Classification of Categorical

Subspaces of Locally Noetherian Schemes

Ryo Kanda^{1}

Received: January 8, 2015 Revised: October 30, 2015

Communicated by Stefan Schwede

Abstract. We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. In order to give the classification, we introduce the notion of a local filter of subobjects of the structure sheaf. The essential part of the argument is given as results on a Grothendieck category with certain properties. We also classify the localizing subcategories, the closed subcategories, and the bilocalizing subcategories in terms of filters.

2010 Mathematics Subject Classification: 18F20 (Primary), 18E15, 16D90, 14A22, 13C05 (Secondary) Keywords and Phrases: Locally noetherian scheme; prelocalizing subcategory; localizing subcategory; closed subcategory; local filter

Contents

1. Introduction 1404 2. Acknowledgement 1408 3. Atom spectrum 1408 4. Subcategories and quotient categories 1413 5. Atom spectra of quotient categories and localization 1422 6. Grothendieck categories with enough atoms 1426 7. The atom spectra of locally noetherian schemes 1430 8. Localization of prelocalizing subcategories and localizing

subcategories 1435 9. Classification of prelocalizing subcategories 1444 10. Classification of localizing subcategories 1451 11. Classification of closed subcategories 1455 12. Classification of bilocalizing subcategories 1460 References 1464

1The author is a Research Fellow of Japan Society for the Promotion of Science. This work is supported by Grant-in-Aid for JSPS Fellows 25·249.

1. Introduction

Gabriel [Gab62] introduced a classification theory of subcategories of the category of modules over a ring. The theory relates several classes of subcategories to collections of ideals, and it reveals that geometry of the prime spectrum of a commutative ring is reflected in the structure of subcategories of modules. In this paper, we extend Gabriel’s result (Theorem 1.2) to an arbitrary locally noetherian scheme X and give a systematic classification of subcategories of the category QCoh X of quasi-coherent sheaves on X. We deal with the following classes of subcategories.

Definition 1.1. Let A be a Grothendieck category.

(1) A prelocalizing subcategory X of A is a full subcategory of A closed under subobjects, quotient objects, and arbitrary direct sums.

(2) A localizing subcategory of A is a prelocalizing subcategory of A closed under extensions.

(3) A closed subcategory of A is a prelocalizing subcategory of A closed under arbitrary direct products.

(4) A bilocalizing subcategory of A is a prelocalizing subcategory of A which is both localizing and closed.

It is known that a full subcategory X of a Grothendieck category A is prelocalizing (resp. closed) if and only if X is closed under subobjects and quotient objects, and the inclusion functor X → A has a right adjoint (resp. both a right and a left adjoint). See Proposition 4.3 and Proposition 11.2. The notion of closed subcategories can be regarded as the categorical reformulation of closed subschemes of a given scheme. In fact, for every ring Λ, Rosenberg [Ros95] showed that there is a bijection

{ two-sided ideals of Λ } → { closed subcategories of Mod Λ }

given by I ↦ { M ∈ Mod Λ | MI = 0 }. It has been shown that the analogous results hold for every noetherian scheme with an ample line bundle ([Smi02, Theorem 4.1]) and for every separated scheme ([Bra14, Proposition 3.18]). One of the aims of this paper is to classify the closed subcategories of QCoh X for a locally noetherian scheme X. In more generality, we classify the prelocalizing subcategories of QCoh X by giving an analog of the following famous theorem by Gabriel [Gab62].

Theorem 1.2 ([Gab62, Lemma V.2.1]; Theorem 9.3). Let Λ be a ring. There is a bijection

{ prelocalizing subcategories of Mod Λ }

→ { prelocalizing filters of right ideals of Λ }

given by Y ↦ { L ⊂ Λ in Mod Λ | Λ/L ∈ Y }.

Note that the prelocalizing filters of right ideals of Λ bijectively correspond to the right linear topologies on Λ (see [Ste75, section VI.4]).

For a locally noetherian scheme X, there exist too many filters of quasi-coherent subsheaves of O_{X} compared with the prelocalizing subcategories of QCoh X. Hence we need to consider a suitable class of filters, which we call local filters (Definition 9.5). By using local filters, we obtain a classification of prelocalizing subcategories, and as a consequence, we deduce classifications of localizing subcategories, closed subcategories, and bilocalizing subcategories.

Theorem 1.3 (Theorem 9.14, Corollary 10.9, Theorem 11.9, Corollary 12.7, Theorem 11.11, and Corollary 12.11). Let X be a locally noetherian scheme. There is a bijection

{ prelocalizing subcategories of QCoh X }

→ { local filters of quasi-coherent subsheaves of O_{X} }

given by

Y ↦ \{ I ⊂ O_{X} in QCoh X | \frac{O_{X}}{I} ∈ Y \}.

This bijection restricts to bijections

{ localizing subcategories of QCoh X }

→ \{ \begin{matrix}local filters of quasi-coherent subsheaves of O_{X} \\ closed under products \end{matrix} \},

{ closed subcategories of QCoh X }

→ { principal filters of quasi-coherent subsheaves of O_{X} },

and

{ bilocalizing subcategories of QCoh X }

→ \{ \begin{matrix}principal filters of quasi-coherent subsheaves of O_{X} \\ closed under products \end{matrix} \}.

In particular, there exists a bijection between the closed subcategories of QCoh X and the closed subschemes of X, and it restricts to a bijection between the bilocalizing subcategories of QCoh X and the subsets of X which are open and closed.

The key of the proof of Theorem 1.3 is to reduce the problem to open affine subschemes, and this part is in fact a consequence of the general theory of Grothendieck categories (Theorem 8.11). The notion of atom spectrum plays a crucial role in this process, and it clarifies the essential properties of the Grothendieck category QCoh X. The atom spectrum ASpec A of a Grothendieck category A is the set of atoms in A which were introduced by Storrer [Sto72] (Definition 3.6). It is regarded as the collection of structural elements of the Grothendieck category in our previous studies [Kan12, Kan15b, Kan15a]. An atom is a generalization of a prime ideal of a commutative ring. Indeed, for every commutative ring R, there exists a canonical bijection between ASpec(Mod R) and Spec R (Proposition 3.7). For a locally noetherian scheme X, it is shown in this paper that there exists a canonical bijection between ASpec(QCoh X) and the underlying space of X (Theorem 7.6). Several fundamental notions of commutative rings and locally noetherian schemes are generalized to Grothendieck categories in terms of atom spectrum as summarized in Table 1. In this paper, we also generalize another kind of Gabriel’s classification of localizing subcategories. Gabriel [Gab62] showed that for a noetherian scheme X, the localizing subcategories of QCoh X bijectively correspond to the specialization-closed subsets of the underlying space of X ([Gab62, Proposition VI.2.4 (b)]). This result has been generalized by a number of authors. (For example, [Hov01], [Kra08], [GP08a], [GP08b], [Tak08], [Tak09], [Her97], [Kra97], [Kan12], and [Kan15a] to some abelian categories. See [GP08a] or [Tak09] for generalizations to derived categories.) By combining the theory of atom spectrum and the description of the atom spectrum of QCoh X for a locally noetherian scheme X (Theorem 7.6), we obtain the following result.

Theorem 1.4 (Theorem 7.8). Let X be a locally noetherian scheme. There is a bijection

{ localizing subcategories of QCoh X } → { specialization-closed subsets of X }

given by X ↦ Supp X. Its inverse is given by Φ ↦ Supp^{−1} Φ.

This paper is organized as follows. In section 3, we recall the definition of the atom spectrum and fundamental notions and results on it. Section 4 is devoted to preliminary results on subcategories and quotient categories by localizing subcategories. In section 5, we summarize results on the atom spectrum and the localization at an atom. In section 6, we introduce the class of Grothendieck categories with enough atoms and show that the localizing subcategories are classified in terms of the atom spectrum for a Grothendieck category with enough atoms (Theorem 6.8). In section 7, we describe the atom spectrum of the Grothendieck category QCoh X for a locally noetherian scheme X and show that QCoh X has enough atoms (Theorem 7.6). In section 8, we investigate a Grothendieck category A with some properties and relate the prelocalizing subcategories (resp. localizing subcategories) of A with the prelocalizing subcategories (resp. localizing subcategories) of quotient categories of A. For a locally noetherian scheme X, the prelocalizing subcategories, the localizing subcategories, the closed subcategories, and the bilocalizing subcategories of QCoh X are classified in section 9, section 10, section 11, and section 12, respectively.

Remark 1.5. In this paper, we use the words “prelocalizing”, “localizing”, and “bilocalizing” subcategories in the same way as in [Pop73]. Some authors use different terminology on these subcategories and also on “closed” subcategories, which are summarized below. Note that we always work inside a Grothendieck category.

Classification of Categorical … 1407

Table 1. Corresponding notions on ASpec A, Spec R, and X

Grothendieck category A  Commutative ring R  Locally noetherian scheme X
Atom spectrum ASpec A  Prime spectrum Spec R  Underlying space |X|
Atom α in A  Prime ideal 𝔭 of R  Point x ∈ X
Associated atoms AAss M  Associated primes Ass M  Associated points Ass M
Atom support ASupp M  Support Supp M  Support Supp M
Open subsets of ASpec A Specialization-closed subsets of Spec R Specialization-closed subsets of X
\overline{\{α\}} for α ∈ ASpec A  \{ 𝔮 ∈ Spec R | 𝔮 ⊂ 𝔭 \} for 𝔭 ∈ Spec R  \{ y ∈ X | x ∈ \overline{\{y\}} \} for x ∈ X
α_{1} ≤ α_{2}  𝔭_{1} ⊂ 𝔭_{2}  \overline{\{x_{1}\}} ∋ x_{2}
Maximal atoms in A  Maximal ideals of R  Closed points in X
Open points in ASpec A  Maximal ideals of R  Closed points in X
Minimal atoms in A  Minimal prime ideals of R  Points in X of height 0

(=Closed points in ASpec A)

Generic point in ASpec A  Unique maximal ideal of R  Unique closed point in X
Injective envelope E(α)  Injective envelope E(R/𝔭)  j_{x}_{∗}E(x)
Residue field k(α)  Residue field k(𝔭)  Residue field k(x)
Atomic object H(α)  Residue field k(𝔭)  j_{x}_{∗}k(x)
Localization A_{α}  Mod R_{𝔭}  Mod O_{X,x}

Documenta Mathematica 20 (2015) 1403–1465

(1) Prelocalizing subcategories are often called weakly closed subcategories. This terminology was introduced by Van den Bergh [VdB01], and followed by [Smi02] and [Pap02], for example. Closed subcategories in [VdB01] were defined in the same way as we do.

