ HomC_{X}, and let [S]_{•} be the intersection of all saturated multiplicative systems containing S. Then Supp(S) = Supp([S]_{•}).

(b) For any set {S_{i} ⊆ HomC_{X} | i ∈ J},

Supp( ⋃_{i∈J} S_{i}) = ⋃_{i∈J} Supp(S_{i}).

(c) Let S_{1} ⊆ HomC_{X} ⊇ S_{2}. Suppose that Iso(C_{X}) is contained either in S_{1}, or in S_{2}. Then

Supp(S_{1}S_{2}) = Supp(S_{1}) ⋃ Supp(S_{2}).

Proof. It suffices to verify these assertions for the support in S^{𝔰}M(X).

(a) The equality Supp_{𝔏}(S) = Supp_{𝔏}([S]_{•}) is obvious.

(b) For any set {S_{i} ⊆ HomC_{X} | i ∈ J}, we have:

\begin{aligned}Supp_{𝔏}( ⋃_{i∈J} S_{i}) & = \{Σ ∈ S^{𝔰}M(X) | ⋃_{i∈J} S_{i} ⊈ Σ\} \\ & = ⋃_{i∈J}\{Σ ∈ S^{𝔰}M(X) | S_{i} ⊈ Σ\} = ⋃_{i∈J} Supp_{𝔏}(S_{i}). \end{aligned}

(c) The argument is similar to that of 9.2. Details are left to the reader.

9.4.2. Weakly associated points in S^{𝔰}M(X). For any Σ ∈ S^{𝔰}M(X), let Σ^{⋆} denote the intersection of all saturated multiplicative systems which contain Σ properly. For any S ⊆ HomC_{X}, we set

Ass_{𝔏}^{1}(S) = {Σ ∈ S^{𝔰}M(X) | S ∩ Σ^{⋆} ⊈ Σ}.  (1)

The elements of Ass_{𝔏}^{1}(S) will be called weakly associated points of S in S^{𝔰}M(X).

9.4.3. Proposition. (i) For any S ⊆ HomC_{X}, Ass_{𝔏}^{1}(S) ⊆ Supp_{𝔏}^{1}(S). In particular, Ass_{𝔏}^{1}(S) ⊆ Spec_{𝔏}^{1}(X).

(ii) For any set {S_{i} ⊆ HomC_{X} | i ∈ J},

Ass_{𝔏}^{1}( ⋃_{i∈J} S_{i}) = ⋃_{i∈J} Ass_{𝔏}^{1}(S_{i}).

Proof. (i) In fact, Spec_{𝔏}^{1}(X) consists of all saturated multiplicative systems Σ such that Σ ≠ Σ^{⋆}. Therefore Ass_{𝔏}^{1}(S) ⊆ Spec_{𝔏}^{1}(X) which implies the inclusion Ass_{𝔏}^{1}(S) ⊆ Supp_{𝔏}^{1}(S).

(ii) We have:

Ass_{𝔏}^{1}( ⋃_{i∈J} S_{i}) = {Σ ∈ S^{𝔰}M(X) | Σ^{⋆} ⋂ ( ⋃_{i∈J} S_{i}) ≠ Σ}

= {Σ ∈ S^{𝔰}M(X) | ⋃_{i∈J}(Σ^{⋆} ∩ S_{i}) ⊈ Σ̂}

= ⋃_{i∈J}{Σ ∈ S^{𝔰}M(X) | Σ^{⋆} ∩ S_{i} ≠ Σ} = ⋃_{i∈J} Ass_{𝔏}^{1}(S_{i})

for any set {S_{i} ⊆ HomC_{X} | i ∈ J}.

9.4.4. Proposition. For any S ⊆ HomC_{X},

Supp_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ∈ Supp_{𝔏}^{1}(S)}  (2)

and

Ass_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ∈ Ass_{𝔏}^{1}(S)}.  (3)

Proof. The equality (2) follows from the definitions of Supp_{𝔏}(S) and Supp_{𝔏}^{1}(S). By definition, Ass_{𝔏}(S) = {Σ ∈ Spec_{𝔏}^{0}(X) | (Σ ∨ Σ̂) ∩ S ⊈ Σ̂} (cf. 9.3). On the other

hand, {Σ ∈ Spec_{𝔏}^{0}(X) | Σ̂ ∈ Ass_{𝔏}^{1}(S)} = {Σ ∈ Spec_{𝔏}^{0}(X) | S ∩ (Σ̂)^{⋆} ⊈ Σ̂}. Notice that, for any Σ ∈ Spec_{𝔏}^{0}(X), (Σ̂)^{⋆} coincides with the smallest saturated multiplicative system, Σ̂ ∨ Σ, containing Σ ∪ Σ̂, hence the equality (3).

For any S ⊆ HomC_{X}, let U_{𝔏}(S) denote {Σ ∈ S^{𝔰}M(X) | S ⊆ Σ}.

9.4.5. Proposition. Let X ⟶^{u} Y be an exact localization with an inverse image functor u^{∗}. Then for any S ⊆ HomC_{Y}, the localization u induces

(i) a bijection of Ass_{𝔏}^{1}(S) ∩ U_{𝔏}(Σ_{u^{∗}}) onto Ass_{𝔏}^{1}(u^{∗}(S)); (ii) a bijection of Supp_{𝔏}^{1}(S) ∩ U_{𝔏}(Σ_{u^{∗}}) onto Supp_{𝔏}^{1}(u^{∗}(S)).

Here Σ_{u^{∗}} = {t ∈ HomC_{Y} | u^{∗}(t) ∈ Iso(C_{X})}.

Proof. By (the argument of) 4.1, the localization functor C_{Y} ⟶^{u^{∗}} C_{X} induces a map

Spec_{𝔏}^{1}(X) ⟶ Spec_{𝔏}^{1}(Y), Σ ⟼ u^{∗^{−1}}(Σ).  (4)

This map induces an isomorphism

Spec_{𝔏}^{1}(X) ⟶ U_{𝔏}(Σ_{u^{∗}}).

(i) By definition 9.4.2,

\begin{aligned}Ass_{𝔏}^{1}(u^{∗}(S)) & = \{Σ ∈ S^{𝔰}M(X) | u^{∗}(S) ∩ Σ^{⋆} ⊈ Σ\} \\ & = \{Σ ∈ S^{𝔰}M(X) | S ∩ u^{∗^{−1}}(Σ^{⋆}) ⊈ u^{∗^{−1}}(Σ)\}. \end{aligned}  (5)

Since u^{∗^{−1}}(Σ^{⋆}) = (u^{∗^{−1}}(Σ))^{⋆}, it follows from (5) that

Ass_{𝔏}^{1}(u^{∗}(S)) = {Σ ∈ S^{𝔰}M(X) | S ∩ (u^{∗^{−1}}(Σ))^{⋆} ⊈ u^{∗^{−1}}(Σ)}.  (6)

The map Σ ⟼ u^{∗^{−1}}(Σ) (see (4) above) induces a bijection of the right hand side of the equality (6) onto Ass_{𝔏}^{1}(S) ∩ U_{𝔏}(Σ_{u^{∗}}).

(ii) Similarly,

\begin{aligned}Supp_{𝔏}^{1}(u^{∗}(S)) & = \{Σ ∈ S^{𝔰}M(X) | u^{∗}(S) ⊈ Σ\} \\ & = \{Σ ∈ S^{𝔰}M(X) | S ⊈ u^{∗^{−1}}(Σ)\}. \end{aligned}  (7)

The map (4) induces a bijection of the right hand side of (7) onto Supp_{𝔏}^{1}(S) ∩ U_{𝔏}(Σ_{u^{∗}}).

9.5. Supports and weakly associated points in other spectra. The other spectra considered in this work (and in [R]) are contained (or embedded) in Spec_{𝔏}^{0}(X), or in Spec_{𝔏}^{1}(X). If Spec_{?}(X) is one of those spectra, we define, for any S ⊆ HomC_{X}, the support, Supp_{?}(S), and weakly associated points, Ass_{?}(S), of S in Spec_{?}(X) as the intersection (pull-back) of resp. Supp_{𝔏}^{1}(S) and Ass_{𝔏}^{1}(S) with Spec_{?}(X). Thus, we have the flat support, Supp_{𝔣𝔏}^{1}(S) = Supp_{𝔏}^{1}(S) ∩ Spec_{𝔣𝔏}^{1}(X), and flat weakly associated points, Ass_{𝔣𝔏}^{1}(S) = Ass_{𝔏}^{1}(S) ∩ Spec_{𝔣𝔏}^{1}(X). Similarly, we have support and weakly associated points of S in Spec_{𝔖}^{0}(X): Supp_{𝔖}(S) = Supp_{𝔏}(S)∩Spec_{𝔖}^{0}(X) and Ass_{𝔖}(S) = Ass_{𝔏}(S)∩ Spec_{𝔖}^{0}(X).

9.6. Supports and weakly associated points of objects. For any object M of the category C_{X}, let E_{M} denote the family of all quotient objects M ⟶ L. We define the support of M in S^{𝔰}M(X) as the support, Supp_{𝔏}(E_{M}), of E_{M} in S^{𝔰}M(X), and the support of M in Spec_{?}(X) as the support of E_{M} in Spec_{?}(X). Here Spec_{?}(X) is one of the spectra considered in this work. The notation remains the same: Supp_{?}(M).

Similarly, we define weakly associated points of an object M, as associated points, Ass_{?}(E_{M}), of E_{M}, and denote it by Ass_{?}(M).

10. Application to abelian categories.

Let C_{X} be an abelian category. Then saturated multiplicative systems are represented by thick subcategories of the category C_{X}, and Spec_{𝔏}^{0}(X) (resp. Spec_{𝔏}^{1}(X), resp. Spec_{𝔖}^{0}(X)) is identified with Spec_{𝔗𝔥}^{0}(X) (resp. Spec_{𝔗𝔥}^{1}(X), resp. Spec_{𝔰}^{0}(X)).

For any S ⊆ HomC_{X}, the isomorphism S^{𝔰}M(X) ⟶^{∼} 𝔗𝔥(X) (cf. 6.1) identifies Supp_{𝔏}(S) with Supp_{𝔗}(S) = {𝕋 ∈ 𝔗𝔥(X) | S ⊈ Σ_{𝕋}}, where

Σ_{𝕋} = {s ∈ HomC_{X} | Ker(s) ∈ Ob𝕋 ∋ Cok(s)}.

In other words, 𝕋 ∈ Supp_{𝔗}(S) iff there is an s ∈ S such that either the kernel of s, or the cokernel of s does not belong to 𝕋.

Recall that, for any thick subcategory, 𝕋 of the category C_{X}, we denote by 𝕋^{⋆} the residue category of 𝕋 defined as the intersection of all thick subcategories of C_{X} properly containing 𝕋. The complete spectrum, Spec_{𝔗𝔥}^{1}(X), consists of all thick subcategories 𝕋 of the category C_{X} such that 𝕋^{⋆} ≠ 𝕋.

10.1. Proposition. (a) For any M ∈ ObC_{X},

Supp_{𝔗}(M) = {𝕋 ∈ 𝔗𝔥(X) | M ∉ 𝕋}.

In particular,

Supp_{𝔗𝔥}^{1}(M) = Supp_{𝔗}(M) ⋂ Spec_{𝔗𝔥}^{1}(X) = {P ∈ Spec_{𝔗𝔥}^{1}(X) | M ∉ ObP}.

(b) Ass_{𝔗𝔥}^{1}(M) consists of all P ∈ Spec_{𝔗𝔥}^{1}(X) such that M has a subobject which belongs to ObP^{⋆} − ObP.

(c) Suppose X has the property (sup) (cf. 6.5). Then Ass_{𝔰}(M) consists of all [P]_{•}, where P ∈ Spec_{𝔰}^{0}(X) is a quotient of a subobject of M by its ⟨P ⟩_{•}-torsion.

Proof. (a) If an object M of C_{X} does not belong to a thick subcategory 𝕋, then the morphism M → 0 does not belong to the saturated multiplicative system Σ_{𝕋} associated with 𝕋, hence E_{M} ⊈ Σ_{𝕋}.

Conversely, M ∈ Ob𝕋 iff E_{M} is a subcategory of 𝕋. Thus, {𝕋 ∈ 𝔗𝔥(X) | M ∉ Ob𝕋} = {𝕋 ∈ 𝔗𝔥(X) | E_{M} ⊈ Σ_{𝕋}}, hence the assertion.

(b) By definition (9.4.2),

Ass_{𝔗𝔥}^{1}(M) = Ass_{𝔗𝔥}^{1}(E_{M}) = {P ∈ 𝔗𝔥(X) | E_{M} ∩ (Σ_{P})^{⋆} ⊈ Σ_{P}}.

Here (Σ_{P})^{⋆} is the intersection of all saturated multiplicative systems properly containing Σ_{P}. Due to the isomorphism between the preorder S^{𝔰}M(X) of saturated multiplicative systems in C_{X} and the preorder 𝔗𝔥(X) of thick subcategories of C_{X}, the equality (Σ_{P})^{⋆} = Σ_{P^{⋆}} holds. Therefore, E_{M} ∩ (Σ_{P})^{⋆} ⊈ Σ_{P} iff there is an epimorphism M ⟶^{f} L which belongs to Σ_{P^{⋆}} − Σ_{P}. This means precisely that Ker(f) ∈ ObP^{⋆} − ObP.

(c) By definition, Ass_{𝔰}(M) = {P ∈ Spec_{𝔰}^{0}(X) | P ∈ Ass_{𝔰}^{1}(M)}. Therefore, it follows from (b) above that Ass_{𝔰}(M) consists of all P ∈ Spec_{𝔰}^{0}(X) such that M has a subobject, L, which belongs to ObP̂^{⋆} − ObP̂. Notice that P̂^{⋆} is the thick envelope of P ∪ P̂; i.e. P̂^{⋆} = P ∨ P̂. Since X has the property (sup), the Serre subcategory P̂ is coreflective; in particular, the object L has a P̂-torsion, i.e. a subobject, L′ which belong to P̂ and such that the quotient object L/L′ is P̂-torsion free. Since L/L′ is a nonzero object of the smallest Serre subcategory, (P ∪ P̂)^{−}, containing P ∪ P̂. Therefore, L/L′ has a nonzero subobject, L′′, which belongs to P ∪ P̂. Since L/L′ is P̂-torsion free, the subobject L′′ belongs to P. Let L̃ be the pull-back of L′′ in L, that is L̃ = L′′ ×_{L/L′} L. Thus L̃ is a subobject of L which contains L′ and such that L̃/L′ = L′′ is a P̂-free nonzero object of the subcategory P; in particular, L′′ ∈ ObP − ObP̂. Therefore, P̂ = ⟨L′′⟩_{•}, or, equivalently, [L]_{•} = P.

10.2. Proposition. (i) Let P ∈ Spec_{𝔗𝔥}^{1}X, and let M ∈ ObP^{⋆} − ObP. Then Supp_{𝔗𝔥}^{1}(M) = {P′ ∈ Spec_{𝔗𝔥}^{1} | P′ ⊆ P}.

(ii) If M is a P-torsion free nonzero object of P^{⋆}, then Ass_{𝔗𝔥}^{1}(M) = {P}.

Proof. (i) Let P′ ∈ Supp_{𝔗𝔥}^{1}(M). The quotient category P′/(P′ ∩ P) is (identified with) a thick subcategory of C_{X}/P. If P′ were not contained in P, then P′/(P′ ∩ P) would be a nonzero thick subcategory of C_{X}/P, hence it should contain the subcategory P^{⋆}/P which contradicts to the fact that the image of the object M in C_{X}/P does not belong to P′/(P′ ∩ P). Thus, P′/(P′ ∩ P) = 𝕆, i.e. P′ ⊆ P.

(ii) Let P̃ ∈ Ass_{𝔗𝔥}^{1}(M). By 10.1(b), this means that M has a subobject, L, which belongs to ObP̃^{⋆} − ObP̃. By hypothesis, M is a P-torsion free object of P^{⋆}, hence every nonzero subobject of M, in particular L, is a P-torsion free object of P^{⋆}.

Since P̃ ∈ Supp_{𝔗𝔥}^{1}(M), by (i) above, P̃ ⊆ P. Thus, L is a P̃-torsion free object of P̃^{⋆}. Since P ∈ Supp_{𝔗𝔥}^{1}(L), by the argument above, P ⊆ P̃, hence P̃ = P.

10.3. Proposition. For any exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,
Supp_{𝔗𝔥}^{1}(M) = Supp_{𝔗𝔥}^{1}(M′) ⋃ Supp_{𝔗𝔥}^{1}(M′′)  (1)

and

Ass_{𝔗𝔥}^{1}(M′) ⊆ Ass_{𝔗𝔥}^{1}(M) ⊆ Ass_{𝔗𝔥}^{1}(M′) ⋃ Ass_{𝔗𝔥}^{1}(M′′).  (2)

Proof. (a) If 𝕋 is a thick subcategory, then M ∈ Ob𝕋 iff M′ and M′′ are objects of 𝕋. Equivalently, M ∉ Ob𝕋 iff either M′ ∉ Ob𝕋, or M′′ ∉ Ob𝕋.

(b) Clearly, Ass_{𝔗𝔥}^{1}(M′) ⊆ Ass_{𝔗𝔥}^{1}(M).

Let P ∈ Ass_{𝔗𝔥}^{1}(M), i.e. M has a subobject, L, which belongs to ObP^{⋆} − ObP. Consider the short exact sequence

0 ⟶ L′ = L ∩ M′ ⟶ L ⟶ L′′ ⟶ 0.

Since L ∈ ObP^{⋆}, both L′ and L′′ belong to the subcategory P^{⋆}. Since L does not belong to P, either L′ ∉ ObP, and then P ∈ Ass_{𝔗𝔥}^{1}(L′), or L′′ ∉ ObP, and then P ∈ Ass_{𝔗𝔥}^{1}(L′′). Notice that L′ is a subobject of M′ and L′′ is a subobject of the object M′′. In particular, Ass_{𝔗𝔥}^{1}(L′) ⊆ Ass_{𝔗𝔥}^{1}(M′) and Ass_{𝔗𝔥}^{1}(L′′) ⊆ Ass_{𝔗𝔥}^{1}(M′′). This proves the assertion.