(2) Van den Bergh used different terminology in the preprint version [VdB98]. Weakly closed (resp. closed) subcategories in the published version [VdB01] were called closed (resp. biclosed) subcategories in [VdB98]. This fits into Gabriel’s terminology [Gab62].

(3) In the context of torsion theory, such as in [Ste75, Chapter VI], prelocalizing subcategories, localizing subcategories, and bilocalizing subcategories in this paper are called hereditary pretorsion class, hereditary torsion class, and TTF-class (TTF indicates “torsion torsionfree”), respectively.

(4) In [Bra14], our prelocalizing subcategories (resp. closed subcategories) are called topologizing subcategories (resp. reflective topologizing subcategories). This preprint is aimed at modifying a theory of Rosenberg [Ros98], and the definition of topologizing subcategories was also changed. In Rosenberg’s paper [Ros98], our prelocalizing subcategories (resp. closed subcategories) are called coreflective topologizing subcategories (resp. reflective topologizing subcategories), and they are also called closed subcategories (resp. left closed subcategories) in [Ros95].

Conventions 1.6. Throughout this paper, we fix a Grothendieck universe. A set is called small if it is an element of the universe. For every category C, the collection Ob C (resp. Mor C) of objects (resp. morphisms) in C is a set, and Hom_{C}(X, Y) is supposed to be small for all objects X and Y in C. A category C is called skeletally small if the set of isomorphism classes of objects in C is in bijection with a small set. The index set of each limit and colimit is assumed to be skeletally small. Rings, modules over rings, schemes, and sheaves on schemes are assumed to be small. Every ring is associative and has an identity element.

2. Acknowledgement

The author would like to express his deep gratitude to Osamu Iyama for his elaborate guidance. The author thanks Mitsuyasu Hashimoto, S. Paul Smith, and Ryo Takahashi for their valuable comments.

3. Atom spectrum

In this section, we recall the definition of the atom spectrum of a Grothendieck category and fundamental results. We start with the definition of a Grothendieck category.

Definition 3.1.

(1) An abelian category A is called a Grothendieck category if it satisfies the following conditions.

(a) A admits arbitrary direct sums (and hence arbitrary direct limits), and for every direct system of short exact sequences in A, its direct limit is also a short exact sequence.

(b) A has a generator G, that is, every object in A is isomorphic to a

quotient object of the direct sum of some copies of G.

(2) A Grothendieck category is called locally noetherian if it admits a small generating set consisting of noetherian objects.

The exactness of direct limits has the following characterizations.

Proposition 3.2. Let A be an abelian category with arbitrary direct sums. Then the following assertions are equivalent.

(1) For every direct system of short exact sequences in A, its direct limit is also a short exact sequence.

(2) Let M be an object in A. For each subobject L of M and each family N of subobjects of M such that every finite subfamily of N has an upper bound in N, we have

L ∩ ∑_{N∈N} N = ∑_{N∈N}(L ∩ N).

(3) For every family {M_{λ}}_{λ∈Λ} of objects in A and every subobject L of ⨁_{λ∈Λ} M_{λ},

L = ∑_{Λ′∈S}(L ∩ ⨁_{λ∈Λ′} M_{λ}),

where S is the set of finite subsets of Λ.

Proof. [Pop73, Theorem 2.8.6]. □

From now on, we deal with a Grothendieck category A. The atom spectrum of a Grothendieck category is defined by using monoform objects defined as follows.

Definition 3.3.

(1) A nonzero object H in A is called monoform if for each nonzero subobject L of H, no nonzero subobject of H is isomorphic to a subobject of H/L.

(2) For monoform objects H_{1} and H_{2} in A, we say that H_{1} is atomequivalent to H_{2} if there exists a nonzero subobject of H_{1} which is isomorphic to a subobject of H_{2}.

We recall the definitions of essential subobjects and uniform objects. These are also important notions in a Grothendieck category and related to monoform objects.

Definition 3.4.

(1) Let M be an object in A. A subobject L of M is called essential if for every nonzero subobject L′ of M, we have L ∩ L′ ≠ 0.

(2) A nonzero object U in A is called uniform if every nonzero subobject of U is essential.

In other words, a nonzero object U in A is uniform if and only if for all nonzero subobjects L_{1} and L_{2} of U, we have L_{1} ∩ L_{2} ≠ 0. It is easy to show that each nonzero subobject of a uniform object is uniform. This type of result also holds for monoform objects.

Proposition 3.5.

(1) Each nonzero subobject of a monoform object is monoform.

(2) Every monoform object is uniform.

(3) Every nonzero noetherian object has a monoform subobject.

Proof. (1) [Kan12, Proposition 2.2].

(2) [Kan12, Proposition 2.6].

(3) [Kan12, Theorem 2.9]. □

It follows from Proposition 3.5 (2) that atom equivalence is an equivalence relation on the set of monoform objects in A ([Kan12, Proposition 2.8]). The atom spectrum is defined by using this relation.

Definition 3.6. Let A be a Grothendieck category. Denote by ASpec A the quotient set of the set of monoform objects in A by atom equivalence. We call it the atom spectrum of A. Each element of ASpec A is called an atom in A. For each monoform object H in A, the equivalence class of H is denoted by \overline{H}.

It is shown in [Kan15b, Proposition 2.7 (2)] that the atom spectrum ASpec A of a Grothendieck category A is in bijection with a small set. The following result shows that the atom spectrum of a Grothendieck category is a generalization of the prime spectrum of a commutative ring.

Proposition 3.7. Let R be a commutative ring.

(1) ([Sto72, Lemma 1.5]) Let 𝔞 be an ideal of R. Then R/𝔞 is a monoform object in Mod R if and only if 𝔞 is a prime ideal.

(2) ([Sto72, p. 631]) There is a bijection Spec R → ASpec(Mod R) given by 𝔭 ↦ \overline{R/𝔭}.

We can also generalize the notions of supports and associated primes in commutative ring theory.

Definition 3.8. Let M be an object in A.

(1) Define the subset AAss M of ASpec A by

AAss M = \{ α ∈ ASpec A | α = \overline{H} for some monoform subobject H of M \}.

We call each element of AAss M an associated atom of M.

(2) Define the subset ASupp M of ASpec A by

ASupp M =\{ α ∈ ASpec A |α = \overline{H} for some monoform subquotient H of M \}.

We call it the atom support of M.

Proposition 3.9. Let R be a commutative ring, and let M be an R-module. Then the bijection Spec R → ASpec(Mod R) in Proposition 3.7 (2) restricts to bijections Ass M → AAss M and Supp M → ASupp M.

Proof. [Kan15b, Proposition 2.13]. □

The following results are generalizations of fundamental results in commutative ring theory.

Proposition 3.10. Let 0 → L → M → N → 0 be an exact sequence in A. Then

AAss L ⊂ AAss M ⊂ AAss L ∪ AAss N,

and ASupp M = ASupp L ∪ ASupp N.

Proof. [Kan12, Proposition 3.5] and [Kan12, Proposition 3.3]. □

Proposition 3.11.

(1) Let {M_{λ}}_{λ∈Λ} be a family of objects in A. Then

AAss ⨁_{λ∈Λ} M_{λ} = ⋃_{λ∈Λ} AAss M_{λ},

and

ASupp ⨁_{λ∈Λ} M_{λ} = ⋃_{λ∈Λ} ASupp M_{λ}.

(2) Let M be an object in A, and let {L_{λ}}_{λ∈Λ} be a family of subobjects of

M. Then

ASupp ∑_{λ∈Λ} L_{λ} = ⋃_{λ∈Λ} ASupp L_{λ}.

Proof. (1) [Kan15b, Proposition 2.12].

(2) Since we have the canonical epimorphism ⨁_{λ∈Λ} L_{λ} ↠ ∑_{λ∈Λ} L_{λ} and the inclusion L_{μ} ⊂ ∑_{λ∈Λ} L_{λ} for each μ ∈ Λ, we obtain

ASupp L_{μ} ⊂ ASupp ∑_{λ∈Λ} L_{λ} ⊂ ⋃_{λ∈Λ} ASupp L_{λ}

by (1). Hence the claim follows. □

Similarly to the case of commutative rings, we have the following results on the associated atoms of uniform objects and essential subobjects.

Proposition 3.12.

(1) Let U be a uniform object in A. Then AAss U consists of at most one element. In particular, for every monoform object H in A, we have AAss H = \{\overline{H}\}.

(2) Let M be an object in A, and let L be an essential subobject of M. Then AAss L = AAss M.

Proof. (1) [Kan15b, Proposition 2.15 (1)].

(2) [Kan15b, Proposition 2.16]. □

We introduce a topology on the atom spectrum.

Definition 3.13. We call a subset Φ of ASpec A a localizing subset if there exists an object M in A such that Φ = ASupp M.

Proposition 3.14. The set of localizing subsets of ASpec A satisfies the axioms of open subsets of ASpec A.

Proof. [Kan12, Proposition 3.8]. □

We call the topology on ASpec A defined by the set of localizing subsets of ASpec A the localizing topology. Throughout this paper, we regard ASpec A as a topological space in this way. For a commutative ring R, the localizing subsets of ASpec(Mod R) define a different topology from the Zariski topology on Spec R. Recall that a subset Φ of Spec R is said to be closed under specialization if for every 𝔭, 𝔮 ∈ Spec R, the conditions 𝔭 ∈ Φ and 𝔭 ⊂ 𝔮 imply 𝔮 ∈ Φ.

Proposition 3.15. Let R be a commutative ring, and let Φ be a subset of Spec R. Then the corresponding subset

\{ \overline{(\frac{R}{𝔭})} ∈ ASpec(Mod R) | 𝔭 ∈ Φ \}

of ASpec(Mod R) is localizing if and only if Φ is closed under specialization.

Proof. [Kan12, Proposition 7.2 (2)]. □

For each α ∈ ASpec A, let Λ(α) be the topological closure of {α} in ASpec A. We introduce a partial order on the atom spectrum.

Definition 3.16. For α, β ∈ ASpec A, we write α ≤ β if α ∈ Λ(β).

The relation ≤ is called the specialization order on the topological space ASpec A with respect to the localizing topology. This is in fact a partial order on ASpec A since the topological space ASpec A is a Kolmogorov space ([Kan15b, Proposition 3.5]). By definition, Λ(β) = { α ∈ ASpec A | α ≤ β } for each β ∈ ASpec A. The partial order has the following descriptions.

Proposition 3.17. Let α, β ∈ ASpec A. Then the following assertions are equivalent.

(1) α ≤ β, that is, α ∈ Λ(β).

(2) For every object M in A, the condition α ∈ ASupp M implies β ∈ ASupp M.

(3) For every monoform object H in A with \overline{H} = α, we have β ∈ ASupp H.

Proof. [Kan15b, Proposition 4.2]. □

The following result claims that the partial order ≤ on ASpec A is a generalization of the inclusion relation between prime ideals of a commutative ring.

Proposition 3.18. Let R be a commutative ring and 𝔭, 𝔮 ∈ Spec R. Then \overline{R/𝔭} ≤ \overline{R/𝔮} in ASpec(Mod R) if and only if 𝔭 ⊂ 𝔮. In other words, the bijection Spec R → ASpec(Mod R) in Proposition 3.7 (2) is an isomorphism between the partially ordered sets (Spec R, ⊂) and (ASpec(Mod R), ≤).

Proof. [Kan15b, Proposition 4.3]. □

4. Subcategories and quotient categories

In this section, we show preliminary results on subcategories and quotient categories of a Grothendieck category A. We start with defining some classes of subcategories, which are the main objects in this paper.

Definition 4.1.

(1) For full subcategories X_{1} and X_{2} of A, we denote by X_{1} ∗ X_{2} the full subcategory of A consisting of all objects M admitting an exact sequence

0 → M_{1} → M → M_{2} → 0

in A, where M_{i} belongs to X_{i} for each i = 1, 2.

(2) We say that a full subcategory X of A is closed under extension if X ∗ X ⊂ X, that is, for every exact sequence 0 → L → M → N → 0 in A, the condition L, N ∈ X implies M ∈ X.

(3) A full subcategory X of A is called a prelocalizing subcategory (or weakly closed subcategory in [VdB01]) of A if X is closed under subobjects, quotient objects, and arbitrary direct sums.

(4) A prelocalizing subcategory X of A is called a localizing subcategory of A if X is also closed under extensions.

(5) For a full subcategory X of A, denote by ⟨X ⟩_{preloc} (resp. ⟨X ⟩_{loc}) the smallest prelocalizing (resp. localizing) subcategory of A containing X. For an object M in A, let ⟨M⟩_{preloc} = ⟨{M}⟩_{preloc} and ⟨M⟩_{loc} = ⟨{M}⟩_{loc}.

Proposition 4.2.

(1) Let X_{1}, X_{2}, and X_{3} be full subcategories of A. Then

(X_{1} ∗ X_{2}) ∗ X_{3} = X_{1} ∗ (X_{2} ∗ X_{3}).

(2) Let X_{1} and X_{2} be prelocalizing subcategories of A. Then X_{1} ∗ X_{2} is also a prelocalizing subcategory of A.

Proof. (1) [Kan12, Proposition 2.4 (2)].

(2) [Pop73, Lemma 4.8.11]. □

Prelocalizing subcategories are characterized as follows.

Proposition 4.3. Let A be a Grothendieck category (or more generally, an abelian category admitting arbitrary direct sums), and let X be a full subcategory of A closed under subobjects and quotient objects. Then the following assertions are equivalent.

(1) X is closed under arbitrary direct sums, that is, X is a prelocalizing subcategory of A.

(2) The inclusion functor X ↪ A has a right adjoint.

(3) For each object M in A, there exists a largest subobject L of M which belongs to X.

Proof. Assume (3). Then the functor A → X which sends each object M to its largest subobject belonging to X and each morphism to the induced morphism is a right adjoint of the inclusion functor X ↪ A. The dual statement of (2)⇒(1) is essentially shown in [Ste75, Proposition X.1.2]. (1)⇒(3) follows from the next remark. □

Remark 4.4. Let X be a full subcategory of A closed under quotient objects and arbitrary direct sums, and let M be an object in A. Since the sum L = ∑_{λ∈Λ} L_{λ} of all subobjects of M which belong to X is a quotient object of the direct sum ⨁_{λ∈Λ} L_{λ}, the subobject L of M also belongs to X. Hence L is the largest subobject of M which belongs to X.

The operation in Remark 4.4 of taking the subobject L from M is used throughout this paper. The following result shows that this operation commutes with taking arbitrary direct sums.

Proposition 4.5. Let A be a Grothendieck category, and let X be a full subcategory of A closed under quotient objects and arbitrary direct sums. Let {M_{λ}}_{λ∈Λ} be a family of objects in A, and take L_{λ} to be the largest subobject of M_{λ} which belongs to X for each λ ∈ Λ. Then ⨁_{λ∈Λ} L_{λ} is the largest subobject of ⨁_{λ∈Λ} M_{λ} which belongs to X.

Proof. Let N be the largest subobject of ⨁_{λ∈Λ} M_{λ} which belongs to X. It suffices to show that N ⊂ ⨁_{λ∈Λ} L_{λ}. We show the claim in the case where Λ = {1, …, n} for some n ∈ ℤ_{≥1}. Let π_{i}: M_{1} ⊕ ⋯ ⊕ M_{n} ↠ M_{i} be the projection for each i ∈ {1, …, n}. Since π_{i}(N) is a quotient object of N, it belongs to X. By the maximality of L_{i}, we have π_{i}(N) ⊂ L_{i}. Hence

N ⊂ π_{1}(N) ⊕ ⋯ ⊕ π_{n}(N) ⊂ L_{1} ⊕ ⋯ ⊕ L_{n}

as subobjects of M_{1} ⊕ ⋯ ⊕ M_{n}. In the general case, let S be the set of finite subsets of Λ. Then by Proposition 3.2,

N = ∑_{Λ′∈S}(N ∩ ⨁_{λ∈Λ′} M_{λ}) ⊂ ∑_{Λ′∈S} ⨁_{λ∈Λ′} L_{λ} = ⨁_{λ∈Λ} L_{λ}.  □

For a localizing subcategory X of A, we have the quotient category A/X of A by X. It is a Grothendieck category together with a canonical (covariant) functor A → A/X ([Pop73, Corollary 4.6.2]). We refer the reader to [Kan15b, Definition 5.2] for the explicit definition of the quotient category. Instead, we state a universal property of the quotient category.

Theorem 4.6. Let A be a Grothendieck category, and let X be a localizing subcategory of A. The canonical functor is denoted by F: A → A/X.

(1) The functor F: A → A/X is exact and has a right adjoint A/X → A. For every object M in A, we have F (M) = 0 if and only if M belongs to X.

(2) Let B be an abelian category together with an exact functor Q: A → B with Q(M) = 0 for each object M in X. Then there exists a unique functor \overline{Q}: A/X → B such that \overline{Q}F = Q. Moreover, the functor \overline{Q} is exact.

Proof. (1) [Pop73, Proposition 4.6.3], [Pop73, Theorem 4.3.8], and [Pop73, Lemma 4.3.4].

(2) [Pop73, Corollary 4.3.11] and [Pop73, Corollary 4.3.12]. □

Every object M in a Grothendieck category A has an injective envelope E(M) ([Gab62, Theorem II.6.2], see also [Pop73, Theorem 3.10.10]). By definition, the object M is an essential subobject of the injective object E(M). The object E(M) is also denoted by E_{A}(M) in order to specify the category explicitly. Let X be a localizing subcategory of A. An object M in A is called X - torsionfree if M has no nonzero subobject belonging to X. Note that every subobject of an X -torsionfree object is X -torsionfree.

Proposition 4.7. Let X be a localizing subcategory of A. Let M be an object in A, and let L be the largest subobject of M which belongs to X. Then M/L is X -torsionfree.

Proof. Assume that M/L is not X -torsionfree. Then there exists a subobject L′ of M such that L ⊊ L′, and L′/L belongs to X. The subobject L′ of M also belongs to X. This contradicts the maximality of L. □

For an object M in A, it is also important to consider the torsionfreeness of E(M)/M.

Proposition 4.8. Let X be a localizing subcategory of A, and let

0 → L → M → N → 0

be an exact sequence in A. If M and E(L)/L are X -torsionfree, then N is X -torsionfree.

Proof. This can be shown similarly to the proof of [Pop73, Proposition 4.5.5]. □

We state important properties of the canonical functor to a quotient category and its right adjoint by using the notion of torsionfreeness.

Proposition 4.9. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) F is surjective, that is, each object in A/X is of the form F (M), where M is some object in A.

(2) The counit morphism ε: F G → 1_{A/X} is an isomorphism. Hence G is fully faithful.

(3) Let η: 1_{A} → GF be the unit morphism. Then for each object M in A, the subobject Ker η_{M} of M is the largest subobject belonging to X, the subobject Im η_{M} of GF (M) is essential, and Cok η_{M} belongs to X. The objects GF (M) and E(GF (M))/GF (M) are X -torsionfree.

(4) Let M′ be an object in A/X. Then G(M′) and E(G(M′))/G(M′) are X -torsionfree.

Proof. (1) This is obvious from the definition of the canonical functor F. It also follows from Theorem 4.6.

(2) [Pop73, Proposition 4.4.3 (1)].

(3) This follows from [Pop73, Proposition 4.4.3 (2)] and the proof of [Pop73, Proposition 4.4.5].

(4) This follows from (1) and (3). □

The next result is necessary to describe subobjects of an object in a quotient category.

Proposition 4.10. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. Let M be an object in A. For each subobject L′ of F (M), there exists a largest subobject L of M satisfying F (L) ⊂ L′ as a subobject of F (M). Moreover, it holds that F (L) = L′, and the quotient object M/L is X -torsionfree. The quotient object F (M)/L′ of F (M) is equal to F (M/L).

Proof. Since G is left exact, the object G(L′) can be regarded as a subobject of GF (M). Let η: 1_{A} → GF be the unit morphism. There is a commutative diagram

\begin{matrix}0 & η_{M}^{−1}(G(L′)) & M & \frac{M}{η_{M}^{−1}(G(L′))} & 0 \\ 0 & G(L′) & GF (M) & \frac{GF (M)}{G(L′)} & 0 \end{matrix}_{η_{M}} .