10.4. Corollary. For any finite set, {M_{i} | i ∈ J}, of objects of C_{X},

Ass_{𝔗𝔥}^{1}(⊕_{i∈J}M_{i}) = ⋃_{i∈J} Ass_{𝔗𝔥}^{1}(M_{i}).  (3)

Proof. It suffices to verify the equality (3) for a set consisting of two objects, M_{1} and M_{2}. By 10.3, we have the inclusions

Ass_{𝔗𝔥}^{1}(M_{1}) ⋃ Ass_{𝔗𝔥}^{1}(M_{2}) ⊆ Ass_{𝔗𝔥}^{1}(M_{1} ⊕ M_{2}) ⊆ Ass_{𝔗𝔥}^{1}(M_{1}) ⋃ Ass_{𝔗𝔥}^{1}(M_{2}),

hence the assertion.

10.5. Corollary. Let {L_{i} | i ∈ J} be a finite set of subobjects of an object M such that ⋂_{i∈J} L_{i} = 0. Then

Ass_{𝔗𝔥}^{1}(M) ⊆ ⋃_{i∈J} Ass_{𝔗𝔥}^{1}(M/L_{i}).  (4)

Proof. Since ⋂_{i∈J} L_{i} = 0, the natural morphism M ⟶ ⊕_{i∈J}M/L_{i} is a monomorphism. Therefore, by 10.3, we have the inclusion Ass_{𝔗𝔥}^{1}(M) ⊆ Ass_{𝔗𝔥}^{1}(⊕_{i∈J}M/L_{i}). By 10.4, Ass_{𝔗𝔥}^{1}(⊕_{i∈J}M/L_{i}) = ⋃_{i∈J} Ass_{𝔗𝔥}^{1}(M/L_{i}).

For every subcategory 𝕋 of the category C_{X}, let U_{𝔗𝔥}(𝕋) denote the family of all P ∈ 𝔗𝔥(X) which contain 𝕋.

10.6. Proposition. Let C_{Y} be an abelian category, and let X ⟶^{u} Y be an exact localization with an inverse image functor u^{∗}. Then for any object M of the category C_{Y}, the localization u induces

(i) a bijection of Ass_{𝔗𝔥}^{1}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) onto Ass_{𝔗𝔥}^{1}(u^{∗}(M)); (ii) a bijection of Supp_{𝔗𝔥}^{1}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) onto Supp_{𝔗𝔥}^{1}(u^{∗}(M)).

Proof. The assertion can be deduced from 9.4.5. For the reader’s convenience, we give below a direct proof.

The localization functor C_{Y} ⟶^{u^{∗}} C_{X} induces an isomorphism

U_{𝔗𝔥}(Ker(u^{∗})) \underrightarrow{u^{∗}} Spec_{𝔗𝔥}^{1}X

with the inverse isomorphism given by P ⟼ u^{∗^{−1}}(P).

(i) Let P ∈ Ass_{𝔗𝔥}^{1}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})), i.e. Ker(u^{∗}) ⊆ P and the object M has a subobject, L, which belongs to ObP^{⋆}−ObP. It follows that u^{∗}(L) ∈ Obu^{∗}(P^{⋆})−Obu^{∗}(P) = Ob(u^{∗}P)^{⋆} − Obu^{∗}(P). Since the functor u^{∗} is left exact, u^{∗}(L) is a subobject of the object u^{∗}(M). This shows that u^{∗}(P) ∈ Ass_{𝔗𝔥}^{1}(u^{∗}(M)); i.e. the isomorphism u^{∗} induces an injective map from Ass_{𝔗𝔥}^{1}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) to Ass_{𝔗𝔥}^{1}(u^{∗}(M)). This map is bijective.

In fact, let P_{x} be an element of Ass_{𝔗𝔥}^{1}(u^{∗}(M)), i.e. there exists a monomorphism, L′ ⟶ u^{∗}(M), such that L′ belongs to ObP_{x}^{⋆} − ObP_{x}. Since u^{∗} is an exact localization (it suffices for u^{∗} to be left exact) and the category C_{Y} has finite limits, the monomorphism L′ ⟶ u^{∗}(M) is the composition of an isomorphism L′ ⟶^{∼} u^{∗}(L) and a morphism

u^{∗}(L) ⟶^{u^{∗}(𝔧)} u^{∗}(M), where L ⟶^{𝔧} M is a monomorphism. Thus, u^{∗}(L) ∈ ObP_{x}^{⋆} − ObP_{x}, or, equivalently, L ∈ Ob(u^{∗^{−1}}(P_{x}^{⋆})) − Ob(u^{∗^{−1}}(P_{x})) = Ob(u^{∗^{−1}}(P_{x}))^{⋆} − Ob(u^{∗^{−1}}(P_{x})). This means that, u^{∗^{−1}}(P_{x}) ∈ Ass_{𝔗𝔥}^{1}(M).

(ii) The assertion (ii) follows directly from the definition of the support. Details are left to the reader.

10.7. The support and weakly associated points in Serre subcategories. Let 𝔖𝔢(X) denote the preorder of Serre subcategories of the category C_{X}, that is

𝔖𝔢(X) = ({𝕋 ∈ 𝔗𝔥(X) | 𝕋^{−} = 𝕋}, ⊆ )

(cf. 6.3). For any M ∈ ObC_{X}, we define the support of M in 𝔖𝔢(X) and the weakly associated points of M in 𝔖𝔢(X) by respectively

Supp_{𝔖𝔢}(M) = Supp_{𝔗} ⋂ 𝔖𝔢(X) and Ass_{𝔖𝔢}(M) = Ass_{𝔗𝔥}^{1} ⋂ 𝔖𝔢(X).

10.7.1. Proposition. (a) For any exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,
Supp_{𝔖𝔢}(M) = Supp_{𝔖𝔢}(M′) ⋃ Supp_{𝔖𝔢}(M′′)  (5)

and

Ass_{𝔖𝔢}(M′) ⊆ Ass_{𝔖𝔢}(M) ⊆ Ass_{𝔖𝔢}(M′) ⋃ Ass_{𝔖𝔢}(M′′).  (6)

(b) Suppose X has the property (sup) (cf. 6.5). And let an object M of C_{X} is a supremum of an ascending family, Ξ, of its subobjects. Then

Supp_{𝔖𝔢}(M) = ⋃_{M′∈Ξ} Supp_{𝔖𝔢}(M′) and Ass_{𝔖𝔢}(M) = ⋃_{M′∈Ξ} Ass_{𝔖𝔢}(M′).

Proof. The assertion (a) follows from 10.3.

(b) The inclusions

⋃_{M′∈Ξ} Supp_{𝔖𝔢}(M′) ⊆ Supp_{𝔖𝔢}(M) and ⋃_{M′∈Ξ} Ass_{𝔖𝔢}(M′) ⊆ Ass_{𝔖𝔢}(M)

follow from (a). Suppose 𝕋 ∈ Supp_{𝔖𝔢}(M), i.e. M ∉ Ob𝕋 and 𝕋 = 𝕋^{−}. Since X has the property (sup), any Serre subcategory of C_{X} is closed under taking supremums of subobject. In particular, since M = sup{M′|M′ ∈ Ξ}, M ∉ Ob𝕋 iff M′ ∉ Ob𝕋 for some M′ ∈ Ξ. This proves the inverse inclusion, Supp_{𝔖𝔢}(M) ⊆ ⋃_{M′∈Ξ} Supp_{𝔖𝔢}(M′).

Let P ∈ Ass_{𝔖𝔢}(M), i.e. M has a subobject L which belongs to P^{⋆} −P. Thanks to the property (sup), L = sup{M′∩L|M′ ∈ Ξ}. Since P is a Serre subcategory, it is closed under taking supremums of subobjects. Therefore, L ∈ ObP^{⋆} − ObP iff L ∩ M′ ∈ ObP^{⋆} − ObP for some M′ ∈ Ξ. This proves the inclusion Ass_{𝔖𝔢}(M) ⊆ ⋃_{M′∈Ξ} Ass_{𝔖𝔢}(M′), hence the assertion.

10.7.2. Corollary. Let X have the property (sup). Then for any set, {M_{i} | i ∈ J}, of objects of C_{X},

Ass_{𝔖𝔢}(⊕_{i∈J}M_{i}) = ⋃_{i∈J} Ass_{𝔖𝔢}(M_{i}).  (7)

Proof. If the set {M_{i} | i ∈ J} is finite, then the assertion is a consequence of 10.4. Since ⊕_{i∈J}M_{i} is the supremum of coproducts ⊕_{i∈J′}M_{i}, where J′ runs through finite subsets of J, the fact follows from 10.7.1(b).

10.7.3. Corollary. Let {L_{i} | i ∈ J} be a finite set of subobjects of an object M such that ⋂_{i∈J} L_{i} = 0. Then

Ass_{𝔖𝔢}(M) ⊆ ⋃_{i∈J} Ass_{𝔖𝔢}(M/L_{i}).  (8)

Proof. The assertion follows from 10.5.

10.7.4. Proposition. Let C_{Y} be an abelian category, and let X ⟶^{u} Y be an exact localization with an inverse image functor u^{∗}. Then for any object M of the category C_{Y}, the localization u induces

(i) a bijection of Ass_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) onto Ass_{𝔖𝔢}(u^{∗}(M)); (ii) a bijection of Supp_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) onto Supp_{𝔖𝔢}(u^{∗}(M)).

Proof. The assertion follows from 10.6.

10.7.5. Proposition. Let Y ⟶^{u} X be an exact localization such that 𝕊 = Ker(u^{∗}) is a coreflective (in particular, Serre) subcategory of the category C_{X}. Let M ∈ ObC_{X}, and let 𝔱_{𝕊}M denote the 𝕊-torsion of M and M_{𝕊} the quotient object M/𝔱_{𝕊}M. Then

Ass_{𝔖𝔢}(𝔱_{𝕊}M) = Ass_{𝔖𝔢}(M) − U_{𝔗𝔥}(𝕊) = Ass_{𝔖𝔢}(M) − Ass_{𝔖𝔢}(M_{𝕊})

and

\begin{aligned}Ass_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(𝕊) & = Ass_{𝔖𝔢}(M_{𝕊}) ∩ U_{𝔗𝔥}(𝕊) \\ & = \{P ∈ Ass_{𝔖𝔢}(M_{𝕊}) | P ⊇ 𝕊 = Ker(u^{∗})\} ⊆ Ass_{𝔖𝔢}(M_{𝕊}). \end{aligned}

Proof. By 10.7.4, an inverse image (localization) functor C_{X} ⟶^{u^{∗}} C_{Y} of the morphism u induces an isomorphism between Ass_{𝔖𝔢}(u^{∗}(M)) and Ass_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(𝕊). Notice that the functor u^{∗} maps the natural epimorphism M ⟶ M_{𝕊} to an isomorphism, i.e.

Ass_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(𝕊) ≃ Ass_{𝔖𝔢}(u^{∗}(M)) ≃ Ass_{𝔖𝔢}(u^{∗}(M_{𝕊})) ≃ Ass_{𝔖𝔢}(M_{𝕊}) ∩ U_{𝔗𝔥}(𝕊), (9)

and Ass_{𝔖𝔢}(𝔱_{𝕊}M) ⊆ Ass_{𝔖𝔢}(M) ∩ V_{𝔰}^{1}(𝕊), where V_{𝔰}^{1}(𝕊) = {P ∈ Spec_{𝔰}^{1}(X) | 𝕊 ⊈ P}. It follows from the short exact sequence

0 ⟶ 𝔱_{𝕊}M ⟶ M ⟶ M_{𝕊} ⟶ 0,

(9), and 10.7.1 that

\begin{aligned}Ass_{𝔖𝔢}(𝔱_{𝕊}M) ⊆ Ass_{𝔖𝔢}(M) & = (Ass_{𝔖𝔢}(M) ∩ V_{𝔰}^{1}(𝕊)) ⋃ (Ass_{𝔖𝔢}(M) ∩ U_{𝔗𝔥}(𝕊)) \\ & = (Ass_{𝔖𝔢}(M) ∩ V_{𝔰}^{1}(𝕊)) ⋃ (Ass_{𝔖𝔢}(M_{𝕊}) ∩ U_{𝔗𝔥}(𝕊)) \\ & ⊆ Ass_{𝔖𝔢}(𝔱_{𝕊}M) ⋃ Ass_{𝔖𝔢}(M_{𝕊}). \end{aligned}  (10)

Since all unions in (10) are disjoint, we obtain that

Ass_{𝔖𝔢}(𝔱_{𝕊}M) = Ass_{𝔖𝔢}(M) ∩ V_{𝔰}^{1}(𝕊) = Ass_{𝔖𝔢}(M) − U_{𝔗𝔥}(𝕊) = Ass_{𝔖𝔢}(M) − Ass_{𝔖𝔢}(M_{𝕊}).

10.8. Associated points in a proper sense. For every M ∈ ObC_{X}, we set 𝔄𝔰𝔰_{𝔏}(M) = {P ∈ 𝔗(X) | there exists a nonzero subobject, L, of M which belongs to P^{⋆} and is P-torsion free}. We call elements of 𝔄𝔰𝔰_{𝔏}(M) associated points of M.

10.8.1. Lemma. 𝔄𝔰𝔰_{𝔏}(M) ⊆ Ass_{𝔖𝔢}(M).

Proof. Let P ∈ 𝔄𝔰𝔰_{𝔏}(M), i.e. there exists a nonzero subobject L of M which P-torsion free and belongs to P^{⋆}. But, if L is P-torsion free, it is P^{−}-torsion free.

Indeed, if an object has a nonzero subobject which belongs to P^{−}, then the latter should have a nonzero subobject which belongs to P.

If P^{−} ≠ P, then P^{⋆} ⊆ P^{−} which contradicts to that L is a nonzero P^{−}-torsion free object of P^{⋆}.

General properties of 𝔄𝔰𝔰_{𝔏}(−) are similar to those of Ass_{𝔗𝔥}^{1}(−), except for a weaker version of functoriality (see 10.8.2(c) below).

10.8.2. Proposition. (a) For any exact sequence

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0,
𝔄𝔰𝔰_{𝔏}(M′) ⊆ 𝔄𝔰𝔰_{𝔏}(M) ⊆ 𝔄𝔰𝔰_{𝔏}(M′) ⋃ 𝔄𝔰𝔰_{𝔏}(M′′).

(b) Suppose X has the property (sup) (cf. 6.5). Let an object M of C_{X} be a supremum of an ascending family, Ξ, of its subobjects. Then

𝔄𝔰𝔰_{𝔏}(M) = ⋃_{M′∈Ξ} 𝔄𝔰𝔰_{𝔏}(M′).

(c) For every object M of C_{X}, any exact localization, Y ⟶^{u} X, induces an injective map 𝔄𝔰𝔰_{𝔏}(M) ∩ U_{𝔗𝔥}(Ker(u^{∗})) ⟶ 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)).

(d) Let Y ⟶^{u} X be an exact localization such that 𝕊 = Ker(u^{∗}) is a coreflective (in particular, Serre) subcategory of the category C_{X}. Let M ∈ ObC_{X}, and let 𝔱_{𝕊}M be the 𝕊-torsion of M and M_{𝕊} the quotient object M/𝔱_{𝕊}M. Then the bijective map

U_{𝔗𝔥}(𝕊) ⟶ 𝔗𝔥(Y), P ⟼ P/𝕊,

induces an isomorphism

𝔄𝔰𝔰_{𝔏}(M_{𝕊}) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(u^{∗}(M))

and

𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) = 𝔄𝔰𝔰_{𝔏}(M) − U_{𝔗𝔥}(𝕊) = 𝔄𝔰𝔰_{𝔏}(M) − 𝔄𝔰𝔰_{𝔏}(M_{𝕊})  (11)

Proof. (a) The inclusion 𝔄𝔰𝔰_{𝔏}(M′) ⊆ 𝔄𝔰𝔰_{𝔏}(M) follows from definitions.

Let P ∈ 𝔄𝔰𝔰_{𝔏}(M), i.e. there exists a nonzero subobject, L, of M which belongs to P^{⋆} and is P-torsion free. Suppose L′ = L ∩ M′ ≠ 0. Then L′ is a nonzero subobject of M′ which belongs to P^{⋆} and is P-torsion free, i.e. P ∈ 𝔄𝔰𝔰_{𝔏}(M′). If L′ = 0, then the composition of L ↪ M and the canonical epimorphism M ⟶ M′′ is a monomorphism, hence P ∈ 𝔄𝔰𝔰_{𝔏}(M′′). This proves the inclusion 𝔄𝔰𝔰_{𝔏}(M) ⊆ 𝔄𝔰𝔰_{𝔏}(M′) ⋃ 𝔄𝔰𝔰_{𝔏}(M′′).

(b) It follows from (a) that the inclusion 𝔄𝔰𝔰_{𝔏}(M) ⊇ ⋃_{M′∈Ξ} 𝔄𝔰𝔰_{𝔏}(M′) holds without any additional conditions on X.

Let P ∈ 𝔄𝔰𝔰_{𝔏}(M), i.e. M has a nonzero subobject, L, which belongs to P^{⋆} and is P-torsion free. Since X has the property (sup), L ∩ M′ ≠ 0 for some M′ ∈ Ξ. Then P ∈ 𝔄𝔰𝔰_{𝔏}(M′) (see the argument in (a) above). This verifies the inverse inclusion, 𝔄𝔰𝔰_{𝔏}(M) ⊆ ⋃_{M′∈Ξ} 𝔄𝔰𝔰_{𝔏}(M′).

(c) Let u^{∗} be an inverse image functor of Y ⟶^{u} X. Set Ker(u^{∗}) = 𝕊. The claim is that the bijective map U_{𝔗𝔥}(X) ⟶ 𝔗𝔥(Y), P ⟶ P/𝕊, induces a (forcibly injective) map 𝔄𝔰𝔰_{𝔏}(M) ∩ U_{𝔗𝔥}(𝕊) ⟶ 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)).