By applying F to this diagram, we obtain the commutative diagram

\begin{matrix}0 & F (η_{M}^{−1}(G(L′))) & F (M) & F (\frac{M}{η_{M}^{−1}(G(L′))}) & 0 \\ 0 & F G(L′) & F GF (M) & F (\frac{GF (M)}{G(L′)}) & 0 \\ 0 & L′ & F (M) & F (\frac{GF (M)}{G(L′)}) & 0 \end{matrix} ≅ ≅ ≅ ≅ ≅

by Proposition 4.9 (2) and Proposition 4.9 (3). Hence the subobject L := η_{M}^{−1}(G(L′)) of M satisfies F (L) = L′, and F (M)/L′ = F (M/L). By Proposition 4.8, the object GF (M)/G(L′) is X -torsionfree, and hence M/L is also X -torsionfree. Let L̃ be a subobject of M such that F (L̃) ⊂ L′. Since we have the commutative diagram

\begin{matrix}L̃ & M \\ GF (L̃) & GF (M) \end{matrix} ; ηL̃ ; η_{M} ,

it holds that η_{M}(L̃) ⊂ GF (L̃). Therefore

L̃ ⊂ η_{M}^{−1}(GF (L̃)) ⊂ η_{M}^{−1}(G(L′)) = L.  □

Several properties of objects are preserved by the canonical functor to a quotient category and its right adjoint as in the following results.

Proposition 4.11. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) Let M′ be an object in A/X, and let L′ be an essential subobject of M′. Then G(L′) is an essential subobject of G(M′).

(2) Let U′ be a uniform object in A/X. Then G(U′) is a uniform object in A.

(3) Let H′ be a monoform object in A/X. Then G(H′) is a monoform object in A.

(4) Let I′ be an injective object in A/X. Then G(I′) is an injective object in A.

(5) Let M′ be an indecomposable object in A/X. Then G(M′) is an indecomposable object in A.

Proof. (1) [Pop73, Corollary 4.4.7].

(2) Let L be a nonzero subobject of G(U′). We have a commutative diagram

\begin{matrix}L & G(U′) \\ GF (L) & GF G(U′) \end{matrix} ,

and the morphism G(U′) → GF G(U′) is an isomorphism by Proposition 4.9 (2). Hence the morphism L → GF (L) is a monomorphism, and in particular F (L) is a nonzero subobject of F G(U′) ≅ U′. By the uniformness of U′ and (1), GF (L) is an essential subobject of GF G(U′). Since L is essential as a subobject of GF (L) by Proposition 4.9 (3), L is an essential subobject of G(U′).

(3) [Kan15b, Lemma 5.14 (1)].

(4) [Pop73, Corollary 4.4.7].

(5) This follows from Proposition 4.9 (2). □

Proposition 4.12. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) Let M be an X -torsionfree object in A, and let L be an essential subobject of M. Then F (L) is an essential subobject of F (M).

(2) Let U be a uniform X -torsionfree object in A. Then F (U) is a uniform object in A/X.

(3) Let H be a monoform X -torsionfree object in A. Then F (H) is a monoform object in A/X.

(4) Let I be an injective X -torsionfree object in A. Then F (I) is an injective object in A/X.

Proof. (1) [Pop73, Lemma 4.4.6 (3)].

(2) This follows from Proposition 4.10 and (1).

(3) [Kan15b, Lemma 5.14 (2)].

(4) [Pop73, Lemma 4.5.1 (2)]. □

The prelocalizing subcategories of A and those of quotient categories are related by the following operations.

Proposition 4.13. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) For each prelocalizing subcategory Y′ of A/X, the full subcategory

F^{−1}(Y′) := { M ∈ A | F (M) ∈ Y′ }

of A is a prelocalizing subcategory, and

X ∗ F^{−1}(Y′) ∗ X = F^{−1}(Y′).

(2) For each prelocalizing subcategory Y of A, the full subcategory

F (Y) := \{ N ∈ \frac{A}{X} | N ≅ F (M) for some M ∈ Y \}

of A/X is a prelocalizing subcategory.

(3) Let Y_{1} and Y_{2} be prelocalizing subcategories of A. Then

F (Y_{1} ∗ X ∗ Y_{2}) = F (Y_{1}) ∗ F (Y_{2}).

Proof. (1) Since F is exact and commutes with arbitrary direct sums, the full subcategory F^{−1}(Y′) is a prelocalizing subcategory. The inclusion F^{−1}(Y′) ⊂ X ∗ F^{−1}(Y′) ∗ X is obvious. By Theorem 4.6 (1),

F (X ∗ F^{−1}(Y′) ∗ X) ⊂ F (X) ∗ F (F^{−1}(Y′)) ∗ F (X) ⊂ Y′.

Hence X ∗ F^{−1}(Y′) ∗ X ⊂ F^{−1}(Y′).

(2) By Proposition 4.10, the full subcategory F (Y) of A/X is closed under subobjects and quotient objects. It is also closed under arbitrary direct sums since F commutes with arbitrary direct sums.

(3) Since F is exact, F (Y_{1} ∗ X ∗ Y_{2}) ⊂ F (Y_{1}) ∗ F (Y_{2}) by Theorem 4.6 (1). Let M′ be an object in A/X which belongs to F (Y_{1}) ∗ F (Y_{2}). Then there exists an exact sequence

0 → F (M_{1}) → M′ → F (M_{2}) → 0

where M_{i} is an object in A which belongs to Y_{i} for each i = 1, 2. Since G is left exact, we have the exact sequence

0 → GF (M_{1}) → G(M′) → GF (M_{2}).

Let η: 1_{A} → GF be the unit morphism, and let B be the image of the morphism G(M′) → GF (M_{2}). Then we obtain a commutative diagram

\begin{matrix}0 & GF (M_{1}) & M & B ∩ Im η_{M_{2}} & 0 \\ 0 & GF (M_{1}) & G(M′) & B & 0 \end{matrix} ,

where M is an object in A. Let N be the cokernel of the composite Im η_{M_{1}} ↪ GF (M_{1}) ↪ G(M′). There is a commutative diagram

\begin{matrix}0 & Im η_{M_{1}} & M & N & 0 \\ 0 & GF (M_{1}) & M & B ∩ Im η_{M_{2}} & 0 \end{matrix} .

By the snake lemma, we have an exact sequence

0 → Cok η_{M_{1}} → N → B ∩ Im η_{M_{2}} → 0.

By Proposition 4.9 (3), the object Cok η_{M_{i}} belongs to X for each i = 1, 2. Hence F (Cok η_{M_{1}}) = 0, and

F (\frac{B}{B ∩ Im η_{M_{2}}} ) ≅ F (\frac{B + Im η_{M_{2}}}{Im η_{M_{2}}} ) ⊂ F (\frac{GF (M_{2})}{Im η_{M_{2}}}) = 0.

By applying F to the morphisms B ∩ Im η_{M_{2}} ↪ B and Im η_{M_{1}} ↪ GF (M_{1}), we obtain F (B ∩ Im η_{M_{2}}) \underrightarrow{∼} F (B) and F (Im η_{M_{1}}) \underrightarrow{∼} F GF (M_{1}) \underrightarrow{∼} F (M_{1}).

Hence we have the commutative diagram

\begin{matrix}0 & F (Im η_{M_{1}}) & F (M) & F (N) & 0 \\ 0 & F GF (M_{1}) & F (M) & F (B ∩ Im η_{M_{2}}) & 0 \\ 0 & F GF (M_{1}) & F G(M′) & F (B) & 0 \\ 0 & F (M_{1}) & M′ & F (M_{2}) & 0 \end{matrix} ; ≅ ; ≅ ; ≅ ; ≅ ; ≅ ; ≅ ; ≅ .

For each i = 1, 2, the quotient object Im η_{M_{i}} of M_{i} belongs to Y_{i}, and hence N belongs to X ∗ Y_{2}. Therefore M′ belongs to F (Y_{1} ∗ X ∗ Y_{2}). □

Proposition 4.14. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) There is a bijection

\{ \begin{matrix}prelocalizing subcategories Y of A \\ satisfying X ∗ Y ∗ X = Y \end{matrix} \} → \{ prelocalizing subcategories of \frac{A}{X} \}

given by Y ↦ F (Y). Its inverse is given by Y′ ↦ F^{−1}(Y′).

(2) For each i = 1, 2, let Y_{i} be a prelocalizing subcategory of A such that X ∗ Y_{i} ∗ X = Y_{i}. Then

F (Y_{1} ∗ Y_{2}) = F (Y_{1}) ∗ F (Y_{2}).

(3) The bijection in (1) restricts to a bijection

\{ \begin{matrix}localizing subcategories Y of A \\ satisfying X ⊂ Y \end{matrix} \} → \{ localizing subcategories of \frac{A}{X} \}.

Proof. (1) By Proposition 4.13 (1) and Proposition 4.13 (2), these maps are well-defined. Let η: 1_{A} → GF be the unit morphism. Let Y be a prelocalizing subcategory of A satisfying X ∗Y ∗X = Y. It is obvious that Y ⊂ F^{−1}F (Y). Let M be an object in A which belongs to F^{−1}F (Y). Then there exists an object N in A which belongs to Y such that F (M) ≅ F (N). There is an exact sequence

0 → Im η_{N} → GF (N) → Cok η_{N} → 0.

The quotient object Im η_{N} of N belongs to Y. By Proposition 4.9 (3), the object Cok η_{N} belongs to X. Hence GF (M) ≅ GF (N) belongs to Y ∗ X. By Proposition 4.2 (2), the subobject Im η_{M} of GF (M) belongs to Y ∗ X. There is an exact sequence

0 → Ker η_{M} → M → Im η_{M} → 0,

where Ker η_{M} belongs to X. Therefore M belongs to X ∗ Y ∗ X = Y. This shows that F^{−1}F (Y) ⊂ Y.

Let Y′ be a prelocalizing subcategory of A/X. It is obvious that F F^{−1}(Y′) ⊂ Y′. Let M′ be an object in A/X which belongs to Y′. Then by Proposition 4.9 (1), there exists an object M in A such that F (M) = M′. Since M belongs to F^{−1}(Y′), the object M′ = F (M) belongs to F F^{−1}(Y′). This shows that Y′ ⊂ F F^{−1}(Y′).

(2) By Proposition 4.13 (3),

F (Y_{1} ∗ Y_{2}) = F (Y_{1} ∗ X ∗ Y_{2}) = F (Y_{1}) ∗ F (Y_{2}).

(3) This follows from (2). □

Remark 4.15. In the setting of Proposition 4.13 (3), the assertion F (Y_{1} ∗ X ∗ Y_{2}) = F (Y_{1} ∗ Y_{2}) does not necessarily hold. The next example gives a counter-example.