Let P ∈ 𝔄𝔰𝔰_{𝔏}(M) ∩ U_{𝔗𝔥}(𝕊), that is 𝕊 ⊆ P, and there exists a nonzero subobject, L, of M which is P-torsion free and belongs to P^{⋆}. Since 𝕊 ⊆ P, the object L is 𝕊-torsion free. Therefore, u^{∗}(L) is a nonzero subobject of u^{∗}(M) which belongs to P^{⋆}/𝕊 = (P/𝕊)^{⋆} and is P/𝕊-torsion free; that is P/𝕊 ∈ 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)).

(d) (i) Notice that 𝔄𝔰𝔰_{𝔏}(M_{𝕊}) = 𝔄𝔰𝔰_{𝔏}(M_{𝕊}) ∩ U_{𝔏}(𝕊).

In fact, let P ∈ 𝔄𝔰𝔰_{𝔏}(M_{𝕊}), i.e. M_{𝕊} has a nonzero subobject, L, such that L ∈ P^{⋆} and L is P-torsion free. If 𝕊 ⊈ P, then P^{⋆} ⊆ P ∨ 𝕊. Therefore, L ∈ ObP ∨ 𝕊. Since P ∨ 𝕊 ⊆ (P ∪ 𝕊)^{−}, L contains a nonzero subobject, L′, which belongs to P ∪ 𝕊. Since L is P-torsion free, the object L′ belongs to 𝕊. But, L is 𝕊-torsion free as well, hence L′ = 0 which contradicts the choice of L′. Thus, 𝕊 ⊆ P.

This shows that 𝔄𝔰𝔰_{𝔏}(M_{𝕊}) = 𝔄𝔰𝔰_{𝔏}(M_{𝕊})∩U_{𝔗𝔥}(𝕊). Therefore, by (c), the bijective map

U_{𝔗𝔥}(𝕊) ⟶^{∼} 𝔗𝔥(Y), P ⟼ P/𝕊,

induces an embedding 𝔄𝔰𝔰_{𝔏}(M_{𝕊}) ⟶ 𝔄𝔰𝔰_{𝔏}(u^{∗}(M_{𝕊})). The claim is that this embedding is surjective (hence it is bijective).

Indeed, let P ∈ U_{𝔗𝔥}(𝕊) be such that P/𝕊 ∈ 𝔄𝔰𝔰_{𝔏}(u^{∗}(M_{𝕊})), i.e. there exists a nonzero subobject, L′, of u^{∗}(M_{𝕊})) which belongs to the subcategory (P/𝕊)^{⋆} = P^{⋆}/𝕊 and is P/𝕊torsion free. Replacing L′ by an isomorphic object, we can assume that L′ = u^{∗}(L). A monomorphism u^{∗}(L′) ⟶ u^{∗}(M_{𝕊})) is represented by a diagram L′ ⟵^{t} L′′ ⟶^{f} M_{𝕊} such that u^{∗}(t) is an isomorphism and u^{∗}(f) is a monomorphism. The morphism f can be represented as a composition of an epimorphism, L′′ ⟶^{𝔢_{f}} L, and a monomorphism,

L ⟶^{𝔧_{f}} M_{𝕊}. Since the functor u^{∗} is exact, in particular right exact, u^{∗}(𝔢_{f}) is an epimorphism. On the other hand, u^{∗}(𝔧_{f}) ∘ u^{∗}(𝔢_{f}) = u^{∗}(f) is a monomorphism, hence u^{∗}(𝔢_{f}) is a

monomorphism. Thus, u^{∗}(𝔢_{f}) is an isomorphism which implies that u^{∗}(L) ⟶^{u^{∗}(𝔧_{f})} u^{∗}(M_{𝕊}) is a nonzero monomorphism.

Since 𝕊 ⊆ P^{⋆} and u^{∗}(L) ∈ P^{⋆}/𝕊, the object L belongs to P^{⋆}. Notice that L is Ptorsion free, because u^{∗}(L) is P/𝕊-torsion free and L, being a subobject of an 𝕊-torsion free object M_{𝕊}, is 𝕊-torsion free. This means that P ∈ 𝔄𝔰𝔰_{𝔏}(M_{𝕊}).

The functor u^{∗} transforms the epimorphism M ⟶ M_{𝕊} into an isomorphism. Therefore, we have isomorphisms

𝔄𝔰𝔰_{𝔏}(M_{𝕊}) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(u^{∗}(M_{𝕊})) ⟵^{∼} 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)).

(ii) If N ∈ Ob𝕊, then Supp_{𝔗𝔥}(N) ∩ U_{𝔗𝔥}(𝕊) = ∅; in particular, 𝔄𝔰𝔰_{𝔏}(N) ∩ U_{𝔗𝔥}(𝕊) = ∅. Therefore, we have the inclusion 𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) ⊆ 𝔄𝔰𝔰_{𝔏}(M) − U_{𝔗𝔥}(𝕊). Applying the

assertion (a) to the exact sequence

0 ⟶ 𝔱_{𝕊}M ⟶ M ⟶ M_{𝕊} ⟶ 0,

we obtain the inclusions

𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) ⊆ 𝔄𝔰𝔰_{𝔏}(M) − U_{𝔗𝔥}(𝕊) ⊆ M ⊆ 𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) ⋃ 𝔄𝔰𝔰_{𝔏}(M_{𝕊}).

Since, by (i), 𝔄𝔰𝔰_{𝔏}(M_{𝕊}) ⊆ U_{𝔗𝔥}(𝕊), this implies the equalities (11).

10.8.3. Corollary. (i) For any finite set, {M_{i} | i ∈ J}, of objects of C_{X},

𝔄𝔰𝔰_{𝔏}(⊕_{i∈J}M_{i}) = ⋃_{i∈J} 𝔄𝔰𝔰_{𝔏}(M_{i}).

If X has the property (sup), then the finiteness condition can be dropped.

(ii) Let {L_{i} | i ∈ J} be a finite set of subobjects of an object M such that ⋂_{i∈J} L_{i} = 0. Then

𝔄𝔰𝔰_{𝔏}(M/(∩_{i∈J}L_{i})) ⊆ ⋃_{i∈J} 𝔄𝔰𝔰_{𝔏}(M/L_{i}).

Proof. (i) For a finite set {M_{i} | i ∈ J}, the assertion follows from 10.8.2(a). The infinite case is a consequence of 10.8.2(b) (see the argument of 10.7.2).

(ii) The assertion follows from (i) and 10.8.2(a) applied to the canonical monomorphism M/(∩_{i∈J}L_{i}) ⟶ ⊕_{i∈J}M/L_{i}.

10.8.4. Corollary. The full subcategory, C_{X_{𝔄𝔰𝔰}^{∅}}, of the category C_{X} whose objects, M, have no associated points, 𝔄𝔰𝔰_{𝔏}(M) = ∅, is closed under extensions, taking subobjects, and colimits of filtered diagrams of monoarrows.

Proof. The assertion is a consequence of 10.8.2(a) and (b).

10.8.5. Corollary. Let Y ⟶^{u} X be a continuous localization, i.e. u has a direct image functor, u_{∗}. Then for any object, M, of the category C_{Y}, the map

U_{𝔗𝔥}(Ker(u^{∗})) ⟶ 𝔗𝔥(Y), P ⟼ P/Ker(u^{∗}),

induces an isomorphism 𝔄𝔰𝔰_{𝔏}(M) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(u_{∗}(M)).

Proof. If u is continuous than 𝕊 = Ker(u^{∗}) is a coreflective thick subcategory of the category C_{X}: the 𝕊-torsion of an object L is the kernel of an adjunction morphism L ⟶ u_{∗}u^{∗}(L). Because u is a localization, for any object M of C_{Y}, an adjunction arrow u^{∗}u_{∗}(M) ⟶ M is an isomorphism, in particular, the object u_{∗}(M) is 𝕊-torsion free. By 10.8.2(d), the map P ⟼ P/𝕊 induces a bijection 𝔄𝔰𝔰_{𝔏}(u_{∗}(M)) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(u^{∗}u_{∗}(M)), and 𝔄𝔰𝔰_{𝔏}(u^{∗}u_{∗}(M)) ≃ 𝔄𝔰𝔰_{𝔏}(M).

10.8.6. Proposition. Let Y ⟶^{u} X be an exact localization such that 𝕊 = Ker(u^{∗}) is a coreflective subcategory of the category C_{X}. Let P ∈ Spec_{𝔰}^{1}(X) and 𝕊 ⊆ P. Let M be an object of C_{X} such that 𝔄𝔰𝔰_{𝔏}(L) ≠ ∅ for any nonzero subobject, L, of M. Then the following conditions are equivalent:

(a) 𝔄𝔰𝔰_{𝔏}(M) = {P};

(b) 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)) = {P/𝕊} and M is 𝕊-torsion free.

Proof. (a)⇒(b). Let 𝔱_{𝕊}M denote the 𝕊-torsion of M. If 𝔱_{𝕊}M ≠ 0, then, by hypothesis, 𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) ≠ ∅, i.e. 𝔄𝔰𝔰_{𝔏}(𝔱_{𝕊}M) = {P}. The latter means that 𝔱_{𝕊}M has a nonzero subobject, L, which belongs to P^{⋆} and is P-torsion free; in particular, it is 𝕊-torsion free which contradicts to that L is a nonzero object of the subcategory 𝕊.

Since M is 𝕊-torsion free, it follows from 10.8.2(d) that 𝔄𝔰𝔰_{𝔏}(u^{∗}(M)) = {P/𝕊}. (b)⇒(a). This implication follows from 10.8.2(d). Details are left to the reader.

10.8.7. Proposition. Suppose X has the property (sup). Let M ∈ ObC_{X}, and let Φ be a subset of 𝔄𝔰𝔰_{𝔏}(M). Then there exists a subobject L ⟶ M such that

𝔄𝔰𝔰_{𝔏}(M/L) = 𝔄𝔰𝔰_{𝔏}(M) − Φ and 𝔄𝔰𝔰_{𝔏}(L) = Φ.  (12)

Proof. (a) Let 𝔇_{Φ} be the set of subobjects, M′, of M such that 𝔄𝔰𝔰_{𝔏}(M′) ⊆ Φ. The set 𝔇_{Φ} is not empty, because it contains the zero subobject. It follows from 10.8.2(b) that supΞ ∈ 𝔇_{Φ} for every filtered subset Ξ of 𝔇_{Φ}. Therefore, by Zorn’s lemma, there exists a maximal element (subobject), L, in 𝔇_{Φ}. We claim that the subobject L satisfies the conditions (12). Thanks to 10.8.2(a), it suffices to show that 𝔄𝔰𝔰_{𝔏}(M/L) ⊆ 𝔄𝔰𝔰_{𝔏}(M) − Φ.

(b) Let P ∈ 𝔄𝔰𝔰_{𝔏}(M/L), i.e. M/L has a subobject, N ⟶ M/L, which belongs to P^{⋆} and is P-torsion free. Consider the associated with N ⟶ M/L short exact sequence

0 ⟶ L ⟶ Ñ = M ×_{M/L} N ⟶ N ⟶ 0.  (10)

By 10.8.2(a), 𝔄𝔰𝔰_{𝔏}(Ñ) ⊆ 𝔄𝔰𝔰_{𝔏}(L) ⋃ 𝔄𝔰𝔰_{𝔏}(N). By 10.2(ii), 𝔄𝔰𝔰_{𝔏}(N) = {P}. Since L is a maximal element of 𝔇_{Φ} and a proper subobject of Ñ, the latter does not belong to 𝔇_{Φ}. Therefore P ∈ 𝔄𝔰𝔰_{𝔏}(Ñ) − Φ.

10.9. Primary decomposition.

10.9.1. Definition. Let M be an object of an abelian category C_{X}. We call a subobject N of M primary, or P-primary, if 𝔄𝔰𝔰_{𝔏}(M/N) consists of one element, P.

10.9.2. Proposition. Let {N_{i} | i ∈ J} be a finite set of P-primary subobjects of an object M of an abelian category C_{X}. Then ⋂_{i∈J} N_{i} is a P-primary subobject of M.

Proof. The fact follows from 10.8.3(ii).

10.9.3. Definition. Let N be a subobject of an object M of the category C_{X}. A primary decomposition of N ↪ M is a finite set, {N_{i} | i ∈ J}, of primary subobjects of M such that N is a subobject of ⋂_{i∈J} N_{i} and 𝔄𝔰𝔰_{𝔏}(⋂_{i∈J} N_{i}/N) = ∅.

10.9.3.1. Note. It follows from this definition and 10.8.3(ii) that if a subobject N of M has a primary decomposition, then 𝔄𝔰𝔰_{𝔏}(M/N) is a subset of {P_{i} | i ∈ J}, in particular, 𝔄𝔰𝔰_{𝔏}(M/N is finite. Here 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = {P_{i}}.

10.9.4. Proposition. Let N be a subobject of an object M such that 𝔄𝔰𝔰_{𝔏}(M/N) is finite. Then there exists a primary decomposition, {N_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N)}, such that N_{P} is P-primary for every P ∈ 𝔄𝔰𝔰_{𝔏}(M/N).

Proof. Replacing M by M/N, we can and will assume that N = 0. By 10.8.7, for every P ∈ 𝔄𝔰𝔰_{𝔏}(M), there exists a subobject N_{P} of M such that 𝔄𝔰𝔰_{𝔏}(M/N_{P}) = {P} and 𝔄𝔰𝔰_{𝔏}(N_{P}) = 𝔄𝔰𝔰_{𝔏}(M) − {P}. Set M_{0} = ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M)} N_{P}. For each P ∈ 𝔄𝔰𝔰_{𝔏}(M), 𝔄𝔰𝔰_{𝔏}(M_{0}) ⊆ 𝔄𝔰𝔰_{𝔏}(N_{P}), hence 𝔄𝔰𝔰_{𝔏}(M_{0}) = ∅.

10.9.5. Definition. Let N be a subobject of an object M such that 𝔄𝔰𝔰_{𝔏}(M/N) is finite. Let {N_{i} | i ∈ J} be a primary decomposition of N in M with 𝔄𝔰𝔰_{𝔏}(M/N_{i} = {P_{i}}. The primary decompsition {N_{i} | i ∈ J} is called reduced if

(a) for any i ∈ J, 𝔄𝔰𝔰_{𝔏}(⋂_{J∋j≠i} N_{j}/ ⋂_{j∈J} N_{j}) ≠ ∅; in particular, the intersection

⋂_{J∋j≠i} N_{j} is not a subobject of N_{i};

(b) if i ≠ j, then P_{i} ≠ P_{j}.

10.9.5.1. Note. Starting with an arbitrary primary decomposition, one can obtain a reduced primary decomposition as follows. Let {N_{i} | i ∈ J} be any primary decomposition of N ↪ M with 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = {P_{i}}, i ∈ J. Set Φ = {P_{i} | i ∈ J}. Let J_{0} is a minimal element of the set of subsets, I, of J such that {N_{i} | i ∈ I} is a primary decomposition. Clearly, {N_{i} | i ∈ J_{0}} satisfies the condition (a). For each P ∈ Φ, let N_{P} = ⋂_{P_{i}=P} N_{i}. By 10.9.2, N_{P} ↪ M is P-primary. Since ⋂_{P∈Φ} N_{P} = ⋂_{i∈J} N_{i}, the set of subobjects {N_{P} | P ∈ Φ} is a reduced primary decomposition of N ↪ M.

10.9.6. Proposition. Let N be a subobject of an object M such that 𝔄𝔰𝔰_{𝔏}(M/N) is finite. Let {N_{i} | i ∈ J} be a primary decomposition of N in M with 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = {P_{i}}.

(i) The following conditions are equivalent:

(a) The decomposition {N_{i} | i ∈ J} is reduced.

(b) All P_{i} belong to 𝔄𝔰𝔰_{𝔏}(M/N) and P_{i} ≠ P_{j} if i ≠ j. (ii) If the equivalent conditions (a), (b) are fulfilled, then

𝔄𝔰𝔰_{𝔏}(M/N) = {P_{i} | i ∈ J} and
𝔄𝔰𝔰_{𝔏}(N_{i}/N) = {P_{j} | j ∈ J, j ≠ i} for all i ∈ J.

Proof. (a)⇒(b). Let {N_{i} | i ∈ J} be a reduced primary decomposition. By 10.9.3.1, 𝔄𝔰𝔰_{𝔏}(M/N) is a subset of {P_{i} | i ∈ J}.

Set N_{i}^{∨} = ⋂_{J∋j≠i} N_{j}. We can and will assume that N = ⋂_{j∈J} N_{j} = N_{i}^{∨} ∩ N_{i}. Since the decomposition {N_{i} | i ∈ J} is reduced, 𝔄𝔰𝔰_{𝔏}(N_{i}^{∨}/N) ≠ ∅. Because N_{i}^{∨}/N is isomorphic to the subobject sup(N_{i}^{∨}, N_{i})/N_{i} of M/N_{i}, this implies that 𝔄𝔰𝔰_{𝔏}(N_{i}^{∨}/N) = {P_{i}}. This shows the inverse inclusion: {P_{i} | i ∈ J} ⊆ 𝔄𝔰𝔰_{𝔏}(M/N).

(b)⇒(a). If the condition (b) holds, {N_{j} | j ∈ J − {i}} cannot be a primary decomposition, because this would imply that P_{i} ∉ 𝔄𝔰𝔰_{𝔏}(M/N). Therefore the primary decomposition {N_{i} | i ∈ J} of N ↪ M is reduced.

The equality 𝔄𝔰𝔰_{𝔏}(M/N) = {P_{i} | i ∈ J} is already established. It remains to show that for any i ∈ J, 𝔄𝔰𝔰_{𝔏}(N_{i}/N) = {P_{j} | j ∈ J, j ≠ i}. Applying 10.8.2(a) to the exact sequence

0 ⟶ N_{i}/N ⟶ M/N ⟶ M/N_{i} ⟶ 0,

we obtain inclusions

𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⊆ 𝔄𝔰𝔰_{𝔏}(M/N) ⊆ 𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⋃ 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = 𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⋃{P_{i}}.

This and the equality 𝔄𝔰𝔰_{𝔏}(M/N) = {P_{j} | j ∈ J} imply that

{P_{j} | j ∈ J − {i}} ⊆ 𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⊆ {P_{j} | j ∈ J}.

On the other hand, since N = ⋂_{j∈J−{i}}(N_{i} ∩ N_{j}), we have an inclusion

𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⊆ ⋃_{j∈J−{i}} 𝔄𝔰𝔰_{𝔏}(N_{i}/(N_{i} ∩ N_{j})).

But, N_{i}/(N_{i} ∩ N_{j}) is isomorphic to the subobject sup(N_{i}, N_{j})/N_{j} of the object M/N_{j}, hence 𝔄𝔰𝔰_{𝔏}(N_{i}/(N_{i} ∩ N_{j})) ⊆ 𝔄𝔰𝔰_{𝔏}(M/N_{j}) = {P_{j}}. This gives the inverse inclusion: 𝔄𝔰𝔰_{𝔏}(N_{i}/N) ⊆ {P_{j} | j ∈ J − {i}}.

10.9.7. Corollary. Let {N_{i} | i ∈ J} be a primary decomposition of a subobject N of an object M. Then Card(𝔄𝔰𝔰_{𝔏}(M/N)) ≤ Card(J). The decomposition {N_{i} | i ∈ J} is reduced iff Card(𝔄𝔰𝔰_{𝔏}(M/N)) = Card(J).

Proof. Following the procedure of 10.9.5.1, one can obtain, starting from {N_{i} | i ∈ J}, a reduced primary decomposition, {Ñ_{j} | j ∈ I} such that Card(I) ≤ Card(J). The rest follows from 10.9.6.

For any object M of the category C_{X}, let 𝔇_{℘}(M) denote the set of reduced primary decompositions of 0 ↪ M. By 10.9.6, each element of 𝔇_{℘}(M) is a set, {N_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M)}

of subobjects of M such that 𝔄𝔰𝔰_{𝔏}(M/N_{P}) = {P} and 𝔄𝔰𝔰_{𝔏}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M)} N_{P}) = ∅.

10.9.8. Proposition. Let {N_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M)} and {Ñ_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M)} be two elements of 𝔇_{℘}(M), and let Φ be a subset of 𝔄𝔰𝔰_{𝔏}(M). Then {N_{P} | P ∈ Φ} ⋃{Ñ_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M) − Φ} is an element of 𝔇_{℘}(M).

Proof. Set N_{Φ} = ⋂_{P∈Φ} N_{P} and Ñ_{Φ}^{∨} = ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M)−Φ} Ñ_{P}. Since 𝔄𝔰𝔰_{𝔏}(M/N_{P}) = {P}

and 𝔄𝔰𝔰_{𝔏}(M/Ñ_{P}) = {P} for all P ∈ 𝔄𝔰𝔰_{𝔏}(M), it suffices to verify (thanks to 10.9.6) that 𝔄𝔰𝔰_{𝔏}(N_{Φ} ⋂ Ñ_{Φ}^{∨}) = ∅.

By 10.9.6(ii), 𝔄𝔰𝔰_{𝔏}(N_{P}) = 𝔄𝔰𝔰_{𝔏}(M) − {P}, in particular, P ∉ 𝔄𝔰𝔰_{𝔏}(N_{P}). Therefore, every element of Φ does not belong to 𝔄𝔰𝔰_{𝔏}(N_{Φ}), i.e. Φ ⋂ 𝔄𝔰𝔰_{𝔏}(N_{Φ}) = ∅. Similarly (𝔄𝔰𝔰_{𝔏}(M) − Φ) ⋂ 𝔄𝔰𝔰_{𝔏}(Ñ_{Φ}^{∨}) = ∅. Thus, 𝔄𝔰𝔰_{𝔏}(N_{Φ} ⋂ Ñ_{Φ}^{∨}) ⊆ Φ ⋂(𝔄𝔰𝔰_{𝔏}(M) − Φ) = ∅.

Let Y ⟶^{u} X be an exact localization such that 𝕊 = Ker(u^{∗}) is a coreflective subcategory of the category C_{X}. Let M ∈ ObC_{X}. For every subobject N of the object M, the preimage in M, N^{𝕊} = M ×_{M/N} 𝔱_{𝕊}(M/N), of the 𝕊-torsion of M/N is called 𝕊-saturation of N in M.

10.9.9. Proposition. (a) Let Y ⟶^{u} X be an exact localization such that 𝕊 = Ker(u^{∗}) is a coreflective subcategory of the category C_{X}.

(a) Let M be an object of the category C_{X}, and let {N_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N)} be a reduced primary decomposition of a subobject N of M. Then

(i) {N_{P}^{𝕊} | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N), 𝕊 ⊆ P} is a reduced primary decomposition of the 𝕊saturation, N^{𝕊}, of N in M;

(ii) {u^{∗}(N_{P}) | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N), Ker(u^{∗}) ⊆ P} is a reduced primary decomposition of u^{∗}(N) ↪ u^{∗}(M).

(b) Suppose Y ⟶^{u} X is a continuous localization, i.e. it has a direct image functor, C_{Y} ⟶^{u_{∗}} C_{X}. Let M be an object of C_{Y}, and let {N_{P} | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N)} be a reduced primary decomposition of a subobject N of M. Then {u_{∗}(N_{P}) | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N)} is a reduced primary decomposition of u_{∗}(N) ↪ u_{∗}(M).

Proof. (a) Because M/N^{𝕊} is 𝕊-torsion free, it follows from 10.8.2(d) that

𝔄𝔰𝔰_{𝔏}(M/N^{𝕊}) = {P ∈ 𝔄𝔰𝔰_{𝔏}(M) | 𝕊 ⊆ P}

and the map P ⟼ P/𝕊 induces a bijective map 𝔄𝔰𝔰_{𝔏}(M/N^{𝕊}) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(u^{∗}(M/N^{𝕊})). In particular, 𝔄𝔰𝔰_{𝔏}(M/N_{P}^{𝕊}) = {P} and 𝔄𝔰𝔰_{𝔏}(u^{∗}(M/N_{P}^{𝕊})) = {P/𝕊}. By 10.9.6(ii),

𝔄𝔰𝔰_{𝔏}(N_{P}/N) = 𝔄𝔰𝔰_{𝔏}(M/N) − {P}.

This implies that if P ∈ 𝔄𝔰𝔰_{𝔏}(M/N) is such that 𝕊 ⊆ P, then 𝔄𝔰𝔰_{𝔏}(u^{∗}(N_{P}/N)) = 𝔄𝔰𝔰_{𝔏}(u^{∗}(M/N)) − {P/𝕊}. By 10.8.2(d), this equality is equivalent to the equality

𝔄𝔰𝔰_{𝔏}(N_{P}^{𝕊}/N^{𝕊}) = 𝔄𝔰𝔰_{𝔏}(M/N^{𝕊}) − {P}.

Since the functor u^{∗} is left exact,

u^{∗}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M/N^{𝕊})} N_{P}^{𝕊}/N^{𝕊}) ≃ u^{∗}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M/N^{𝕊})} N_{P}/N) ≃ ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M/N^{𝕊})} u^{∗}(N_{P}/N).

It follows from the argument above that 𝔄𝔰𝔰_{𝔏}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(M/N^{𝕊})} u^{∗}(N_{P}/N)) = ∅. Therefore,

{u^{∗}(N_{P}) | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N^{𝕊}) is a reduced primary decomposition of u^{∗}(N) ↪ u^{∗}(M). This implies, by 10.8.2(d), that {N_{P}^{𝕊} | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N^{𝕊}) is a reduced primary decomposition of N^{𝕊} ↪ M.

(b) By 10.8.5, the bijective map U_{𝔗𝔥}(𝕊) ⟶^{∼} 𝔗𝔥(Y), P ⟼ P/𝕊, induces a bijective

map 𝔄𝔰𝔰_{𝔏}(u_{∗}(M)) ⟶^{∼} 𝔄𝔰𝔰_{𝔏}(M).

Let {N_{P} | P/𝕊 ∈ 𝔄𝔰𝔰_{𝔏}(M)} be a reduced primary decomposition of N ↪ M. Since the functor u^{∗} is left exact, for every P ∈ 𝔄𝔰𝔰_{𝔏}(u_{∗}(M)) (i.e. such P that P/𝕊 ∈ 𝔄𝔰𝔰_{𝔏}(M)) u_{∗}(N_{P})/u_{∗}(N) is a subobject of u_{∗}(N_{P}/N), and the functor u^{∗} maps the canonical morphism u_{∗}(N_{P})/u_{∗}(N) ⟶ u_{∗}(N_{P}/N) to an isomorphism. The latter implies (by 10.8.5) that 𝔄𝔰𝔰_{𝔏}(u_{∗}(N_{P})/u_{∗}(N)) = 𝔄𝔰𝔰_{𝔏}(u_{∗}(N_{P})/N)) = {P/𝕊}. Since u_{∗} is left exact,

u_{∗}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(u_{∗}(M))} N_{P}/N ) ≃ ⋂_{P∈𝔄𝔰𝔰_{𝔏}(u_{∗}(M))} u_{∗}(N_{P}/N).

Therefore,

𝔄𝔰𝔰_{𝔏}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(u_{∗}(M))} u_{∗}(N_{P})/u_{∗}(N)) ⊆ 𝔄𝔰𝔰_{𝔏}( ⋂_{P∈𝔄𝔰𝔰_{𝔏}(u_{∗}(M))} u^{∗}(N_{P}/N)) = ∅.

Thanks to 10.9.6, this proves that {u_{∗}(N_{P}) | P ∈ 𝔄𝔰𝔰_{𝔏}(M/N)} is a reduced primary decomposition of u_{∗}(N) ↪ u_{∗}(M).

10.9.10. Proposition. Suppose X has the property (sup). Let N be a subobject of an object M such that 𝔄𝔰𝔰_{𝔏}(L) ≠ ∅ for any nonzero subquotient of M/N. Suppose that 𝔄𝔰𝔰_{𝔏}(M/N) is finite, and let {N_{i} | i ∈ J} be a reduced primary decomposition of N ↪ M with 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = {P_{i}}. If P_{i} is a maximal element of 𝔄𝔰𝔰_{𝔏}(M/N), then N_{i} equals to P_{i}-saturation of N, i.e. N_{i}/N coincides with P_{i}-torsion of M/N.

Proof. Replacing M by M/N, we assume that N = 0. let {N_{i} | i ∈ J} be a reduced primary decomposition of 0 ↪ M with 𝔄𝔰𝔰_{𝔏}(M/N_{i}) = {P_{i}}.

Since X has the property (sup), any Serre subcategory of C_{X} is coreflective. In particular, every P_{i}, i ∈ J, is coreflective.

By 10.9.9(a)(i), {N_{j}^{P_{i}} | P_{i} ⊆ P_{j}} is the rediced primary decomposition of the Ptorsion, 𝔱_{P_{i}}M, of M. In particular, 𝔄𝔰𝔰_{𝔏}(M^{P_{i}}) = {P_{j} | P_{i} ⊆ P_{j}} = {P_{i}}, because P_{i} is maximal. Here M^{P_{i}} = M/𝔱_{P_{i}}M. Thus, 𝔱_{P_{i}}M ↪ M is a P_{i}-primary subobject. By 10.9.2, the intersection Ñ_{i} = N_{i} ∩ 𝔱_{P_{i}}M is a P_{i}-primary subobject. By 10.8.6, the object M/Ñ_{i} is P_{i}-torsion free for every i ∈ J. Since Ñ_{i} is an object of P_{i}, this implies that Ñ_{i} = 𝔱_{P_{i}}M, i.e. 𝔱_{P_{i}}M ↪ N_{i}. Thus, {N_{i}} is a reduced primary decomposition of 𝔱_{P_{i}}M ↪ N_{i}. Applying to this primary decomposition 10.9.6(ii), we obtain that 𝔄𝔰𝔰_{𝔏}(N_{i}/𝔱_{P_{i}}M) = ∅. Therefore, by hypothesis, N_{i}/𝔱_{P_{i}}M = 0.

10.10. Supports and associated points of objects with a Gabriel-Krull dimension. We fix a ’space’ X such that C_{X} is an abelian category. Let {C_{X_{α}}} be the Gabriel filtration of X (see 8.7).

10.10.1. Proposition. Suppose M ∈ ObC_{X} has a Gabriel-Krull dimension. Then (i) 𝔄𝔰𝔰_{𝔏}(M) ≠ ∅. In particular, Ass_{𝔖𝔢}(M) is not empty. (ii) Every element of Supp_{𝔖𝔢}(M) is contained in some element of Ass_{𝔖𝔢}(M).

Proof. (i) Let β be the biggest ordinal such that the object M is C_{X_{β}}-torsion free. Then the image, M_{β}, of M in C_{X/X_{β}} = C_{X}/C_{X_{β}} has a simple subobject, L. Let P be the preimage of ⟨L⟩ in C_{X}. This is a Serre subcategory. We claim that P ∈ 𝔄𝔰𝔰_{𝔏}(M).

In fact, there exists a subobject, L̃, of M such that the image of L̃ in C_{X}/C_{X_{β}} is isomorphic to L. The object L̃ is P-torsion free: if L′ is a nonzero subobject of L̃, then its image in C_{X}/C_{X_{β}} is a nonzero subobject of L, and L is ⟨L⟩-torsion free.

On the other hand, L̃ ∈ ObP^{⋆}, because P^{⋆} is the preimage of ⟨L⟩^{⋆} = [L]_{•} ∨ ⟨L⟩, and the image of L̃ belongs to [L]_{•}.

(ii) Let 𝕊 ∈ Supp_{𝔖𝔢}(M); i.e. 𝕊 is a Serre subcategory of the category C_{X} such that M ∉ Ob𝕊. The latter means that the image, M̃, of M in the quotient category C_{X}/𝕊 is nonzero. Since M has a Gabriel-Krull dimension, the object M̃ has a Gabriel-Krull dimension too. By (i), 𝔄𝔰𝔰_{𝔏}(M̃) ≠ ∅. Let P̃ ∈ 𝔄𝔰𝔰_{𝔏}(M̃), and let P be the preimage of P̃ in C_{X}. By construction, 𝕊 ⊆ P. We claim that P ∈ Ass_{𝔖𝔢}(M).

In fact, there exists a nonzero subobject, L̃, of the object M̃ which is P̃-torsion free and belongs to P̃^{⋆}. There exists a nonzero subobject, L, of the object M such that L̃ is isomorphic to the image of L in C_{X}/𝕊. Since P^{⋆} is the preimage of P̃^{⋆}, the object L belongs to P^{⋆} and does not belong to P.

10.10.2. Canonical filtrations. Let X have the property (sup). Then all Serre subcategories of C_{X}, in particular the Serre subcategories {C_{X_{α}}} of the Gabriel filtration, are coreflective.

Let 𝕋 be a topologizing subcategory of the category C_{X}. To any ordinal, α, we assign a full subcategory, 𝕋_{α}, of C_{X} defined as follows:

𝕋_{0} = 𝕆, and Ob𝕋_{1} consists of all objects of C_{X} which are supremums of their subobjects from 𝕋; i.e. 𝕋_{1} is the smallest coreflective subcategory of C_{X} containing 𝕋.

If α is not a limit ordinal, then 𝕋_{α} = 𝕋_{α−1} •_{X} 𝕋_{1}.

If β is a limit ordinal, then 𝕋_{β} is the smallest coreflective subcategory containing ⋃_{α<β} 𝕋_{α}, that is 𝕋_{β} = ( ⋃_{α<β} 𝕋_{α})_{1}.

Since the category 𝕋_{1} is coreflective and the Gabriel product of coreflective categories is coreflective, it follows that all subcategories 𝕋_{α} are coreflective. Their union coincides with the Serre subcategory, 𝕋^{−}, spanned by 𝕋.

We use this construction to obtain a refinement of the Gabriel filtration, {C_{X_{α}}}. We set X_{α}^{0} = X_{α}, that is C_{X_{α}^{0}} = C_{X_{α}}. Let C_{X_{α}^{1}} denote the preimage in C_{X} of the socle of the quotient category C_{X}/C_{X_{α}}

which is, by definition, the full subcategory, (C_{X}/C_{X_{α}})_{1}, of the category C_{X}/C_{X_{α}} spanned by semisimple objects. Since (C_{X}/C_{X_{α}})_{1} is a coreflective topologizing subcategory of C_{X}/C_{X_{α}}, its preimage is a coreflective topologizing subcategory of the category C_{X}. For β ≥ 1, we define the subcategories C_{X_{α}^{β}} by setting C_{X_{α}^{β}} = (C_{X_{α}^{1}})_{β}.

It follows that every C_{X_{α}^{β}} is a coreflective subcategory of the Serre subcategory C_{X_{α+1}}, and the union ⋃_{β} C_{X_{α}^{β}} coincides with C_{X_{α+1}}.

Thus, we have obtained an increasing filtration, {C_{X_{α}^{β}} | α, β ≥ 0} of the category C_{X} by coreflective topologizing subcategories such that ⋃_{β} C_{X_{α}^{β}} = C_{X_{α}} for every ordinal α, and ⋃_{α} ⋃_{β} C_{X_{α}^{β}} = C_{X_{ω}}, where C_{X_{ω}} is the full subcategory of the category C_{X} spanned by all objects having a Gabriel-Krull dimension.

10.10.2.1. Proposition. Suppose X has the property (sup). Then

Spec_{𝔰}^{1}(X_{ω}) = Spec^{−}(X_{ω}) ≃ ⋃_{α} Spec^{−}(X_{α}^{1}) ≃ ∐_{α} Spec(X_{α}^{1}/X_{α}).  (1)

Here C_{X_{α}^{1}/X_{α}} = C_{X_{α}^{1}}/C_{X_{α}}.

Proof. The equality Spec_{𝔰}^{1}(X_{ω}) = Spec^{−}(X_{ω}) is a consequence of 8.7.1.