Example 4.16. Let K be a field, and let Λ be the ring

Λ = [\begin{matrix}K & 0 & 0 \\ K & K & 0 \\ K & K & K \end{matrix}]

of 3×3 lower triangular matrices. Define simple Λ-modules S_{i} for each i = 1, 2, 3 by

S_{1} = [K 0 0],
S_{2} = \begin{matrix}[ & K K 0 & ] \\ [ & K 0 0 & ] \end{matrix},

S_{3} = \begin{matrix}[ & K K K & ] \\ [ & K K 0 & ] \end{matrix},

and let X_{i} be the localizing subcategory of Mod Λ consisting of arbitrary direct sums of copies of S_{i}. Let F: A → A/X_{2} and G: A/X_{2} → A denote the canonical functors. Since the Λ-module

M = [K K K]

belongs to X_{1}∗X_{2}∗X_{3}, it follows that M ≅ GF (M) belongs to GF (X_{1}∗X_{2}∗X_{3}). On the other hand, every Λ-module belonging to X_{1} ∗ X_{3} is the direct sum of some object in X_{1} and some object in X_{3}. Since Mod Λ is a locally noetherian Grothendieck category, by [Pop73, Proposition 5.8.12], the functor G commutes with arbitrary direct sums. Hence every Λ-module belonging to GF (X_{1} ∗ X_{3}) is the direct sum of some object in GF (X_{1}) = X_{1} ∗ X_{2} and some object in GF (X_{3}) = X_{3}. Since M is indecomposable and belongs to neither X_{1} ∗ X_{2} nor X_{3}, the Λ-module M does not belong to GF (X_{1} ∗ X_{3}). This shows that F (X_{1} ∗ X_{2} ∗ X_{3}) ⊄ F (X_{1} ∗ X_{3}).

The following result gives a characterization of a quotient category.

Proposition 4.17. Let A and B be Grothendieck categories, and let Q: A → B be an exact functor with a fully faithful right adjoint B → A. Then the full subcategory

X = { M ∈ A | Q(M) = 0 }

of A is a localizing subcategory, and there exists a unique equivalence \overline{Q}: A/X \underrightarrow{∼} B such that \overline{Q}F = Q, where F: A → A/X is the canonical functor.

Proof. [Pop73, Theorem 4.4.9]. □

We state some facts on the image of a localizing subcategory in a quotient category.

Proposition 4.18. Let X and Y be localizing subcategories of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) It holds that ⟨F (Y)⟩_{loc} = F (⟨X ∪ Y⟩_{loc}).

(2) If X ⊂ Y, then the composite Y → A → A/X induces an equivalence

\frac{Y}{X} \underrightarrow{∼} F (Y).

(3) If X ⊂ Y, then the composite

A → A/X → \frac{A/X}{F (Y)}

induces an equivalence

\frac{A}{Y} \underrightarrow{∼} \frac{A/X}{F (Y)}.

Proof. (1) It is obvious that ⟨F (Y)⟩_{loc} ⊂ F (⟨X ∪ Y⟩_{loc}). Since F is exact and commutes with arbitrary direct sums,

F (⟨X ∪ Y⟩_{loc}) ⊂ ⟨F (X ∪ Y)⟩_{loc} = ⟨F (X) ∪ F (Y)⟩_{loc} = ⟨F (Y)⟩_{loc}

by Theorem 4.6 (1).

(2) The equivalence follows from the construction of A/X (see [Gab62, p. 365] or [Kan15b, Definition 5.2]).

(3) By Proposition 4.14 (3), the full subcategory F (Y) of A/X is a localizing subcategory, and F^{−1}F (Y) = Y. By Proposition 4.9 (2), the composite is an exact functor with a fully faithful right adjoint. Hence by Proposition 4.17, it induces an equivalence

\frac{A}{Y} = \frac{A}{F^{−1}F (Y)} \underrightarrow{∼} \frac{A/X}{F (Y)}.  □

5. Atom spectra of quotient categories and localization

Throughout this section, let A be a Grothendieck category. We recall a description of the atom spectrum of a quotient category of A and fundamental results on the localization of A at an atom. We start with relating localizing subcategories of A and localizing subsets of ASpec A.

Definition 5.1.

(1) For a full subcategory X of A, define the subset ASupp X of ASpec A by

ASupp X = ⋃_{M∈X} ASupp M.

(2) For a subset Φ of ASpec A, define the full subcategory ASupp^{−1} Φ of

A by

ASupp^{−1} Φ = { M ∈ A | ASupp M ⊂ Φ }.

Proposition 5.2.

(1) For every full subcategory X of A, the subset ASupp X of ASpec A is a localizing subset.

(2) For every subset Φ of ASpec A, the full subcategory ASupp^{−1} Φ of A is a localizing subcategory.

Proof. (1) Recall that ASpec A is in bijection with a small set. For each α ∈ ASupp X, choose an object M(α) in A which belongs to X such that α ∈ ASupp M(α). Then

ASupp ⨁_{α∈ASupp X} M(α) = ⋃_{α∈ASupp X} ASupp M(α) = ASupp X

by Proposition 3.11 (1).

(2) This follows from Proposition 3.10 and Proposition 3.11 (1). □

The following result shows that a localizing subset of ASpec A is determined by the corresponding localizing subcategory of A.

Proposition 5.3. For every localizing subset Φ of ASpec A,

ASupp(ASupp^{−1} Φ) = Φ.

Proof. This follows from the proof of [Kan12, Theorem 4.3]. □

If A is a locally noetherian Grothendieck category, we also have ASupp^{−1}(ASupp X) = X for every localizing subcategory X of A, and these correspondences establish a bijection between the localizing subcategories of A and the localizing subsets of ASpec A ([Kan12, Theorem 5.5]). We generalize this result later as Theorem 6.8. We describe the atom spectrum of the quotient category by a localizing subcategory.

Theorem 5.4. Let A be a Grothendieck category, and let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. Then the map ASpec A \ ASupp X → ASpec(A/X) given by \overline{H} ↦ \overline{F (H)} is a homeomorphism. Its inverse is given by \overline{H′} ↦ \overline{G(H′)}.

Proof. [Kan15b, Theorem 5.17]. □

Remark 5.5. Every localizing subcategory X of A is a Grothendieck category, and ASpec X is homeomorphic to the localizing subset ASupp X of ASpec A by the correspondence \overline{H} ↦ \overline{H} ([Kan15b, Proposition 5.12]). We identify ASpec X with ASupp X, and ASpec(A/X) with ASpec A \ ASupp X via the homeomorphism in Theorem 5.4. Then

ASpec A = ASpec X ∪ ASpec \frac{A}{X} ,

and

ASpec X ∩ ASpec \frac{A}{X} = ∅.

We describe atom supports and associated atoms in a quotient category.

Proposition 5.6. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) For every object M′ in A/X,

AAss G(M′) = AAss M′,

and

ASupp G(M′) \ ASupp X = ASupp M′.

(2) For every object M in A,

AAss F (M) ⊃ AAss M \ ASupp X ,

and ASupp F (M) = ASupp M \ ASupp X .

Proof. These follow from [Kan15b, Lemma 5.16]. By considering Proposition 4.9 (4), the assertion AAss G(M′) = AAss M′ also follows. □

The atom spectrum of the image of a localizing subcategory in a quotient category is described as follows.

Proposition 5.7. Let X and Y be localizing subcategories of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. Then

ASpec ⟨F (Y)⟩_{loc} = ASpec F (⟨X ∪ Y⟩_{loc})

= ASpec Y ∩ ASpec \frac{A}{X}

= ASpec Y \ ASpec X ,

and

ASpec \frac{A/X}{⟨F (Y)⟩_{loc}} = ASpec \frac{A}{⟨X ∪ Y⟩_{loc}}

= ASpec \frac{A}{X} ∩ ASpec \frac{A}{Y}

= ASpec A \ (ASpec X ∪ ASpec Y).

Proof. This follows from ASupp ⟨X ∪ Y⟩_{loc} = ASupp X ∪ ASupp Y and Proposition 4.18. □

Definition 5.8. Let A be a Grothendieck category and α ∈ ASpec A. Define a localizing subcategory X (α) of A by X (α) = ASupp^{−1}(ASpec A \ Λ(α)). Define the localization A_{α} of A at α by A_{α} = A/X (α). The canonical functor A → A_{α} is denoted by (−)_{α}.

In Definition 5.8, the subset ASpec A\Λ(α) of ASpec A is localizing. By Proposition 5.3, ASupp X (α) = ASpec A \ Λ(α). Therefore we have the following result.

Theorem 5.9. Let A be a Grothendieck category and α ∈ ASpec A. Then ASpec A_{α} = Λ(α). In particular, the partially ordered set ASpec A has the largest element α.

Proof. [Kan15b, Proposition 6.6 (1)]. □

We obtain the following description of atom supports.

Proposition 5.10.

(1) For every α ∈ ASpec A,

X (α) = { M ∈ A | α ∉ ASupp M }.

(2) For every object M in A,

ASupp M = { α ∈ ASpec A | M_{α} ≠ 0 }.

Proof. [Kan15b, Proposition 6.2]. □

We show that the localization of a Grothendieck category at an atom is “local” in the following sense.

Definition 5.11. Let A be a Grothendieck category.

(1) We say that A is local if there exists a simple object in A such that E(S) is a cogenerator of A.

(2) A localizing subcategory X of A is called prime if A/X is a local Grothendieck category.

Theorem 5.12. Let A be a Grothendieck category. Then the following assertions are equivalent.

(1) A is local.

(2) There exists α ∈ ASpec A such that for every nonzero object M in A, we have α ∈ ASupp M.

(3) There exists α ∈ ASpec A such that the canonical functor A → A_{α} is an equivalence.

Proof. [Kan15b, Proposition 6.4 (1)] and [Kan15b, Proposition 6.6 (2)]. □

In the case of where A is a locally noetherian Grothendieck category, the localness of A is characterized as follows.

Proposition 5.13. Let A be a Grothendieck category. If A is local, then all simple objects in A are isomorphic to each other. In the case where A is a nonzero locally noetherian Grothendieck category, the converse also holds.

Proof. [Kan15b, Proposition 6.4 (2)]. □

Theorem 5.12 shows that the localizing subcategory X (α) is prime for every α ∈ ASpec A. This correspondence gives the following bijection.

Theorem 5.14. Let A be a Grothendieck category. There is a bijection

ASpec A → { prime localizing subcategories of A }

given by α ↦ X (α). For each α, β ∈ ASpec A, we have α ≤ β if and only if X (α) ⊃ X (β).

Proof. [Kan15b, Theorem 6.8]. □

We consider the localization of a quotient category.

Proposition 5.15. Let X be a localizing subcategory of A and α ∈ ASpec A \ ASupp X. Then the composite of the canonical functors A → A/X and A/X → (A/X)_{α} induces an equivalence A_{α} \underrightarrow{∼} (A/X)_{α}.

Proof. By Proposition 5.10 (1), X ⊂ X (α). Hence the claim follows from Proposition 5.6 (2) and Proposition 4.18 (3). □

In the setting of Proposition 5.15, we identify A_{α} and (A/X)_{α}. The following result shows that the localization of a Grothendieck category at an atom is a generalization of the localization a commutative ring at a prime ideal.

Proposition 5.16. Let R be a commutative ring.

(1) Let 𝔭 ∈ Spec R. Denote by α the corresponding atom \overline{R/𝔭} in Mod R. Then the functor − ⊗_{R} R_{𝔭}: Mod R → Mod R_{𝔭} induces an equivalence (Mod R)_{α} \underrightarrow{∼} Mod R_{𝔭}.

(2) The Grothendieck category Mod R is local if and only if the commutative ring R is local.

Proof. (1) [Kan15b, Proposition 6.9].

(2) This follows from Theorem 5.12 and (1). □

6. Grothendieck categories with enough atoms

The purpose of this paper is to investigate the category QCoh X of quasicoherent sheaves on a locally noetherian scheme X. In general, the category QCoh X is a Grothendieck category but not necessarily locally noetherian (see Remark 7.5). In this section, we introduce the notion of a Grothendieck category with enough atoms and investigate its properties. It is shown later that QCoh X is a Grothendieck category with enough atoms. Let A be a Grothendieck category. Recall that every monoform object in A is uniform (Proposition 3.5 (2)). We say that uniform objects U_{1} and U_{2} in A are equivalent (denoted by U_{1} ∼ U_{2}) if there exists a nonzero subobject of U_{1} which is isomorphic to a subobject of U_{2}. The equivalence between monoform objects is exactly the same as the atom-equivalence defined in Definition 3.3 (2).

Proposition 6.1. Let U_{1} and U_{2} be uniform objects in A. Then U_{1} is equivalent to U_{2} if and only if E(U_{1}) is isomorphic to E(U_{2}).

Proof. [Kra03, Lemma 2]. □

Since every indecomposable injective object in A is uniform ([Ste75, Proposition V.2.8]), the map

\frac{\{ uniform objects in A \}}{∼} → \frac{\{ indecomposable injective objects in A \}}{≅}

induced by the correspondence U ↦ E(U) is bijective. We consider the restriction of this bijection to ASpec A.

Definition 6.2. Let A be a Grothendieck category. For α ∈ ASpec A, define the injective envelope E(α) of α by E(α) = E(H), where H is a monoform object in A satisfying \overline{H} = α.

Proposition 6.1 implies that the isomorphism class of E(α) in Definition 6.2 does not depend on the choice of the representative H.

Definition 6.3. We say that a Grothendieck category A has enough atoms if A satisfies the following conditions.

(1) Every injective object in A has an indecomposable decomposition.

(2) Each indecomposable injective object in A is isomorphic to E(α) for some α ∈ ASpec A.

Note that an indecomposable decomposition of an injective object is unique in the following sense.

Theorem 6.4. Let A be a Grothendieck category, and let I be an injective object with

I ≅ ⨁_{λ∈Λ} I_{λ} ≅ ⨁_{μ∈Λ′} I′_{μ},

where I_{λ} and I′_{μ} are indecomposable for each λ ∈ Λ and μ ∈ Λ′. Then there exists a bijection φ: Λ → Λ′ such that I_{λ} is isomorphic to I′_{φ(λ)} for each λ ∈ Λ.

Proof. This follows from Krull–Remak–Schmidt–Azumaya’s theorem ([Pop73, Theorem 5.1.3]) and the fact that the endomorphism ring of each indecomposable injective object in A is local ([Pop73, Lemma 4.20.3]). □

The following result shows that a Grothendieck category with enough atoms is a generalization of a locally noetherian Grothendieck category.

Proposition 6.5. Every locally noetherian Grothendieck category has enough atoms.

Proof. This follows from Proposition 3.5 (3) and [Ste75, Proposition V.4.5] since every nonzero object in a locally noetherian Grothendieck category has a nonzero noetherian subobject. □

We show that every quotient category of a Grothendieck category with enough atoms has enough atoms.

Proposition 6.6. Let A be a Grothendieck category, and let X be a localizing subcategory of A.

(1) If every injective object in A has an indecomposable decomposition, then every injective object in A/X has an indecomposable decomposition.

(2) If A has enough atoms, then A/X has enough atoms.

Proof. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) Let I′ be an injective object in A/X. By Proposition 4.11 (4), the object G(I′) in A is injective. Hence G(I′) has an indecomposable decomposition

G(I′) = ⨁_{λ∈Λ} I_{λ}.

We obtain

I′ ≅ F G(I′) ≅ ⨁_{λ∈Λ} F (I_{λ}).

By Proposition 4.9 (4), Proposition 4.12 (2) and Proposition 4.12 (4), the object F (I_{λ}) is an indecomposable injective object in A/X for each λ ∈ Λ.

(2) Let I′ be an indecomposable injective object in A/X. Then by Proposition 4.11 (5) and Proposition 4.11 (4), the object G(I′) in A is indecomposable and injective. Hence there exists α ∈ ASpec A such that G(I′) ≅ E_{A}(α). We obtain I′ ≅ F G(I′) ≅ F (E_{A}(α)). Let H be a monoform subobject of E_{A}(α). By Proposition 4.9 (4), the object H is X -torsionfree. By Proposition 4.12 (3), the object I′ has the monoform subobject F (H). This implies that I′ = E(F (H)) = E_{A/X} (α). □

A Grothendieck category A is called locally uniform^{2} if every nonzero object in A has a uniform subobject. It is shown that this holds whenever A has enough atoms.

Proposition 6.7. Let A be a Grothendieck category with enough atoms. Then every nonzero object in A has a monoform subobject. In particular, the Grothendieck category A is locally uniform.

Proof. Let M be a nonzero object in A. Then there exists a family {α_{λ}}_{λ∈Λ} of atoms in A such that

E(M) ≅ ⨁_{λ∈Λ} E(α_{λ}).

2In [Pop73, p. 330], it is called locally coirreducible since a uniform object is called a coirreducible object.

Hence E(M) has a monoform subobject H. Since M is an essential subobject of E(M), the subobject H ∩ M of M is monoform by Proposition 3.5 (1). The last assertion follows from Proposition 3.5 (2). □

The classification of the localizing subcategories by the atom spectrum we mentioned after Proposition 5.3 is generalized to a Grothendieck category with enough atoms.

Theorem 6.8. Let A be a Grothendieck category with enough atoms. There is a bijection

{ localizing subcategories of A } → { localizing subsets of ASpec A }

given by X ↦ ASupp X. Its inverse is given by Φ ↦ ASupp^{−1} Φ.

Proof. By Proposition 5.2 and Proposition 5.3, it suffices to show that ASupp^{−1}(ASupp X) = X for each localizing subcategory X of A. The inclusion X ⊂ ASupp^{−1}(ASupp X) holds obviously. Let M be an object in A which belongs to ASupp^{−1}(ASupp X), and let L be the largest subobject of M which belongs to X. If M/L is nonzero, then by Proposition 6.7, there exists a monoform subobject H of M/L. Since \overline{H} ∈ ASupp M ⊂ ASupp X, there exists a nonzero subobject H′ of H which belongs to X. Let H′ = L′/L ⊂ M/L. Since L and L′/L belongs to X, the subobject L′ of M also belongs to X. This contradicts the maximality of L. Therefore ASupp^{−1}(ASupp X) = X. □

We show that every localizing subcategory is the intersection of some family of prime localizing subcategories.

Corollary 6.9. Let A be a Grothendieck category with enough atoms. For every localizing subcategory X of A,

X = ⋂_{α∈ASpec A\ASupp X} X (α).

Proof. By Proposition 5.10 (1) and Theorem 6.8,

⋂_{α∈ASpec A\ASupp X}

X (α)

= { M ∈ A | α ∉ ASupp M for each α ∈ ASpec A \ ASupp X }

= { M ∈ A | ASupp M ⊂ ASupp X }
= ASupp^{−1}(ASupp X)
= X .  □

Let A be a Grothendieck category and α ∈ ASpec A. It is shown in the proof of [Kan15a, Theorem 2.5] that the injective envelope E(α) has a largest monoform subobject H(α). The object H(α) is called the atomic object corresponding to α. It is straightforward to show that no monoform object in A has a proper essential subobject isomorphic to H(α). The atomic objects correspond to the simple objects in the localizations.

Proposition 6.10. Let A be a Grothendieck category and α ∈ ASpec A. Denote the canonical functor by F_{α}: A → A_{α} and its right adjoint by G_{α}: A_{α} → A. Let S′ be the simple object in A_{α}.

(1) S′ is the atomic object corresponding to the atom \overline{S′} in A_{α}.

(2) G_{α}(S′) is isomorphic to the atomic object H(α).

(3) The ring End_{A}(H(α)) is isomorphic to the skew field End_{A_{α}}(S′).

Proof. (1) It holds that S′ ⊂ H(\overline{S′}) ⊂ E(\overline{S′}) = E(S′). If S′ ⊊ H(\overline{S′}), then by Theorem 5.12, \overline{S′} ∈ ASupp(H(\overline{S′})/S′), and hence there exist a subobject L of H(\overline{S′}) with S′ ⊂ L and a subobject of H(\overline{S′})/L which is isomorphic to S′. This contradicts the monoformness of H(\overline{S′}). Therefore S′ = H(\overline{S′}).

(2) By Theorem 5.4, the object F_{α}(H(α)) is a monoform object in A_{α}, and G_{α}F_{α}(H(α)) is a monoform object in A. By (1), F_{α}(H(α)) ≅ S′. Since H(α) is X (α)-torsionfree, by Proposition 4.9 (3), the canonical morphism H(α) → G_{α}F_{α}(H(α)) is a monomorphism, and H(α) is essential as a subobject of G_{α}F_{α}(H(α)). Therefore the morphism H(α) → G_{α}F_{α}(H(α)) is an isomorphism, and G_{α}(S′) ≅ G_{α}F_{α}(H(α)) ≅ H(α).