Let P ∈ Spec^{−}(X_{ω}). Since C_{X_{ω}} = ⋃_{α} C_{X_{α}} and P is a proper Serre subcategory of C_{X_{ω}}, there exists an ordinal α such that C_{X_{α}} ⊆ P and C_{X_{α+1}} ⊈ P. Since P is a Serre subcategory, and C_{X_{α+1}} = C_{X_{α}^{1}}^{−}, C_{X_{α+1}} ⊈ P iff C_{X_{α}^{1}} ⊈ P. The latter means that C_{X_{α}^{1}}/C_{X_{α}} ⊈ P/C_{X_{α}}. Since all objects of the category C_{X_{α}^{1}/X_{α}} = C_{X_{α}^{1}}/C_{X_{α}} are semisimple, there exists a simple object, L, of the category C_{X_{ω}}/C_{X_{α}} which does not belong to the Serre subcategory P/C_{X_{α}}. By 8.5.1, this implies that P/C_{X_{α}} = ⟨L⟩; i.e. P/C_{X_{α}} is a closed point of Spec(X_{ω}/X_{α}). Since every simple object of the category C_{X_{ω}}/C_{X_{α}} = C_{X_{ω}/X_{α}} belongs to the subcategory C_{X_{α}^{1}/X_{α}}, the family of closed points of Spec(X_{ω}/X_{α}) is in one-to-one correspondence with Spec(X_{α}^{1}/X_{α}).

10.10.2.2. The canonical filtration of objects. For any M ∈ ObC_{X}, the filtration {C_{X_{α}^{β}}} induces a canonical increasing filtration, {M_{α}^{β}}, of M. Here the subobject M_{α}^{β} is the C_{X_{α}^{β}}-torsion of M. The supremum of the subobjects M_{α}^{β} coincides with the C_{X_{ω}}- torsion of the object M. Let q_{α}^{∗} denote a localization functor C_{X} ⟶ C_{X/X_{α}} = C_{X}/C_{X_{α}}. Then for any α and β, the objects q_{α}^{∗}(M_{α}^{β+1}/M_{α}^{β}) are either zero, or semisimple.

10.10.2.3. Proposition. Suppose X has a Gabriel-Krull dimension (i.e. X = X_{ω}) and the property (sup). Then for any M ∈ ObC_{X},

\begin{matrix}𝔄𝔰𝔰_{𝔏}(M) = & ⋃_{α,β} 𝔄𝔰𝔰_{𝔏}(M_{α}^{β}) ⊆ ⋃_{α} ⋃_{β} 𝔄𝔰𝔰_{𝔏}(M_{α}^{β+1}/M_{α}^{β}). \\ Ass_{𝔖𝔢}(M) = & ⋃_{α,β} Ass_{𝔖𝔢}(M_{α}^{β}) ⊆ ⋃_{α} ⋃_{β} Ass_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β}) = ⋃_{α} ⋃_{β} 𝔄𝔰𝔰_{𝔏}(M_{α}^{β+1}/M_{α}^{β}) \\ Supp_{𝔖𝔢}(M) = & ⋃_{α,β} Supp_{𝔖𝔢}(M_{α}^{β}) = ⋃_{α} ⋃_{β} Supp_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β}) \end{matrix}

Proof. The equality and the inclusion

𝔄𝔰𝔰_{𝔏}(M) = ⋃_{α,β} 𝔄𝔰𝔰_{𝔏}(M_{α}^{β}) ⊆ ⋃_{α} ⋃_{β} 𝔄𝔰𝔰_{𝔏}(M_{α}^{β+1}/M_{α}^{β})

follow from 10.8.2. Similarly, the equality and the inclusion

Ass_{𝔖𝔢}(M) = ⋃_{α,β} Ass_{𝔖𝔢}(M_{α}^{β}) ⊆ ⋃_{α} ⋃_{β} Ass_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β})

and the equalities

Supp_{𝔖𝔢}(M) = ⋃_{α,β} Supp_{𝔖𝔢}(M_{α}^{β}) = ⋃_{α} ⋃_{β} Supp_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β})

follow from 10.7.1. It remains to show that

Ass_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β}) = 𝔄𝔰𝔰_{𝔏}(M_{α}^{β+1}/M_{α}^{β})  (2)

for all α and β. If M_{α}^{β+1}/M_{α}^{β} = 0, the equality (2) holds by a trivial reason. Suppose M_{α}^{β+1}/M_{α}^{β} ≠ 0; and let P ∈ Ass_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β}); i.e. there exists a subobject, L, of M_{α}^{β+1}/M_{α}^{β} which belongs to ObP^{⋆} − ObP. By (the argument of) 10.10.2.1, there exists an ordinal γ such that P is the preimage in C_{X} of the Serre subcategory ⟨N ⟩ for some simple object, N, of the quotient category C_{X/X_{γ}} = C_{X}/C_{X_{γ}}. It follows from this description that γ = α. In fact, it cannot be that γ > α, because in this case the object L belongs to the subcategory C_{X_{γ}} which contradicts to the condition that L ∉ ObP.

The condition L ∈ ObP^{⋆} − ObP implies that the image, q_{γ}^{∗}(L), of L in C_{X}/C_{X_{γ}} belongs to the Serre subcategory, ({N } ∪ ⟨N ⟩)^{−}, containing the subcategory ⟨N ⟩ and the object N, but does not belong to ⟨N ⟩. This implies that N is a subobject of the object q_{γ}^{∗}(L). By a standard argument, there exists a subobject, L_{1}, of L such that q_{γ}^{∗}(L_{1}) ≃ N. In particular, L_{1} ∈ ObC_{X_{γ+1}}. If γ < α, then L_{1} is C_{X_{γ+1}}-torsion free, hence it is equal to zero which is not the case. Thus, γ = α and L_{1} is a nonzero object of the category P^{⋆} which is P-torsion free.

10.10.2.4. Note. Suppose X has the property (sup), and let M ∈ ObC_{X}. Consider the canonical filtration, {M_{α}^{β}} of the object M. If M_{α}^{β+1}/M_{α}^{β} ≠ 0, then the object q_{α}^{∗}(M_{α}^{β+1}/M_{α}^{β}) of the quotient category C_{X}/C_{X_{α}} is semisimple (cf. 10.10.2.2). By a standard argument, this means that there exist monomorphisms L_{i} ⟶ M_{α}^{β+1}/M_{α}^{β}, i ∈ J, such that all objects q_{α}^{∗}(L_{i}) are simple and the natural morphism

⨁_{i∈J} q_{α}^{∗}(L_{i}) ⟶ q_{α}^{∗}(M_{α}^{β+1}/M_{α}^{β})

is an isomorphism. In particular, q_{α}^{∗}(L_{i}) ∩ q_{α}^{∗}(L_{j}) = 0 if i ≠ j. Since the localization q_{α}^{∗} is exact, in particular left exact, q_{α}^{∗}(L_{i}) ∩ q_{α}^{∗}(L_{j}) ≃ q_{α}^{∗}(L_{i} ∩ L_{j}), hence L_{i} ∩ L_{j} belongs to the subcategory C_{X_{α}}. Since the object M_{α}^{β+1}/M_{α}^{β} is C_{X_{α}}-torsion free, this means that L_{i} ∩ L_{j} = 0 whenever i ≠ j. Therefore the monomorphisms L_{i} ⟶ M_{α}^{β+1}/M_{α}^{β}, i ∈ J, induce an essential monomorphism

⨁_{i∈J} L_{i} ⟶ M_{α}^{β+1}/M_{α}^{β}  (3)

whose cokernel belongs to the subcategory C_{X_{α}}. Recall that essential means that every nonzero subobject of M_{α}^{β+1}/M_{α}^{β} has a nonzero intersection with ⨁_{i∈J} L_{i}.

The set Ass_{𝔖𝔢}(M_{α}^{β+1}/M_{α}^{β}) = 𝔄𝔰𝔰_{𝔏}(M_{α}^{β+1}/M_{α}^{β}) consists of preimages in C_{X} of the Serre subcategories ⟨q_{α}^{∗}(L_{i})⟩, i ∈ J.

Notice that if α = 1, then all objects L_{i}, i ∈ J, are simple, and the morphism (3) is an isomorphism.

10.10.3. Proposition. Suppose M ∈ ObC_{X} has a Gabriel-Krull dimension. The following properties are equivalent:

(a) The dimension of M is zero.

(b) Every element of Ass_{𝔖𝔢}(M) is of the form ⟨P ⟩ for some simple object P.

(c) Every element of Supp_{𝔖𝔢}(M) is of the form ⟨P ⟩ for some simple object P.

Proof. The implication (c) ⇒ (b) follows from the inclusion Ass_{𝔖𝔢}(M) ⊆ Supp_{𝔖𝔢}(M).

(b) ⇒ (c). By 10.10.1, every P ∈ Supp_{𝔖𝔢}(M) is contained in an element of Ass_{𝔖𝔢}(M). By hypothesis, the latter is of the form ⟨L⟩ for some simple object L. The inclusion P ⊆ ⟨L⟩ means precisely that L ∉ ObP. Let q^{∗} be a localization functor C_{X} ⟶ C_{X}/P. Since the simple object L does not belong to P its image, q^{∗}(L), in the quotient category C_{X}/P is a simple object. Let N ∈ ObC_{X} − ObP. Since the category C_{X}/P is local and q^{∗}(N) ≠ 0, of the object q^{∗}(N) is ’greater’ than q^{∗}(L), i.e. q^{∗}(N) ≻ q^{∗}(L). This means that L ∈ ObP^{⋆}. By 8.5.1, P = ⟨L⟩.

(a) ⇒ (c). Let C_{X_{0}} be the full subcategory of C_{X} spanned by objects of zero dimension. And let C_{X_{0}^{1}} be the full subcategory of C_{X} spanned by semisimple objects. It follows from 10.10.2.1 that Spec_{𝔰}^{0}(X)_{0} = Spec_{𝔰}(X)_{0}^{1} = SpecX_{0}^{1} = {⟨L⟩ | L ∈ SimpleC_{X}}.

(c) ⇒ (a). If the dimension of M is greater than zero, then M has a quotient which belongs to C_{X_{α}} for α ≥ 1. This implies that Supp_{𝔖𝔢}(M) contains an element which is not of the form ⟨L⟩ for a simple object L.

10.10.4. Corollary. For any object, M, of zero dimension Supp_{𝔖𝔢}(M) = Ass_{𝔖𝔢}(M).

Proof. The fact follows from the argument (b) ⇒ (c) in the proof of 10.10.3.

10.10.5. Proposition. Let M be an object with Gabriel-Krull dimension. Let P ∈ Supp_{𝔖𝔢}(M) and let q^{∗} denote the localization functor C_{X} ⟶ C_{X}/P. The object q^{∗}(M) is of zero dimension iff P is a maximal element of Ass_{𝔖𝔢}(M).

Proof. By 10.7.4, the map P′ ⟶ P′/P induces a bijection {P′ ∈ Ass_{𝔖𝔢}(M) | P ⊆ P′} and Ass_{𝔖𝔢}(q^{∗}(M)). Since the category C_{X}/P is local, it follows from 10.10.3 that the object q^{∗}(M) is of dimension zero iff Ass_{𝔖𝔢}(q^{∗}(M)) = {𝕆}. This happens iff P is a maximal element of Ass_{𝔖𝔢}(M).

10.10.6. Proposition. Let M be an object of finite length. There exists only one (necessarily reduced) primary decomposition of 0 ↪ M.

Proof. By (the argument of) 10.10.3, all elements of Ass_{𝔖𝔢}(M) are maximal. In particular, all elements of 𝔄𝔰𝔰_{𝔏}(M) are maximal. The uniqueness of the primary decomposition follows now from 10.9.10. For every P ∈ 𝔄𝔰𝔰_{𝔏}(M), the P-primary subobject of M coincides with the P-torsion of M.

10.11. Supports and associated points in other spectra.

The other spectra introduced in this work (and in [R]) are contained (or embedded) in one of the spectra considered above: Spec_{𝔗𝔥}^{0}(X), or in Spec_{𝔗𝔥}^{1}(X), or in Spec_{𝔰}^{1}(X). If Spec_{?}(X) is one of those spectra, we define, for any M ∈ ObC_{X}, the support, Supp_{?}(M), and weakly associated points, Ass_{?}(M), of M in Spec_{?}(X) as the intersection (a pull-back) of resp. Supp_{𝔗𝔥}^{1}(M) and Ass_{𝔗𝔥}^{1}(M) with Spec_{?}(X). Thus, we have the flat support,

Supp_{𝔣𝔏}^{1}(M) = Supp_{𝔗𝔥}^{1}(M) ⋂ Spec_{𝔣𝔏}^{1}(X)

and flat weakly associated points,

Ass_{𝔣𝔏}^{1}(M) = Ass_{𝔗𝔥}^{1}(M) ⋂ Spec_{𝔣𝔏}^{1}(X).

Similarly, we have support and weakly associated points of M in Spec_{𝔰}^{0}(X):

Supp_{𝔰}(M) = Supp_{𝔗𝔥}(M) ⋂ Spec_{𝔰}^{0}(X) and Ass_{𝔰}(M) = Ass_{𝔗𝔥}(M) ⋂ Spec_{𝔰}^{0}(X).

Finally, we define Supp(M) and Ass(M) by taking the intersection of Spec(X) (cf. 7.6) with respectively Supp_{𝔗𝔥}(M) and Ass_{𝔗𝔥}(M).

The general properties described in 10.2, 10.3, 10.4, and 10.5 are inherited by Supp_{?} and Ass_{?} for all ?. If Supp_{?} and Ass_{?} are contained in Spec_{𝔰}^{1}(X), the assertion 10.7.1(b) and its corollary 10.7.2 are valid for them.

One should be careful with the functorial properties described by 10.6 and 10.7.4: the functoriality breaks for Supp_{𝔰}(−) and Ass_{𝔰}(−) and for Supp(−) and Ass(−).

IV. Spectra of ’spaces’ represented by triangulated categories.

11. Actions of monoidal categories and their spectra.

We fix a monoidal category C̃_{T} = (C_{T} , ⊙, 𝔞; 1, ϕ_{𝔩}, ϕ_{𝔯}). Here C_{T} × C_{T} ⟶^{⊙} C_{T} is a monoidal structure (’tensor product’); 𝔞 is an associativity constraint, i.e. a functor isomorphism

⊙ ∘ (⊙ × Id_{C_{T}}) ⟶ ⊙ ∘ (Id_{C_{T}} × ⊙)

satisfying certain natural compatibility conditions; 1 denotes the unit object,

1 ⊙ − ⟶^{ϕ_{𝔩}} Id_{C_{T}} ⟵^{ϕ_{𝔯}} − ⊙ 1

are canonical isomorphisms.

11.1. Actions. Let X be a ’space’. Fix an action, C_{T} ×C_{X} ⟶^{γ^{∗}} C_{X}, of the monoidal category C̃_{T} = (C_{T} , ⊙, 1) on the category C_{X}. The functor γ^{∗} induces a functor

C_{T} ⟶^{Γ} End(C_{X}), a ⟼ Γ_{a},

where End(C_{X}) denote the category of functors C_{X} ⟶ C_{X}, and Γ_{a}(M) = γ^{∗}(a, M). The functor γ^{∗} being an ’action’ means precisely that Γ is a monoidal functor, i.e. for any a, b ∈ ObC_{T}, there are natural morphisms

Γ_{a} ∘ Γ_{b} ⟶^{ϕ_{a,b}} Γ_{a⊙b} and Γ_{1} ⟶^{∼} Id_{C_{X}}

related in a natural way between themselves and with associativity constraint on C̃_{T}.

A pair (C_{X}, γ^{∗}), where γ^{∗} is a C̃_{T} -action, is called a C̃_{T} -category. We call a pair (X, γ^{∗}) a C̃_{T} -’space’. A morphism (more precisely, a 1-morphism) between two C̃_{T} -’spaces’, (X, γ^{∗}) ⟶ (Y, γ^{∗}), is given by a pair (F, ϕ), where F is a functor C_{Y} ⟶^{F} C_{X} such that the diagram

C_{T} × C_{Y} ⟶^{Id×F} C_{T} × C_{X}
γ^{∗} ↓  ↓ γ^{∗}
C_{Y}  ⟶^{F}  C_{X}

quasi-commutes, and ϕ is a functor isomorphism γ^{∗} ∘ (Id × F) ⟶^{∼} F ∘ γ^{∗} satisfying a standard cocycle condition. The composition of morphisms is defined naturally.

11.1.1. Note. In the language of ’spaces’, the monoidal structure, C_{T} × C_{T} ⟶^{⊙} C_{T}, can be regarded as an inverse image of a morphism (a coaction) T ⟶ T ∐ T, and the action γ^{∗} as an inverse image functor of a morphism X ⟶^{γ} T ∐ X.

11.1.2. Actions of a monoid, ℤ-categories. Any monoid, G, might be regarded as a discrete category with the monoidal structure given by multiplication. This defines an isomorphism between the category of monoids and the category of discrete ’small’ monoidal categories which allows to define actions of monoids on categories. Thus, a G-category is a pair (C_{X}, γ^{∗}), where γ^{∗} is a monoidal functor from G to the monoidal category 𝔈𝔫𝔡(C_{X}) of functors C_{X} ⟶ C_{X}. If G is a group, the functor γ takes values in the monoidal subcategory Pic(X) of 𝔈𝔫𝔡(C_{X}) formed by invertible functors and isomorphisms between them. The group ℤ is of particular interest because triangulated categories, categories of graded modules, and the category of quasi-coherent sheaves on (commutative, or noncommutative) Proj are ℤ-categories.

11.2. A graded category associated with an action. Suppose the category C_{T} is ’small’. For any pair of objects, L, M, of the category C_{X} and an object a of the category C_{T}, set C_{X}^{a}(L, M) = C_{X}(L, γ^{∗}(a, M)) = C_{X}(L, Γ_{a}(M)). If N ∈ ObC_{X} and b ∈ ObC_{T}, then we define a map

C_{X}^{a}(L, M) × C_{X}^{b}(M, N) ⟶ C_{X}^{a⊙b}(L, N)  (1)

as the composition of the maps

C_{X}(L, Γ_{a}(M)) × C_{X}(M, Γ_{b}(N)) ⟶ C_{X}(L, Γ_{a}(M)) × C_{X}(Γ_{a}(M), Γ_{a} ∘ Γ_{b}(N))
↓
C_{X}(L, Γ_{a}(M)) × C_{X}(Γ_{a}(M), Γ_{a⊙b}(N)),

where the vertical arrow is induced by the functor morphism Γ_{a} ∘ Γ_{b} ⟶^{ϕ_{a,b}} Γ_{a⊙b}, and the composition map

C_{X}(L, Γ_{a}(M)) × C_{X}(Γ_{a}(M), Γ_{a⊙b}(N)) ⟶ C_{X}(L, Γ_{a⊙b}(N)) = C_{X}^{a⊙b}(L, N)

This defines an enriched category, C_{(X,γ^{∗})}, with the same objects as C_{X}; morphisms between objects are C̃_{T} -graded sets. To every morphism, (X, γ^{∗}) ⟶ (Y, γ^{∗}), of C̃_{T} - actions, there corresponds a morphism, C_{(X,γ^{∗})} ⟶ C_{(Y,γ^{∗})}, of enriched categories.