(3) By (2) and Proposition 4.9 (2),

End_{A}(H(α)) ≅ End_{A}(G_{α}(S′))
≅ Hom_{A_{α}}(F_{α}G_{α}(S′), S′)
≅ End_{A_{α}}(S′).

This gives a ring isomorphism End_{A}(H(α)) ≅ End_{A_{α}}(S′). □

The skew field End_{A}(H(α)) is called the residue field of α and denoted by k(α).

7. The atom spectra of locally noetherian schemes

In this section, we describe the atom spectrum of the category of quasi-coherent sheaves on a locally noetherian scheme. Let X be a locally noetherian scheme with the underlying topological space |X| and the structure sheaf O_{X}. It is known that the category Mod X of O_{X}-modules and the category QCoh X of quasi-coherent sheaves on X are Grothendieck categories (see [Har66, Theorem II.7.8] and [Con00, Lemma 2.1.7]). For a commutative ring R, we identify QCoh(Spec R) with Mod R.

Proposition 7.1. Let U be an open affine subscheme of X, and let i: U ↪ X be the immersion. Then the functor i_{∗}: Mod U → Mod X and its left adjoint i^{∗}: Mod X → Mod U induce the functor i_{∗}: QCoh U → QCoh X and its left adjoint i^{∗}: QCoh X → QCoh U.

Proof. [Gro60, 0.4.4.3.1] and [Gro60, Proposition I.9.4.2 (i)]. □

In the rest of this paper, every quasi-coherent sheaf M on X is always regarded as an object in QCoh X, not in Mod X. Hence a subobject of M means a quasicoherent subsheaf of M.

For an open affine subscheme U of X with the immersion i: U ↪ X, the functor i^{∗}: QCoh X → QCoh U is also denoted by (−)|_{U}. The category QCoh U is realized as a quotient category of QCoh X through this functor.

Proposition 7.2. Let U be an open affine subscheme of X. Then the functor (−)|_{U}: QCoh X → QCoh U induces an equivalence (QCoh X)/X_{U} \underrightarrow{∼} QCoh U, where X_{U} is a localizing subcategory of QCoh X defined by

X_{U} = { M ∈ QCoh X | M|_{U} = 0 }.

Proof. Let i: U ↪ X be the immersion. Since the counit functor i^{∗}i_{∗} → 1_{QCoh U} is an isomorphism, the functor i_{∗} is fully faithful. The functor i^{∗} is exact. Hence the claim follows from Proposition 4.17. □

For each object M in QCoh X, the subset Supp M of X is defined by

Supp M = { x ∈ X | M_{x} ≠ 0 }.

For each x ∈ X, let j_{x}: Spec O_{X,x} → X be the canonical morphism. Note that j_{x}^{∗} is equal to the localization (−)_{x}: QCoh X → Mod O_{X,x}. The category Mod O_{X,x} is realized as a quotient category of QCoh X through this morphism.

Proposition 7.3. For every x ∈ X, the full subcategory

X (x) := { M ∈ QCoh X | x ∉ Supp M } = { M ∈ QCoh X | M_{x} = 0 }

of QCoh X is a prime localizing subcategory. The functor (−)_{x}: QCoh X → Mod O_{X,x} induces an equivalence (QCoh X)/X (x) \underrightarrow{∼} Mod O_{X,x}.

Proof. Let i: U ↪ X be the immersion of an open affine subscheme with x ∈ U. Then the functor (−)_{x}: QCoh X → Mod O_{X,x} is equal to the composite of (−)|_{U}: QCoh X → QCoh U and (−)_{x}: QCoh U → Mod O_{X,x}. By Proposition 7.2 and Proposition 5.16 (1), these two functors are exact functors with fully faithful right adjoints. Hence we obtain the equivalence by Proposition 4.17. By Proposition 5.16 (2), the localizing subcategory X (x) is prime. □

For each x ∈ X, denote the unique maximal ideal of O_{X,x} by 𝔪_{x}, the residue field of x by k(x) = O_{X,x}/𝔪_{x}, and an injective envelope of k(x) in Mod O_{X,x} by E(x) = E_{O_{X,x}}(k(x)). We state that every injective object in QCoh X is a direct sum of indecomposable injective objects of this form.

Theorem 7.4 (Hartshorne [Har66]). Let X = (|X|, O_{X}) be a locally noetherian scheme.

(1) For every family {I_{λ}}_{λ∈Λ} of injective objects in QCoh X, the direct sum ⨁_{λ∈Λ} I_{λ} is also injective.

(2) Every injective object in QCoh X has an indecomposable decomposition.

(3) There is a bijection

|X| → \frac{\{ indecomposable injective objects in QCoh X \}}{≅}

given by x ↦ j_{x}_{∗}E(x).

Proof. [Con00, Lemma 2.1.5]. □

Remark 7.5. In [Har66, p. 135], it is shown that there exists a locally noetherian scheme X such that the Grothendieck category QCoh X is not locally noetherian. By combining Theorem 7.4 (1) and [Pop73, Theorem 5.8.7], we deduce that QCoh X is not even (categorically) locally finitely generated. On the other hand, the set of coherent sheaves on X generates QCoh X [Gro60, Corollary I.9.4.9]. Consequently, a coherent sheaf on X is not necessarily a finitely generated object in QCoh X.

We give a description of the atom spectrum of QCoh X.

Theorem 7.6. Let X = (|X|, O_{X}) be a locally noetherian scheme.

(1) For each x ∈ X, the set AAss j_{x}_{∗}E(x) consists of one element, say α_{x}. The injective envelope of α_{x} is E(α_{x}) = j_{x}_{∗}E(x). The atomic object is H(α_{x}) = j_{x}_{∗}k(x). The residue field is k(α_{x}) ≅ k(x).

(2) There is a bijection |X| → ASpec(QCoh X) given by x ↦ α_{x}. Moreover, the Grothendieck category QCoh X has enough atoms.

Proof. (1) By Proposition 7.3 and Proposition 5.6 (1),

AAss j_{x}_{∗}E(x) = AAss E(x) = \{\overline{k(x)}\}.

Since j_{x}_{∗}E(x) is an indecomposable injective object by Theorem 7.4 (3), it is an injective envelope of each of its nonzero subobjects. Hence E(α_{x}) = j_{x}_{∗}E(x). By Proposition 6.10 (2), H(α_{x}) = j_{x}_{∗}k(x). By Proposition 6.10 (3), k(α_{x}) ≅ End_{O_{X,x}}(k(x)) ≅ k(x).

(2) The bijection in Theorem 7.4 (3) is the composite of the map

|X| → ASpec(QCoh X)

given by x ↦ α_{x} and the injection

ASpec(QCoh X) → \frac{\{ indecomposable injective objects in QCoh X \}}{≅}

given by α ↦ E(α). Hence these maps are also bijective. By Theorem 7.4 (2), the Grothendieck category QCoh X has enough atoms. □

A subset Φ of X is said to be closed under specialization if for every x ∈ Φ, we have \overline{\{x\}} ⊂ Φ. Atom supports and related notions in QCoh X are described as follows.

Corollary 7.7.

(1) Let M be an object in QCoh X. Then the bijection |X| → ASpec(QCoh X) in Theorem 7.6 (2) restricts to a bijection Supp M → ASupp M.

(2) For each x ∈ X, we have X (α_{x}) = X (x). The canonical functor QCoh X → Mod O_{X,x} induces an equivalence (QCoh X)_{α_{x}} \underrightarrow{∼} Mod O_{X,x}.

(3) For each subset Φ of X, the corresponding subset

{ α_{x} ∈ ASpec(QCoh X) | x ∈ Φ }

of ASpec(QCoh X) is localizing if and only if Φ is closed under specialization.

(4) Let x, y ∈ X. Then α_{x} ≤ α_{y} if and only if y ∈ \overline{\{x\}}.

Proof. (1) For each x ∈ X, by Proposition 7.3 and Proposition 5.6 (2), α_{x} ∈ ASupp M if and only if \overline{k(x)} ∈ ASupp j_{x}^{∗}M. By Proposition 3.9, this is equivalent to 𝔪_{x} ∈ Supp j_{x}^{∗}M, which means M_{x} = j_{x}^{∗}M ≠ 0.

(2) By (1) and Proposition 5.10 (1), X (α_{x}) = X (x). The equivalence follows from Proposition 7.3.

(3) By (2), it suffices to show that Φ is closed under specialization if and only if there exists an object M in QCoh X satisfying Φ = Supp M. For every object M in QCoh X, it is straightforward to show that Supp M is closed under specialization. Assume that Φ is closed under specialization. For each x ∈ Φ, we have Supp j_{x}_{∗}k(x) = \overline{\{x\}}. Hence

Supp ⨁_{x∈Φ} j_{x}_{∗}k(x) = ⋃_{x∈Φ} Supp j_{x}_{∗}k(x) = Φ.

(4) This follows from (3). □

We specialize Theorem 6.8 to the case of QCoh X. For a full subcategory X of QCoh X, define the specialization-closed subset Supp X of X by

Supp X = ⋃_{M∈X} Supp M.

For a subset Φ of X, define the localizing subcategory Supp^{−1} Φ of QCoh X by

Supp^{−1} Φ = { M ∈ QCoh X | Supp M ⊂ Φ }.

Theorem 7.8. Let X be a locally noetherian scheme. There is a bijection

{ localizing subcategories of QCoh X } → { specialization-closed subsets of X }

given by X ↦ Supp X. Its inverse is given by Φ ↦ Supp^{−1} Φ.

Proof. In Theorem 7.6 (2), we showed that the Grothendieck category QCoh X has enough atoms and described ASpec(QCoh X). Hence the claim follows from Theorem 6.8 and Corollary 7.7 (3). □

Definition 7.9. Let X be a locally noetherian scheme, and let M be an object in QCoh X. The subset Ass M of X is defined by

Ass M = { x ∈ X | 𝔪_{x} ∈ Ass_{O_{X,x}} M_{x} }.

Each element of Ass M is called an associated point of M.

In order to show that associated atoms are generalizations of associated points defined in Definition 7.9, we need the following results.

Proposition 7.10. Let Spec R be an open affine subscheme of X, and let i: Spec R ↪ X be the immersion. For every R-module M, we have Ass i_{∗}M = i(Ass_{R} M).

Proof. [Gro65, Proposition 3.1.13] and [Gro65, Proposition 3.1.2]. □

Lemma 7.11. For each x ∈ X, we have Ass j_{x}_{∗}E(x) = {x} and Supp j_{x}_{∗}E(x) = \overline{\{x\}}.