11.2.1. The case of ’spaces’ over a monoid. Let G be a monoid and (X, γ^{∗}) a G- ’space’ such that C_{X} is a k-linear category for for some commutative ring k. Then C_{(X,γ^{∗})} gives rise to an enriched category over the monoidal category of G-graded k-modules. In particular, a ℤ-’space’ defines an enriched category over the monoidal category of ℤ-graded k-modules.

11.3. Stable saturated multiplicative systems. Let S^{𝔰}M(X, γ^{∗}) denote the family of saturated multiplicative systems in C_{X} which are invariant with respect to the action of C̃_{T}. It follows from the universal property of localizations that for every Σ ∈ S^{𝔰}M(X, γ^{∗})^{⋆}, the ’space’ of fractions, Σ^{−1}X, inherits a C̃_{T} -action uniquely defined by the condition that the canonical morphism Σ^{−1}X ⟶^{q_{Σ}} X is a morphism of actions.

11.3.1. Proposition. (a) If Σ is a C̃_{T} -stable multiplicative system, then its saturation, Σ^{𝔰} is a C̃_{T} -stable multiplicative system too.

(b) If Σ_{1} and Σ_{2} are C̃_{T} -stable saturated multiplicative systems, then the smallest saturated multiplicative system, Σ_{1} ∨ Σ_{2}, spanned by Σ_{1} and Σ_{2} is C̃_{T} -stable.

(c) Suppose C_{X} has finite colimits. Then the intersection of any set of C̃_{T} -stable saturated left multiplicative systems is a C̃_{T} -stable saturated left multiplicative system.

Proof. (a) The saturation, Σ^{𝔰}, of a multiplicative system Σ consists of all morphisms s of C_{X} such that u∘s ∈ Σ ∋ s∘v for some morphisms u and v (see 1.3.3). If Σ is C̃_{T} -stable, then

Γ_{a}(u) ∘ Γ_{a}(s) = Γ_{a}(u ∘ s) ∈ Σ ∋ Γ_{a}(s ∘ v) = Γ_{a}(s) ∘ Γ_{a}(v)

for all a ∈ ObC_{T} (cf. 11.1 for notations); hence Γ_{a}(s) ∈ Σ for all a ∈ ObC_{T}.

(b) We need several steps.

(i) Let Σ_{1} and Σ_{2} be two left multiplicative systems, and let Σ_{1} ⊔ Σ_{2} denote the smallest family of arrows closed under composition and containing Σ_{1} and Σ_{2}. We claim that Σ_{1} ⊔ Σ_{2} is a left multiplicative system.

The family Σ_{1} ⊔ Σ_{2} consists of all possible compositions of arrows of Σ_{1} ⋃ Σ_{2}. Since Σ_{1} and Σ_{2} are closed under composition and contain all identical morphisms, a generic element, s, of Σ_{1} ⊔ Σ_{2} can be represented as a composition

M_{0} ⟶^{s_{1}} L_{1} ⟶^{t_{1}} M_{1} ⟶^{s_{2}} … ⟶^{s_{n}} L_{n} ⟶^{t_{n}} M_{n},  (2)

where s_{i} ∈ Σ_{1} and t_{i} ∈ Σ_{2}, i = 1, ..., n.

Let s be an element of Σ_{1} ⊔Σ_{2} given by the composition (2), and let M_{0} ⟶^{f} M′_{0} be an arbitrary morphism. By the property (SL2) (see 1.3), there exists a commutative square

M_{0}  ⟶^{s_{1}} L_{1}
f ↓  ↓ g_{1}
M′_{0}  ⟶^{s′_{1}} L′_{1}

where s′_{1} ∈ Σ_{1}. Applying the property (SL2) to the pair of morphisms L′_{1} ⟵^{g_{1}} L_{1} ⟶^{t_{1}} M_{1}, we complete it to a commutative diagram

L_{1}  ⟶^{t_{1}} M_{1}
g_{1} ↓  ↓ f_{1}
L′_{1}  ⟶^{t′_{1}} M′_{1}

where t′_{1} ∈ Σ_{2}. Continuing this process, we obtain a commutative diagram

M_{0}  ⟶^{s_{1}} L_{1}  ⟶^{t_{1}} M_{1}  ⟶^{s_{2}} … ⟶^{s_{n}} L_{n}  ⟶^{t_{n}} M_{n}
f ↓  ↓ g_{1}  ↓ f_{1}  …  g_{n} ↓  ↓ f_{n}
M′_{0}  ⟶^{s′_{1}} L′_{1}  ⟶^{t′_{1}} M′_{1}  ⟶^{s′_{2}} … ⟶^{s′_{n}} L′_{n}  ⟶^{t′_{n}} M′_{n}

where s′_{i} ∈ Σ_{1} and t′_{i} ∈ Σ_{2}, i = 1, ..., n. Thus, Σ_{1} ⊔ Σ_{2} has the property (SL2). It remains to verify the property (SL3) (see 1.3).

Let s be an element of Σ_{1} ⊔Σ_{2} presented as the composition (2). And let M_{n} \overrightarrow{⟶}^{f}_{g} N

be a pair of arrows such that f ∘ s = g ∘ s. This equality can be presented as

(f ∘ t_{n} ∘ s_{n} ∘ ... ∘ t_{1}) ∘ s_{1} = (g ∘ t_{n} ∘ s_{n} ∘ ... ∘ t_{1}) ∘ s_{1}.

By the property (SL3), there exists an element s′_{1} ∈ Σ_{1} such that

s′_{1} ∘ (f ∘ t_{n} ∘ s_{n} ∘ ... ∘ t_{1}) = s′_{1} ∘ (g ∘ t_{n} ∘ s_{n} ∘ ... ∘ t_{1}).  (3)

Applying (SL3) to the equality (3) presented in the form

(s′_{1} ∘ f ∘ t_{n} ∘ s_{n} ∘ ... ∘ s_{2}) ∘ t_{1} = (s′_{1} ∘ g ∘ t_{n} ∘ s_{n} ∘ ... ∘ s_{2}) ∘ t_{1},

we find an element t′_{1} ∈ Σ_{2} such that

t′_{1} ∘ (s′_{1} ∘ f ∘ t_{n} ∘ s_{n} ∘ ... ∘ s_{2}) = t′_{1} ∘ (s′_{1} ∘ g ∘ t_{n} ∘ s_{n} ∘ ... ∘ s_{2}).

By an induction argument, we obtain the equality

(t′_{n} ∘ s′_{n} ∘ ... ∘ t′_{1} ∘ s′_{1}) ∘ f = (t′_{n} ∘ s′_{n} ∘ ... ∘ t′_{1} ∘ s′_{1}) ∘ g,

where s′_{i} ∈ Σ_{1} and t′_{i} ∈ Σ_{2}, i = 1, ..., n.

(ii) It follows from the description of Σ_{1} ⊔ Σ_{2} in (i) that if Σ_{1} and Σ_{2} are C̃_{T} -stable, then Σ_{1} ⊔ Σ_{2} is C̃_{T} -stable too.

(iii) The smallest saturated multiplicative system, Σ_{1} ∨ Σ_{2}, spanned by Σ_{1} and Σ_{2} is, evidently, the saturation of Σ_{1} ⊔ Σ_{2}. By (ii), the left multiplicative system Σ_{1} ⊔ Σ_{2} is C̃_{T} -stable. Therefore, by (a), its saturation, Σ_{1} ∨ Σ_{2}, is C̃_{T} -stable.

(c) The assertion follows from 1.4.3.

11.4. Spectra. Clearly, the trivial multiplicative system, Iso(C_{X}), is C̃_{T} -invariant, i.e. Iso(C_{X}) ∈ S^{𝔰}M(X, γ^{∗}). Let S^{𝔰}M(X, γ^{∗})^{⋆} denote the family S^{𝔰}M(X, γ^{∗})−{Iso(C_{X})} of non-trivial C̃_{T} -invariant saturated multiplicative systems.

All notions and facts considered so far in this work are extended to ’spaces’ with an action of a monoidal category C̃_{T} = (C_{T} , ⊙, 1) by simply replacing S^{𝔰}M(X)^{⋆} by S^{𝔰}M(X, γ^{∗})^{⋆} and inserting ”C̃_{T} -invariant” whenever it is required.

Thus, 𝔏-local C̃_{T} -’spaces’ are those C̃_{T} -’spaces’ (X, γ^{∗}) for which the intersection of all Σ ∈ S^{𝔰}M(X, γ^{∗})^{⋆} belongs to S^{𝔰}M(X, γ^{∗})^{⋆}.

The complete 𝔏-spectrum, Spec_{𝔏}^{1}(X, γ^{∗}), of a C̃_{T} -’space’ (X, γ^{∗}) consists of all Σ ∈ S^{𝔰}M(X, γ^{∗})^{⋆} such that the C̃_{T} -’space’ of fractions (Σ^{−1}X, γ^{∗}) is 𝔏-local. In other words, there exists the smallest C̃_{T} -invariant saturated multiplicative system, Σ^{⋆}, properly containing Σ.

The 𝔏-spectrum, Spec_{𝔏}^{0}(X, γ^{∗}), of a C̃_{T} -’space’ (X, γ^{∗}) is formed by Σ ∈ S^{𝔰}M(X, γ^{∗}) such that there exists the biggest C̃_{T} -invariant saturated multiplicative system, Σ̂, which does not contain Σ.

11.4.1. Proposition. The map Σ ⟼ Σ̂ is an injective morphism from Spec_{𝔏}^{0}(X, γ^{∗}) to Spec_{𝔏}^{1}(X, γ^{∗}).

Proof. Let Σ ∈ Spec_{𝔏}^{0}(X, γ^{∗}). By 11.3.1(b), Σ ∨ Σ̂ is the smallest C̃_{T} -invariant saturated multiplicative system which contains Σ and Σ̂. If Σ_{1} is a C̃_{T} -invariant saturated multiplicative system which properly contains Σ̂, then it contains Σ too, hence it contains Σ∨Σ̂. Therefore, Σ∨Σ̂ is the smallest C̃_{T} -invariant saturated multiplicative system which properly contains Σ̂; in particular, Σ̂ ∈ Spec_{𝔏}^{1}(X, γ^{∗}).

Injectivity of the map Σ ⟼ Σ̂ follows from that Σ_{1} ⊆ Σ_{2} iff Σ̂_{1} ⊆ Σ̂_{2}.

The definitions of the remaining spectra are even more straightforward.

11.4.2. Closed spectra and flat spectra. Elements of the closed complete 𝔏spectrum, Spec_{ℭ}^{1}(X, γ^{∗}), are those Σ ∈ Spec_{𝔏}^{1}(X, γ^{∗}) which are closed in the sense of 5.2. The closed 𝔏-spectrum, Spec_{ℭ}^{0}(X, γ^{∗}), consists of all Σ ∈ Spec_{𝔏}^{0}(X, γ^{∗}) such that Σ̂ is closed.

The flat complete 𝔏-spectrum, Spec_{𝔣𝔏}^{1}(X, γ^{∗}), is formed by Σ ∈ Spec_{𝔏}^{1}(X, γ^{∗}) such that the localization Σ^{−1}X ⟶ X is continuous (i.e. it has a direct image functor).

The flat 𝔏-spectrum, Spec_{𝔣𝔏}^{0}(X, γ^{∗}), is formed by Σ ∈ Spec_{𝔏}^{0}(X, γ^{∗}) such that Σ̂ belongs to Spec_{𝔣𝔏}^{1}(X, γ^{∗}).

11.4.2.1. Note. For any Σ ∈ S^{𝔰}M(X, γ^{∗}), the full subcategory of the category C_{X} generated by objects which are left closed for Σ (cf. 5.3) is C̃_{T} -stable. But, the full subcategory of C_{X} whose objects are Σ-torsion free objects of C_{X} is not, in general, C̃_{T} -stable.

11.5. Locally trivial actions and spectra.

11.5.1. Proposition. Let (X, γ^{∗}) be a C̃_{T} -’space’ such that there exists a family {Σ_{i} | i ∈ J} of saturated stable multiplicative systems with the following properties:

(a) ⋂_{i∈J} Σ_{i} = Iso(C_{X});

(b) every Σ ∈ S^{𝔰}M(X) containing some of Σ_{i} is C̃_{T} -stable.

Then the canonical map Spec_{𝔏}^{0}(X) ⟶ Spec_{𝔏}^{1}(X), Σ ↦ Σ̂, takes values in Spec_{𝔏}^{1}(X, γ^{∗}).

Proof. Let U_{i} denote the ’space’ Σ_{i}^{−1}X and u_{i} the canonical morphism U_{i} ⟶ X, i ∈ J. Since each Σ_{i} is stable, the action γ^{∗} induces a C̃_{T} -action, γ_{i}^{∗} on U_{i}. The condition (a) means that the family of localizations {U_{i} ⟶^{q_{i}} X | i ∈ J} is conservative. By 2.2.5, Spec_{𝔏}^{0}(X) = ⋃_{i∈J} Spec_{𝔏}^{0}(U_{i}; X), where Spec_{𝔏}^{0}(U_{i}; X) = {Σ ∈ Spec_{𝔏}^{0}(X) | Σ_{u_{i}} ⊆ Σ̂}. The condition (b) means that for every i ∈ J, all Σ ∈ S^{𝔰}M(U_{i}) are C̃_{T} -stable. In particular, Spec_{𝔏}^{0}(U_{i}, γ_{i}^{∗}) = Spec_{𝔏}^{0}(U_{i}) for every i ∈ J. This implies that for every Σ ∈ Spec_{𝔏}^{0}(X), the multiplicative system Σ̂ is C̃_{T} -stable. Thus, the map Σ ⟼ Σ̂ is an embedding Spec_{𝔏}^{0}(X) ⟶ Spec_{𝔏}^{1}(X, γ^{∗}).

12. Spectra of triangulated categories.

12.1. ’Spaces’ represented by triangulated categories. Triangulated categories are triples (C_{X}, γ; 𝔇), where (C_{X}, γ) is an additive ℤ-category, and 𝔇 a family of ’distinguished’ triangles. We shall denote a triangulated category (C_{X}, γ; 𝔇) by CT_{𝔛} and regard it as a triangulated category representing a ’space’ 𝔛.

12.2. Preliminaries: thick subcategories and saturated multiplicative systems of a triangulated category.

12.2.1. Multiplicative systems in triangulated categories. Fix a triangulated category CT_{𝔛} = (C_{X}, γ; 𝔇). A multiplicative system Σ of (X, γ) is said to be compatible with triangulation if for any pair of distinguished triangles (L, M, N, u, v, w) and (L′, M′, N′, u′, v′, w′) and any commutative diagram

L ⟶^{u} M
s ↓  ↓ s′
L′ ⟶^{u′} M′

where s and s′ are elements of Σ, there exists a morphism N ⟶^{t} N′ which belongs to Σ and such that (s, s′, t) is a morphism of triangles.

We shall use same notations: SM(𝔛) (resp. S^{𝔰}M(𝔛)) for the preorder of multiplicative (resp. saturated multiplicative) systems of the triangulated ’space’ 𝔛. The dualization functor 𝔛 ⟼ 𝔛^{o} induces an isomorphism of preorders

SM(𝔛) ⟶^{∼} SM(𝔛^{o}) and S^{𝔰}M(𝔛) ⟶^{∼} S^{𝔰}M(𝔛^{o}).

12.2.2. Triangulated subcategories. Recall that a full subcategory, 𝕋, of the category C_{X} is called a triangulated subcategory if it is stable by translations, and has a triangulated structure such that the inclusion functor 𝕋 ↪ C_{X} is exact.

Let 𝕋 be a full subcategory of C_{X} stable by translations. The subcategory 𝕋 admits a triangulated structure which makes it a triangulated subcategory of CT_{𝔛} iff for any morphism L ⟶^{f} M of 𝕋, there exists a distinguished triangle (L, M, N, f, g, h) such that N ∈ Ob𝕋.

12.2.3. Definitions. (1) A full triangulated subcategory, 𝕋, of CT_{𝔛} is called saturated (in [Ve2]), if every direct summand (in C_{X}) of an object of 𝕋 belongs to 𝕋.

(2) A full triangulated subcategory, 𝕋, of CT_{𝔛} is called thick (in [Ve1] and everywhere else), if every distinguished triangle (L, M, N, u, v, w) such that N ∈ Ob𝕋 and L ⟶^{u} M factors through an object of 𝕋, belongs to 𝕋 (that is L and M are objects of 𝕋).

These two notions are equivalent: A full triangulated subcategory of a triangulated category is thick iff it is saturated.

12.2.4. Triangulated subcategories and multiplicative systems. For any full triangulated subcategory, 𝕋, of the triangulated category CT_{𝔛}, let Σ_{𝕋} denote the family of all morphisms L ⟶^{u} M of CT_{𝔛} such that there exists a distinguished triangle (L, M, N, u, v, w), where N is an object of 𝕋.

12.2.5. Proposition [Ve2, 2.1.8]. For any full triangulated subcategory 𝕋 of a triangulated category CT_{𝔛}, the family Σ_{𝕋} is a multiplicative system. The system Σ_{𝕋} is saturated iff the subcategory 𝕋 is thick.

For any multiplicative system Σ in the triangulated category CT_{𝔛}, let 𝕋_{Σ} denote the full subcategory of CT_{𝔛} generated by objects N contained in a distinguished triangle (L, M, N, u, v, w) such that L ⟶^{u} M belongs to Σ.

12.2.6. Proposition [Ve1, 2.1]. The map Σ ⟼ 𝕋_{Σ} is an isomorphism of the preorder S^{𝔰}M(𝔛) of saturated multiplicative systems of a triangulated category CT_{𝔛} onto the preorder 𝔗𝔥𝔱(𝔛) of thick triangulated subcategories of CT_{𝔛}. The inverse isomorphism is given by the map 𝕋 ⟼ Σ_{𝕋}.

12.2.6.1. Corollary. The intersection of any set of saturated multiplicative systems of a triangulated category is a saturated multiplicative system.

Proof. The assertion follows from an easily checked fact that the intersection of any set of thick triangulated subcategories of a triangulated category is a thick triangulated subcategory.

12.3. The spectrum Spec_{𝔏}^{0}. Fix a triangulated category CT_{𝔛}. We define Spec_{𝔏}^{0}(𝔛) the same way as Spec_{𝔏}^{0}(X) is defined in 2.1: elements of Spec_{𝔏}^{0}(𝔛) are saturated multiplicative systems, Σ, of 𝔛 such that the union, Σ̂, of all Σ′ ∈ S^{𝔰}M(𝔛) which do not contain Σ is a saturated multiplicative system.

Thanks to the isomorphism between the preorder S^{𝔰}M(𝔛) of saturated multiplicative systems and the preorder 𝔗𝔥𝔱(𝔛) of thick triangulated subcategories of CT_{𝔛}, (see 12.2.6.), one can, like in the abelian case, define spectra in terms of thick triangulated subcategories.

We denote by Spec_{𝔗𝔥𝔱}^{0}(𝔛) the preorder (with respect to the inclusion) of all thick triangulated subcategories P of CT_{𝔛} such that there exists the biggest thick triangulated subcategory, P̂ of CT_{𝔛} which does not contain P. The map P ⟼ Σ_{P} (cf. 12.2.4) induces an isomorphism of preorders Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⟶^{∼} Spec_{𝔏}^{0}(𝔛).

12.3.1. Representatives of elements of Spec_{𝔗𝔥𝔱}^{0}(𝔛). For any object M of the category CT_{𝔛}, let [M]_{𝔱} denote the thick envelope of M which is the smallest thick triangulated subcategory of CT_{𝔛} containing M. And let ⟨M⟩_{𝔱} denote the full subcategory of CT_{𝔛} whose objects are N ∈ ObCT_{𝔛} such that M ∉ Ob[N]_{𝔱}.

The following assertion is a triangulated version of 6.1.3.3.

12.3.2. Proposition. (a) For any object M of the category CT_{𝔛}, the subcategory ⟨M⟩_{𝔱} is the union of all thick triangulated subcategories which do not contain M.

(b) The following conditions are equivalent:

(i) [M]_{𝔱} ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔛);

(ii) ⟨M⟩_{𝔱} is a thick triangulated subcategory.

Proof. (a) If 𝕋 ∈ 𝔗𝔥𝔱(𝔛) and M ∉ Ob𝕋, then M ∉ [N]_{𝔱} for every N ∈ Ob𝕋, hence the assertion.

(b) (i) ⇒ (ii). Clearly, [M]_{𝔱} ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔛) iff there exists the biggest thick triangulated subcategory P′ such that M ∉ ObP′. Therefore ⟨M⟩_{𝔱} ⊆ P′. The inverse inclusion, P′ ⊆ ⟨M⟩_{𝔱}, follows from the definition of the subcategory ⟨M⟩_{𝔱}.

(ii) ⇒ (i). Every thick triangulated subcategory which does not contain M is a subcategory of ⟨M⟩_{𝔱}. Therefore, if ⟨M⟩_{𝔱} ∈ 𝔗𝔥𝔱(𝔛), then it is the biggest thick triangulated subcategory which does not contain M.

We denote by Spec_{𝔗𝔥𝔱}^{0}(𝔛) the family of all objects M of the category CT_{𝔛} satisfying the equivalent conditions of 12.3.1(b).

12.4. 𝔏-Local triangulated categories and the associated spectrum.

12.4.1. 𝔏-Local triangulated categories. A triangulated category CT_{𝔛} is 𝔏-local iff the intersection of all nonzero thick triangulated subcategories of CT_{𝔛} is nonzero. In other words, the preorder 𝔗𝔥𝔱_{⋆}(𝔛) of all nonzero thick triangulated subcategories of CT_{𝔛} has the smallest element.

12.4.2. The spectrum Spec_{𝔗𝔥𝔱}^{1}. The elements of Spec_{𝔗𝔥𝔱}^{1}(𝔛) are thick triangulated subcategories, P, of CT_{𝔛} such that the intersection P^{⋆} of all thick triangulated subcategories of CT_{𝔛} properly containing P is not equal to P.

The map P ⟼ Σ_{P} (cf. 12.2.4) induces an isomorphism of preorders

Spec_{𝔗𝔥𝔱}^{1}(𝔛) ⟶^{∼} Spec_{𝔏}^{1}(𝔛).

12.4.3. Proposition. (a) A thick triangulated subcategory P belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) iff the quotient triangulated category CT_{𝔛}/P is 𝔏-local.

(b) There is an embedding Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⟶ Spec_{𝔗𝔥𝔱}^{1}(𝔛) defined by P ⟼ P̂.

Proof. The assertion (a) is obvious. An argument similar to the argument of 3.2 proves (b).

12.5. Functorial properties. The following proposition (which is a part of [Ve2, 2.3.1]) is a convenient reference for the rest of this section.

12.5.1. Proposition. Let B and A be full triangulated subcategories of a triangulated category CT_{𝔛} such that B ⊆ A.

(a) The canonical functor A/B ⟶ CT_{𝔛}/B is fully faithful and injective on objects. The image of this functor is q_{B}^{∗}(A), where q_{B}^{∗} is the canonical functor CT_{𝔛} ⟶ CT_{𝔛}/B.

The subcategory A is thick iff the subcategory q_{B}^{∗}(A) is thick.

(b) The map A ⟼ q_{B}^{∗}(A) is an isomorphism of the preorder of strictly full triangulated subcategories of CT_{𝔛} containing the kernel, B^{𝔱} of the functor q_{B}^{∗} onto the preorder of strictly full triangular subcategories of CT_{𝔛}/B.

(c) The canonical functor CT_{𝔛}/A ⟶ (CT_{𝔛}/B)/(A/B) is an isomorphism of triangulated categories.

12.5.2. Corollary. Let B and A be full triangulated subcategories of a triangulated category CT_{𝔛} such that B ⊆ A. Let B^{𝔱} be the thick envelope of B in CT_{𝔛}. Then B^{𝔱} ∩ A is the thick envelope of B in A.

Proof. Consider the commutative diagram

B^{𝔱} ⟶ CT_{𝔛} ⟶ CT_{𝔛}/B
↑ ↑ ↑
B_{A}^{𝔱} ⟶ A ⟶ A/B

with exact rows. By 12.5.1(a), the functor A/B⟶CT_{𝔛}/B is faithful. Therefore, the kernel, B_{A}^{𝔱}, of the localization functor A ⟶ A/B, which is the thick envelope of B in A, coincides with B^{𝔱} ∩ A.

12.5.3. Proposition. Let CT_{𝔜} be a full triangulated subcategory of a triangulated category CT_{𝔛}.

(a) The localization CT_{𝔛} ⟶ CT_{𝔛}/CT_{𝔜} = CT_{𝔛/𝔜} induces an isomorphism

U_{𝔗𝔥𝔱}(C_{𝔜}) ⋂ Spec_{𝔗𝔥𝔱}^{1}(𝔛) ⟶^{∼} Spec_{𝔗𝔥𝔱}^{1}(𝔛/𝔜)

and an injective morphism

U_{𝔗𝔥𝔱}(C_{𝔜}) ⋂ Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⟶ Spec_{𝔗𝔥𝔱}^{0}(𝔛/𝔜).

Here U_{𝔗𝔥𝔱}(C_{𝔜}) = {𝕋 ∈ 𝔗𝔥𝔱(𝔛) | C_{𝔜} ⊆ 𝕋}.

(b) Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⋂ ObC_{𝔜} ⊆ Spec_{𝔗𝔥𝔱}^{0}(𝔜).

(c) Suppose the triangulated subcategory CT_{𝔜} is thick. Then the map

𝔗𝔥𝔱(𝔛) ⟶ 𝔗𝔥𝔱(𝔜), 𝕋 ⟼ 𝕋 ∩ CT_{𝔜},

induces an injective morphism

Spec_{𝔗𝔥𝔱}^{0}(𝔛) − U_{𝔗𝔥𝔱}(C_{𝔜}) ⟶ Spec_{𝔗𝔥𝔱}^{0}(𝔜).  (1)

Proof. (a) The assertion follows from 12.5.1 (or 4.3.1).

(b) Let M ∈ ObCT_{𝔜}; and let ⟨M⟩_{𝔱} be the union of all thick triangulated subcategories of the triangulated category CT_{𝔛} which do not contain the object M. Then ⟨M⟩_{𝔱} ⋂ CT_{𝔜} is the union, ⟨M⟩_{𝔱}^{𝔜}, of all thick triangulated subcategories of CT_{𝔜} which do not contain M.

In fact, let 𝕋 be a thick triangulated subcategory of CT_{𝔜} which does not contain M. And let 𝕋^{𝔱} be the thick envelope of 𝕋 in CT_{𝔛}. By 12.5.2, 𝕋^{𝔱} ⋂ CT_{𝔜} = 𝕋; in particular, M ∉ Ob𝕋^{𝔱}. The latter means precisely that 𝕋^{𝔱} ⊆ ⟨M⟩_{𝔱}, hence 𝕋 = 𝕋^{𝔱} ⋂ CT_{𝔜} ⊆ ⟨M⟩_{𝔱} ⋂ CT_{𝔜}. This shows that ⟨M⟩_{𝔱} ⋂ CT_{𝔜} is the union of all thick triangulated subcategories of CT_{𝔜} which do not contain M.

If M ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔛), i.e. ⟨M⟩_{𝔱} is a thick triangulated subcategory of CT_{𝔛}, then the intersection ⟨M⟩_{𝔱} ⋂ CT_{𝔜} is a thick triangulated subcategory of CT_{𝔜}. Therefore M ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔜).

(c) Let M ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔛), i.e. [M]_{𝔱} ∈ Spec_{𝔗𝔥𝔱}^{0}(𝔛). Then

[M ∉ ObC_{𝔜}^{𝔱}] ⇔ [C_{𝔜}^{𝔱} ⊆ ⟨M⟩_{𝔱}] ⇔ [C_{𝔜} ⊆ ⟨M⟩_{𝔱}].

Here C_{𝔜}^{𝔱} is the thick envelope of CT_{𝔜} in C_{𝔜}, and ⟨M⟩_{𝔱} is the biggest thick subcategory which does not contain M (cf. 12.3.1). Thus, if the triangulated subcategory C_{𝔜} is thick, then [M ∈ ObC_{𝔜}] ⇔ [[M]_{𝔱} ⊆ ObC_{𝔜}] ⇔ [C_{𝔜} ⊈ ⟨M⟩_{𝔱}]. The assertion follows now from (a).

12.6. Preliminaries on orthogonality. For any subcategory B of CT_{𝔛}, the left orthogonal, ^{⊥}B, of B is the full subcategory of CT_{𝔛} generated by all objects N such that CT_{𝔛}(N, M) = 0 for all M ∈ ObB. The right orthogonal of B is defined dually and is denoted by B^{⊥}. Its objects are B-torsion free objects of CT_{𝔛}. If B is a triangulated subcategory, then B^{⊥} and ^{⊥}B are thick triangulated subcategories.

12.6.1. Proposition [Ve2, 2.3.3]. Let B be a full triangulated subcategory of a

triangulated category CT_{𝔛} and CT_{𝔛} ⟶^{q_{B}^{∗}} CT_{𝔛}/B the canonical localization functor.

(a) For an object M of CT_{𝔛}, the following conditions are equivalent:

(i) The object M is q∗_{B}-free.

(ii) The object M is left closed for Σ_{B}, i.e. CT_{𝔛}(s, M) is an isomorphism for every s ∈ Σ_{B}. Here Σ_{B} is the multiplicative system corresponding to B (cf. 12.2.4).

(iii) Every morphism M ⟶^{s} N with s ∈ Σ_{B} admits a retraction.

(iv) The object M is B-torsion free, that is for every L ∈ ObB, CT_{𝔛}(L, M) = 0.

(v) For every N ∈ ObCT_{𝔛}, the map

CT_{𝔛}(N, M) ⟶ CT_{𝔛}/B(q∗_{B}(N), q∗_{B}(M))

is an isomorphism.

(b) The full subcategory L(q_{B}) of CT_{𝔛} generated by q∗_{B}-free objects is a thick triangulated subcategory.

(c) The composition of the inclusion functor CT_{L(q_{B})} ↪ CT_{𝔛} and the localizaiton

functor CT_{𝔛} ⟶^{q_{B}^{∗}} CT_{𝔛}/B is a fully faithful functor injective on objects.

(d) Let CT_{𝔇(q_{B∗})} be the full subcategory of the quotient triangulated category CT_{𝔛}/B generated by all objects M such that the functor CT_{𝔛}/B(q∗_{B}(−), M) is representable. The subcategory CT_{𝔇(q_{B∗})} is triangulated and strictly full. If infinite coproducts or products exist in CT_{𝔛}, then CT_{𝔇(q_{B∗})} is thick.

(e) The localization functor q∗_{B} induces an equivalence of categories

CT_{L(q_{B})} ⟶ CT_{𝔇(q_{B∗})}.

(f) An object N of CT_{𝔛} belongs to the preimage, CT_{ℜ(q_{B})} = q∗^{−1}_{B}(CT_{𝔇(q_{B∗})}), of the subcategory CT_{𝔇(q_{B∗})} iff there exists a morphism N ⟶^{s} M such that M is q∗_{B}-free and q∗_{B}(s) is invertible.

(g) The inclusion functor CT_{L(q_{B})} ⟶ CT_{ℜ(q_{B})} has a left adjoint.

Proof. The equivalence of (i), (ii), (iii), and (v) follow from 5.3.3.2, as well as the assertions (c), (e), (f), and (g). The proof of the remaining statements can be found in [Ve2, 2.3.3].

12.6.2. Corollary [Ve1, 6–3]. Let 𝕋 be a thick triangulated subcategory of the triangulated category CT_{𝔛}. The full subcategory of CT_{𝔛} generated by objects which are left closed for Σ_{𝕋}, is the right orthogonal, 𝕋^{⊥}, of the subcategory 𝕋.

Proof. The fact follows from the equivalence of (ii) and (iv) in 12.6.1.

12.6.3. Proposition [Ve1, 6–5]. Let 𝕋 be a thick triangulated subcategory of CT_{𝔛}, and let

𝕋 ⟶^{ι_{𝕋}^{∗}} CT_{𝔛} ⟶^{q_{𝕋}^{∗}} CT_{𝔛}/𝕋

be the inclusion and localization functors. The following properties are equivalent: (a) The functor ι_{𝕋}^{∗} has a right adjoint.

(b) The functor q_{𝕋}^{∗} has a right adjoint.

12.7. The structure of Spec_{𝔗𝔥𝔱}^{1}(𝔛).

12.7.1. Proposition. Let P be a thick triangulated subcategory of CT_{𝔛} such that the intersection P_{0} = P^{⊥} ∩ P^{⋆} is nonzero. Then

(a) The subcategory P is an object of Spec_{𝔗𝔥𝔱}^{1}(𝔛) and P = ^{⊥}P_{0}.

(b) The triangulated category P^{⊥} is 𝔏-local, and P_{0} is its smallest nonzero thick triangulated subcategory.

Proof. (a) The condition P_{0} ≠ 0 implies, obviously, that P^{⋆} contains P properly, i.e. P is an object of Spec_{𝔗𝔥𝔱}^{1}(𝔛)

The inclusion P_{0} ⊆ P^{⊥} is equivalent to the inclusion P ⊆^{⊥} P_{0}. If the (thick triangulated) subcategory ^{⊥}P_{0} contains P properly, then ^{⊥}P_{0}/P is a nonzero thick triangulated subcategory of CT_{𝔛}/P, hence it contains the image, P̃_{0}, of the subcategory P_{0}. This means that for every L ∈ ObP_{0}, there exists an object, M, of ^{⊥}P_{0} and an isomorphism q_{P}^{∗}(M) ⟶^{∼} q_{P}^{∗}(L). The latter is determined by a diagram M ⟶^{s′} K ⟵^{s} L whose both arrows belong to Σ_{P}. Since L is an object of P^{⊥}, it follows from the equivalence of (iii) and

(iv) in 12.6.1 that the morphism K ⟵^{s} L admits a retraction, K ⟶^{s′′} L. Let M ⟶^{t} L be the composition s′′ ∘ s′. Since the morphism t belongs to Σ_{P}, there exists a distinguished triangle M ⟶^{t} L ⟶ N such that N ∈ ObP. In particular, N ∈ Ob^{⊥}P_{0}. Thus, we have a distinguished triangle, M ⟶^{t} L ⟶ N, such that M and N are objects of the thick subcategory ^{⊥}P_{0}. Therefore, L is an object of ^{⊥}P_{0}, which cannot happen, unless L = 0. Thus, ^{⊥}P_{0} cannot contain P properly, i.e. P = ^{⊥}P_{0}.

(b) Let 𝕋 be a nonzero thick triangulated subcategory of P^{⊥}. Then the image, q_{P}^{∗}(𝕋), in the quotient category CT_{𝔛}/P is nonzero, hence its thick envelope contains the subcategory P^{⋆}/P. In particular, it contains the image of the subcategory P_{0}. Since objects of the thick envelope of q_{P}^{∗}(𝕋) are direct summands of objects of q_{P}^{∗}(𝕋), this means that for every object L of P_{0}, there exists an object M of 𝕋 such that q_{P}^{∗}(L) is a direct summand of q_{P}^{∗}(M). Since both objects, L and M, belong to the subcategory P^{⊥} and, by 12.6.1(c) (and 12.6.1(i) ⇔ (iv)), the restriction of the localization functor q_{P}^{∗} to the subcategory P^{⊥} is a fully faithful functor, it follows that L is a direct summand of M. Since 𝕋 is a thick subcategory of CT_{𝔛}, it contains all direct summands of its objects. Thus P_{0} ⊆ 𝕋.

12.7.2. Corollary. Let P be a thick triangulated subcategory of CT_{𝔛} such that the intersection P_{0} = P^{⊥} ∩ P^{⋆} is nonzero. Then P is closed under all colimits which exist in CT_{𝔛}, in particular, P is closed under all coproducts which exist in CT_{𝔛}.

Proof. In fact, by 12.7.1, P = ^{⊥}P_{0}; and the left orthogonal to any subcategory is closed under arbitrary colimits which exist in CT_{𝔛}.

12.7.3. Proposition. Suppose that infinite coproducts or products exist in CT_{𝔛}. Let P be a thick triangulated subcategory of CT_{𝔛}. Then the following properties of are equivalent:

(i) P_{0} = P^{⊥} ∩ P^{⋆} is nonzero;

(ii) P belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) and the composition of the inclusion P_{0} ↪ CT_{𝔛} and

the localization functor CT_{𝔛} ⟶^{q_{P}^{∗}} CT_{𝔛}/P induces an equivalence of triangulated categories

P_{0} ⟶^{∼} P^{⋆}/P.

(iii) P belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) and the inclusion functor P ↪ P^{⋆} has a right adjoint. (iv) P belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) and P^{⊥} is nonzero.

Proof. The implications (iii) ⇐ (ii) ⇒ (i) ⇒ (iv) ⇐ (iii) hold by obvious reasons (see 12.7.1(a)). The implication (iii) ⇒ (ii) follows from 12.6.3 (see also 12.6.2). Thus, (iii) ⇔ (ii) ⇒ (i) ⇒ (iv) without any additional hypothesis on CT_{𝔛}. The existence of infinite coproducts or products is needed for the implication

(iv) ⇒ (ii). Fix an object P of Spec_{𝔗𝔥𝔱}^{1}(𝔛). By 12.6.1(e) (see also 12.6.1(i) ⇔ (iv)), the composition of the localization functor q_{P}^{∗} with the inclusion functor P^{⊥} ↪ CT_{𝔛} induces an equivalence of the triangulated categories P^{⊥} ⟶ CT_{𝔇(q_{P∗})}. Here CT_{𝔇(q_{P∗})} is the full subcategory of the quotient category CT_{𝔛}/P generated by all objects M such that the functor CT_{𝔛}/P(q_{P}^{∗}(−), M) is representable. By 12.6.1(d), if infinite coproducts, or infinite products exist in CT_{𝔛}, then CT_{𝔇(q_{P∗})} is a thick triangulated subcategory of the 𝔏-local triangulated category CT_{𝔛}/P. If P^{⊥} ≠ 0, then (and only then) the subcategory CT_{𝔇(q_{P∗})} is nonzero, hence it contains the (smallest non-trivial thick) subcategory P^{⋆}/P which implies that CT_{𝔇(q_{P∗})} is an 𝔏-local triangulated category having P^{⋆}/P as the smallest nonzero thick subcategory. This, in turn, implies that P_{0} = P^{⊥} ∩ P^{⋆} is nonzero and, moreover, the localization q_{P}^{∗} induces an equivalence between P_{0} and P^{⋆}/P.

12.7.4. Decomposition of Spec_{𝔗𝔥𝔱}^{1}(𝔛). Fix a triangulated category CT_{𝔛}. Let Spec_{𝔗𝔥𝔱}^{1,0}(𝔛) (resp. Spec_{𝔗𝔥𝔱}^{1,1}(𝔛)) denote the full subcategory of Spec_{𝔗𝔥𝔱}^{1}(𝔛) generated by P such that P_{0} = P^{⋆} ∩ P^{⊥} = 0 (resp. P_{0} ≠ 0). Thus, we have the decomposition

Spec_{𝔗𝔥𝔱}^{1}(𝔛) = Spec_{𝔗𝔥𝔱}^{1,0}(𝔛) ⨆ Spec_{𝔗𝔥𝔱}^{1,1}(𝔛).

If ’small’ coproducts or products exist in CT_{𝔛}, then, by 12.7.3(iv), the subcategory Spec_{𝔗𝔥𝔱}^{1,0}(𝔛) is generated by all thick triangulated subcategories P such that P ≠ P^{⋆} and P^{⊥} = 0. The latter equality means, by definition, that P generates the category CT_{𝔛}. Thus, Spec_{𝔗𝔥𝔱}^{1,1}(𝔛) might be regarded as the ’proper’ part of Spec_{𝔗𝔥𝔱}^{1}(𝔛).

In the case of an arbitrary triangulated category CT_{𝔛}, objects of Spec_{𝔗𝔥𝔱}^{1,0}(𝔛) are P such that P generates P^{⋆} (and P ≠ P^{⋆}).

12.8. The intermediate spectrum Spec_{𝔗𝔥𝔱}^{1/2}(𝔛). Let Spec_{𝔗𝔥𝔱}^{1/2}(𝔛) denote the full subpreorder of 𝔗𝔥𝔱(𝔛) whose objects are thick triangulated subcategories Q such that ⊥Q belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) and every thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}Q contains Q; i.e. ^{⊥}Q ∨ Q is the smallest thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}Q.

12.8.1. Proposition. (a) The map Q ⟼ ^{⊥}Q induces an isomorphism of preorders

Spec_{𝔗𝔥𝔱}^{1/2}(𝔛) ⟶^{∼} Spec_{𝔗𝔥𝔱}^{1,1}(𝔛).

(b) If Q is an object of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛), then Q is a minimal nonzero thick triangulated subcategory of CT_{𝔛}.

(c) Suppose that CT_{𝔛} has infinite coproducts or products. Then the following properties of a thick triangulated subcategory Q are equivalent:

(i) Q belongs to Spec_{𝔗𝔥𝔱}^{1/2}(𝔛);

(ii) Q is a minimal nonzero thick triangulated subcategory of CT_{𝔛} such that ^{⊥}Q belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛).

Proof. (a) Let Q be an object of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛). This means that ^{⊥}Q belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛) and (^{⊥}Q)^{⋆} = ^{⊥}Q ∨ Q. Since Q is contained in the intersection

Q_{1} = (^{⊥}Q)^{⊥} ∩ (^{⊥}Q)^{⋆} = (^{⊥}Q)^{⊥} ∩ (^{⊥}Q ∨ Q)

and Q ≠ 0, the subcategory ^{⊥}Q belongs to Spec_{𝔗𝔥𝔱}^{1,1}(𝔛).

By 12.7.1(b), the triangulated category (^{⊥}Q)^{⊥} is 𝔏-local and Q_{1} is its smallest nonzero thick triangulated subcategory. Therefore, Q_{1} = Q. Thus, the composition of the map

Spec_{𝔗𝔥𝔱}^{1/2}(𝔛) ⟶ Spec_{𝔗𝔥𝔱}^{1,1}(𝔛), Q ⟼ ^{⊥}Q,  (1)

with the map P ⟼ P_{0} = P^{⊥} ∩ P^{⋆} is identical. It follows from 12.7.1 that the latter map is a morphism of preorders

Spec_{𝔗𝔥𝔱}^{1,1}(𝔛) ⟶ Spec_{𝔗𝔥𝔱}^{1/2}(𝔛).  (2)

The argument above shows that the map (2) is inverse to the map (1).

(b) If Q is an object of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛), then, by (a), Q is the smallest thick triangulated subcategory of the 𝔏-local category (^{⊥}Q)^{⊥}; in particular, Q is a minimal nonzero thick triangulated subcategory of CT_{𝔛}.

(c) Suppose that CT_{𝔛} has infinite coproducts or products. Let Q be a thick triangulated subcategory of CT_{𝔛} such that ^{⊥}Q belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛). Then (^{⊥}Q)^{⊥} contains a nonzero subcategory Q, hence it is nonzero. By 12.7.3(iv), this is equivalent to that Q_{1} = (^{⊥}Q)^{⊥} ∩ (^{⊥}Q)^{⋆} is nonzero. By 12.7.1(b), Q_{1} is the smallest thick triangulated subcategory of the 𝔏-local triangulated category (^{⊥}Q)^{⊥}. In particular, Q_{1} ⊆ Q. If Q is a minimal nonzero thick triangulated subcategory of CT_{𝔛}, then the inclusion Q_{1} ⊆ Q implies that Q coincides with Q_{1}. The assertion follows now from (a).

12.8.2. Corollary. (a) If Q is an object of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛), then Q = [M]_{𝔱} for any nonzero object M of Q.

(b) The following properties of an object M of the category CT_{𝔛} are equivalent: (i) The thick envelope, [M]_{𝔱}, of M belongs to Spec_{𝔗𝔥𝔱}^{1/2}(𝔛). (ii) ^{⊥}M belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛), and if 𝕋 is a thick triangulated subcategory of CT_{𝔛}

properly containing ^{⊥}M, then M ∈ Ob𝕋.

(iii) [M]_{𝔱} is a minimal nonzero thick subcategory, and ^{⊥}M belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛).

(c) The equivalent conditions (i), (ii), or (iii) imply the following property:

(iv) ^{⊥}M belongs to Spec_{𝔗𝔥𝔱}^{1}(𝔛), and every nonzero thick triangulated subcategory of (^{⊥}M)^{⊥} contains M.

(d) If infinite coproducts or products exist in CT_{𝔛}, then (iv) is equivalent to the properties (i), (ii), and (iii).

Proof. (a) The assertion follows from the minimality of Q (see 12.8.1(b)).

(b) (i) ⇒ (ii). Let Q be an object of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛); and let M be a nonzero object of Q. Since Q = [M]_{𝔱} and ^{⊥}M = ^{⊥}[M]_{𝔱}, the subcategory ^{⊥}M coincides with ^{⊥}Q. Since Q belongs to Spec_{𝔗𝔥𝔱}^{1/2}(𝔛),^{⊥}Q∨Q is the smallest thick subcategory of CT_{𝔛} properly containing ⊥Q. It contains the object M.

(ii) ⇒ (i). The conditions (ii) mean that ^{⊥}M ∨ [M]_{𝔱} is the smallest thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}M = ^{⊥}[M]_{𝔱}.

The implications (ii) ⇔ (iii) follow from 12.5.1.

(c) (iii) ⇒ (iv). Let Q = [M]_{𝔱}. It follows from 12.5.1(ii) that (^{⊥}M)^{⊥} = (^{⊥}Q)^{⊥} is an 𝔏-local triangulated category and Q = [M]_{𝔱} is its smallest nonzero thick subcategory. Clearly M ∈ ObQ.

(d) The implication (iv) ⇒ (iii) follows from 12.5.1(c).

12.9. The structure of Spec_{𝔗𝔥𝔱}^{0}(𝔛). The decomposition

Spec_{𝔗𝔥𝔱}^{1}(𝔛) = Spec_{𝔗𝔥𝔱}^{1,0}(𝔛) ⨆ Spec_{𝔗𝔥𝔱}^{1,1}(𝔛).  (1)

induces a decomposition

Spec_{𝔗𝔥𝔱}^{0}(𝔛) = Spec_{𝔗𝔥𝔱}^{0,0}(𝔛) ⨆ Spec_{𝔗𝔥𝔱}^{0,1}(𝔛).  (2)

Here Spec_{𝔗𝔥𝔱}^{0,i}(𝔛) is the full subcategory of Spec_{𝔗𝔥𝔱}^{0}(𝔛) whose objects are all thick triangulated subcategories P such that P̂ belongs to Spec_{𝔗𝔥𝔱}^{1,i}(𝔛), i = 0, 1.

12.9.1. Proposition. Let P be an object of Spec_{𝔗𝔥𝔱}^{0}(𝔛). The following conditions are equivalent:

(i) P belongs to Spec_{𝔗𝔥𝔱}^{0,1}(𝔛).

(ii) P = P̂^{⊥} ∩ P̂^{⋆} = P̂^{⊥} ∩ (P̂ ∨ P).

(iii) P̂ = ^{⊥}P, i.e. if 𝕋 is a thick triangulated subcategory of CT_{𝔛} such that P ⊈ 𝕋, then 𝕋 ⊆ ^{⊥}P.

Proof. (i) ⇒ (ii). The condition (i) means that P̂ is a thick triangulated subcategory and P̂_{0} = P̂^{⊥}∩P̂^{⋆} is nonzero. In particular, P̂_{0} ⊈ P̂. By the definition of P̂ this implies the inclusion P ⊆ P̂_{0}. By 12.7.1(b), P̂_{0} is the smallest nonzero thick triangulated subcategory of P̂^{⊥}. Therefore, P = P̂_{0}.

(ii) ⇒ (iii). By 12.7.1(a), P̂ = ^{⊥}P̂_{0}. Therefore, since by (ii) P = P̂_{0}, we obtain the equality P̂ = ^{⊥}P.

(iii) ⇒ (i). Suppose P̂ = ^{⊥}P. Then

P̂_{0} = P̂^{⊥} ∩ P̂^{⋆} = (^{⊥}P)^{⊥} ∩ (P̂ ∨ P) = (^{⊥}P)^{⊥} ∩ (^{⊥}P ∨ P).

Notice that (^{⊥}P)^{⊥} ⊇ P ⊆ (^{⊥}P ∨ P), hence P ⊆ (^{⊥}P)^{⊥} ∩ (^{⊥}P ∨ P) ⊆ P̂_{0}. In particular, P̂_{0} ≠ 0 which means, by definition, that P belongs to Spec_{𝔗𝔥𝔱}^{0,1}(𝔛).

12.9.2. Note. It follows from 12.9.1 that

Spec_{𝔗𝔥𝔱}^{0,1}(𝔛) = Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⋂ Spec_{𝔗𝔥𝔱}^{1/2}(𝔛).

12.10. Flat spectra. Let 𝔖𝔢(𝔛) denote the family of all thick triangulated subcategories of the triangulated category CT_{𝔛} which satisfy equivalent conditions of 12.6.3. We define the complete flat spectrum of 𝔛, Spec_{𝔣𝔏}^{1}(𝔛), by setting

Spec_{𝔣𝔏}^{1}(𝔛) = Spec_{𝔗𝔥𝔱}^{1}(𝔛) ⋂ 𝔖𝔢(𝔛).  (1)

We define the flat spectrum of 𝔛 as a full subpreorder, Spec_{𝔣𝔏}^{0}(𝔛), of 𝔗𝔥𝔱(𝔛) whose objects are all P such that P̂ ∈ Spec_{𝔣𝔏}^{1}(𝔛)}.

It follows from these definitions that the map P ⟼ P̂ defines an injective morphism

Spec_{𝔣𝔏}^{0}(𝔛) ⟶ Spec_{𝔣𝔏}^{1}(𝔛).  (2)

Let Spec_{𝔣𝔏}^{1/2}(𝔛) denote the full subpreorder of Spec_{𝔗𝔥𝔱}^{1/2}(𝔛) whose objects are all Q such that ^{⊥}Q belongs to 𝔖𝔢(𝔛).

12.10.1. Proposition. (a) The map

𝔗𝔥𝔱(𝔛) ⟶ 𝔗𝔥𝔱(𝔛), Q ⟼ ^{⊥}Q,

induces an isomorphism

Spec_{𝔣𝔏}^{1/2}(𝔛) ⟶^{∼} Spec_{𝔣𝔏}^{1}(𝔛).  (3)

(b) Spec_{𝔣𝔏}^{0}(𝔛) = Spec_{𝔗𝔥𝔱}^{0}(𝔛) ⋂ Spec_{𝔣𝔏}^{1/2}(𝔛). The canonical morphism (2) is the com-

position of the inclusion Spec_{𝔣𝔏}^{0}(𝔛) ↪ Spec_{𝔣𝔏}^{1/2}(𝔛) and the isomorphism (3).

Proof. Notice that Spec_{𝔣𝔏}^{1}(𝔛) ⊆ Spec_{𝔗𝔥𝔱}^{1,1}(𝔛), hence Spec_{𝔣𝔏}^{0}(𝔛) ⊆ Spec_{𝔗𝔥𝔱}^{0,1}(𝔛). This follows from 12.6.3 and the definitions of these spectra. Now the assertion (a) becomes a consequence of 12.8.1, and (b) follows from 12.9.1 and 12.9.2.

12.10.2. Proposition. (a) If Q is an object of Spec_{𝔣𝔏}^{1/2}(𝔛), then Q = [M]_{𝔱} (hence ⊥Q = ^{⊥}M) for any nonzero object M of Q.

(b) The following properties of an object M of the category CT_{𝔛} are equivalent: (i) The thick envelope, [M]_{𝔱}, of M belongs to Spec_{𝔣𝔏}^{1/2}(𝔛).

(ii) ^{⊥}M belongs to Spec_{𝔣𝔏}^{1}(𝔛), and if 𝕋 is a thick triangulated subcategory of CT_{𝔛} properly containing ^{⊥}M, then M ∈ Ob𝕋.

(iii) [M]_{𝔱} is a minimal nonzero thick subcategory, and ^{⊥}M belongs to Spec_{𝔣𝔏}^{1}(𝔛). (iv) M is a nonzero object which belongs to every nonzero thick triangulated sub-

category of (^{⊥}M)^{⊥} and such that the inclusion functor ^{⊥}M ⟶ CT_{𝔛} has a right adjoint.

Proof. (a) The assertion is a consequence of the minimality of Q (see 12.8.1(b)).

(b) The implications (i) ⇔ (ii) ⇔ (iii) ⇒ (iv) follow from the corresponding implications of 12.8.2.

(iv) ⇒ (iii). By 2.6.3, the inclusion functor ^{⊥}M ⟶ CT_{𝔛} has a right adjoint iff the localization functor CT_{𝔛} ⟶ CT_{𝔛}/^{⊥}M has a right adjoint. The latter implies that the quotient category CT_{𝔛}/^{⊥}M is equivalent to the triangulated category (^{⊥}M)^{⊥}. The condition that M is contained in every nonzero thick triangulated subcategory of (^{⊥}M)^{⊥} means that (^{⊥}M)^{⊥} is 𝔏-local and [M]_{𝔱} is its smallest thick triangulated subcategory. Therefore, ^{⊥}M belongs to Spec_{𝔣𝔏}^{1}(𝔛), and [M]_{𝔱} is a minimal nonzero thick triangulated subcategory of CT_{𝔛}.

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