Proof. Let i: Spec R ↪ X be the immersion of an open affine subscheme such that x = i(𝔭) for some 𝔭 ∈ Spec R. Then the morphism j_{x} is the composite of j: Spec O_{X,x} ≅ Spec R_{𝔭} → Spec R and i: Spec R ↪ X. By [Mat89, Theorem 18.4 (vi)], j_{∗}E(x) = E_{R}(R/𝔭). By Proposition 7.10,

Ass j_{x}_{∗}E(x) = Ass i_{∗}E_{R}(\frac{R}{𝔭}) = i(Ass_{R} E_{R}(\frac{R}{𝔭})) = i(\{𝔭\}) = \{x\}.

By the argument in [Mat89, p. 150], for each 𝔮 ∈ Spec R, we have E_{R}(R/𝔭)_{𝔮} = E_{R_{𝔮}}((R/𝔭)_{𝔮}). Hence we obtain

Supp E_{R}(\frac{R}{𝔭}) = \{ 𝔮 ∈ Spec R | 𝔭 ⊂ 𝔮 \}

and

Supp j_{x}_{∗}E(x) = Supp i_{∗}(E_{R}(\frac{R}{𝔭})) = \overline{\{x\}}.  □

Proposition 7.12. Let M be an object in QCoh X. Then the bijection |X| → ASpec(QCoh X) in Theorem 7.6 (2) restricts to a bijection Ass M → AAss M.

Proof. Assume that α_{x} ∈ AAss M, and let i: U ↪ X be the immersion of an open affine subscheme with x ∈ U. By Proposition 7.2 and Proposition 5.6 (2), α_{x} ∈ AAss i^{∗}M. By Proposition 3.9 and Proposition 7.10, we obtain x ∈ Ass i_{∗}i^{∗}M. Since the canonical morphism M → i_{∗}i^{∗}M induces an isomorphism M_{x} \underrightarrow{∼} (i_{∗}i^{∗}M)_{x}, we deduce that x ∈ Ass M. Conversely, assume that x ∈ Ass M. By Theorem 7.4 (2) and Theorem 7.4 (3), there exists a family {x_{λ}}_{λ∈Λ} of points in X such that

E(M) ≅ ⨁_{λ∈Λ} j_{x_{λ}}_{∗}E(x_{λ}).

By [Gro65, Proposition 3.1.7],

x ∈ Ass M ⊂ Ass E(M) = ⋃_{λ∈Λ} Ass j_{x_{λ}}_{∗}E(x_{λ}).

Hence there exists λ ∈ Λ such that x ∈ Ass j_{x_{λ}}_{∗}E(x_{λ}). By Lemma 7.11, x_{λ} = x. By Proposition 3.12 (2), we deduce that

α_{x} ∈ AAss j_{x}_{∗}E(x) ⊂ AAss E(M) = AAss M.  □

8. Localization of prelocalizing subcategories and localizing subcategories

In order to classify the prelocalizing subcategories of QCoh X for a locally noetherian scheme X, we show that they are determined by their restrictions to open affine subschemes of X. In this section, we prove this claim in a categorical setting (Setting 8.3). We start with two lemmas, which show the setting includes the case of QCoh X.

Lemma 8.1. Let X be a locally noetherian scheme, and let M be an object in QCoh X. Then for each y ∈ Supp M, there exists x ∈ Ass M with y ∈ \overline{\{x\}}.

Proof. By Theorem 7.4 (2) and Theorem 7.4 (3), there exists a family {x_{λ}}_{λ∈Λ} of points in X such that

E(M) ≅ ⨁_{λ∈Λ} j_{x_{λ}}_{∗}E(x_{λ}).

Then

y ∈ Supp M ⊂ Supp E(M) = ⋃_{λ∈Λ} Supp j_{x_{λ}}_{∗}E(x_{λ}).

By Lemma 7.11, y ∈ Supp j_{x_{λ}}_{∗}E(x_{λ}) = \overline{\{x_{λ}\}} for some λ ∈ Λ. By Proposition 7.12 and Proposition 3.12 (2), we obtain

Ass M = Ass E(M) = ⋃_{λ∈Λ} Ass j_{x_{λ}}_{∗}E(x_{λ}) = { x_{λ} | λ ∈ Λ }.

Therefore the claim follows. □

Lemma 8.2. Let R be a commutative ring, and let S be a multiplicatively closed subset of R. Let M be an R-module. Then the R-module M_{S} is a quotient object of the direct sum of some copies of M. In particular, for every 𝔭 ∈ Spec R, the R-module M_{𝔭} belongs to the prelocalizing subcategory ⟨M⟩_{preloc} of Mod R.

Proof. For each s ∈ S, the image of the R-homomorphism M → M_{S} given by x ↦ xs^{−1} is Ms^{−1}. Hence the R-submodule Ms^{−1} of M_{S} is a quotient R-module of M. Since

⨁_{s∈S} Ms^{−1} ↠ ∑_{s∈S} Ms^{−1} = M_{S},

the claim follows. □

In the rest of this section, we investigate localizations of prelocalizing subcategories in the following setting.

Setting 8.3. Let A be a Grothendieck category with enough atoms, and let {X_{λ}}_{λ∈Λ} be a family of localizing subcategories of A. For each λ ∈ Λ, let U_{λ} = A/X_{λ}. Denote the canonical functors and their right adjoints by

• F_{λ}: A → U_{λ} and G_{λ}: U_{λ} → A for each λ ∈ Λ,

• F_{μ}^{λ}: U_{λ} → U_{μ} and G_{μ}^{λ}: U_{μ} → U_{λ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ},
• F_{α}: A → A_{α} and G_{α}: A_{α} → A for each α ∈ ASpec A,

• F_{α}^{λ}: U_{λ} → (U_{λ})_{α} and G_{α}^{λ}: (U_{λ})_{α} → U_{λ} for each λ ∈ Λ and α ∈

ASpec U_{λ}. (Note that (U_{λ})_{α} = A_{α}.)

We assume the following properties.

(1) It holds that

ASpec A = ⋃_{λ∈Λ} ASpec U_{λ}.

Moreover, for each λ_{1}, λ_{2} ∈ Λ and α ∈ ASpec U_{λ_{1}} ∩ ASpec U_{λ_{2}}, there exists μ ∈ Λ such that

α ∈ ASpec U_{μ} ⊂ ASpec U_{λ_{1}} ∩ ASpec U_{λ_{2}}.

In other words, the family {ASpec U_{λ}}_{λ∈Λ} satisfies the axiom of open basis of ASpec A.^{3}

(2) For each object M in A and β ∈ ASupp M, there exists α ∈ AAss M with α ≤ β.

(3) Let λ ∈ Λ, and let M′ be an object in U_{λ} and α ∈ ASpec U_{λ}. Then the object G_{α}^{λ}F_{α}^{λ}(M′) belongs to ⟨M′⟩_{preloc}.

For a locally noetherian scheme X, let {U_{λ}}_{λ∈Λ} be an open affine basis of X (that is, an open basis of X consisting of affine subsets). Then Lemma 8.1 and Lemma 8.2 show that the Grothendieck category QCoh X together with {QCoh U_{λ}}_{λ∈Λ} satisfies the conditions in Setting 8.3. We assume Setting 8.3 in the rest of this section. We show that every quotient category of A also satisfies the same conditions.

Proposition 8.4. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. Then the Grothendieck category A/X together with the family {⟨F (X_{λ})⟩_{loc}}_{λ∈Λ} of localizing subcategories of A/X also satisfies the conditions in Setting 8.3. In particular, for every α ∈ ASpec A, the Grothendieck category A_{α} together with {⟨(X_{λ})_{α}⟩_{loc}}_{λ∈Λ} satisfies the conditions in Setting 8.3.^{4}

Proof. By Proposition 6.6 (2), the Grothendieck category A/X has enough atoms.

(1) By Proposition 5.7,

ASpec \frac{A/X}{⟨F (X_{λ})⟩_{loc}} = ASpec \frac{A}{X_{λ}} ∩ ASpec \frac{A}{X} .

(2) Let M′ be an object in A/X, and let β ∈ ASupp M′. By Proposition 5.6 (1), β ∈ ASupp G(M′). Hence there exists α ∈ AAss G(M′) = AAss M′ with α ≤ β.

3However, we regard ASpec A as a topological space only by the localizing topology. (See Proposition 3.14.)

4It is shown in Proposition 8.15 (3) that ⟨F (X_{λ})⟩_{loc} = F (X_{λ}). In particular, ⟨(X_{λ})_{α}⟩_{loc} = (X_{λ})_{α}.

(3) Let λ ∈ Λ. By Proposition 4.18 (1) and Proposition 4.18 (3),

\frac{A/X}{⟨F (X_{λ})⟩_{loc}} ≅ \frac{A}{⟨X_{λ} ∪ X ⟩_{loc}} ≅ \frac{A/X_{λ}}{⟨F_{λ}(X)⟩_{loc}} .

Let U′_{λ} := U_{λ}/⟨F_{λ}(X)⟩_{loc}. Denote the canonical functors by F′: U_{λ} → U′_{λ}, F′_{α}: U′_{λ} → A_{α}, and their right adjoints by G′: U′_{λ} → U_{λ}, G′_{α}: A_{α} → U′_{λ}, respectively. Let M′′ be an object in U′_{λ}, and let α ∈ ASpec U′_{λ}. Then by the assumption, the object G_{α}^{λ}F_{α}^{λ}G′(M′′) belongs to ⟨G′(M′′)⟩_{preloc}. Since F′ is exact, the object F′G_{α}^{λ}F_{α}^{λ}G′(M′′) belongs to ⟨F′G′(M′′)⟩_{preloc} = ⟨M′′⟩_{preloc}. Since

F′G_{α}^{λ}F_{α}^{λ}G′(M′′) ≅ F′G′G′_{α}F′_{α}F′G′(M′′) ≅ G′_{α}F′_{α}(M′′),

the claim follows. □

Under the assumptions of Setting 8.3, we can show a complemental fact on associated atoms in a quotient category.

Lemma 8.5. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. For every object M in A,

AAss F (M) = AAss M \ ASupp X .

In particular, for every α ∈ ASpec A,

AAss M_{α} = AAss M ∩ Λ(α).

Proof. By Proposition 5.6,

AAss GF (M) = AAss F (M) ⊃ AAss M \ ASupp X .

Let η: 1_{A} → GF be the unit morphism and β ∈ AAss GF (M). Note that β ∉ ASupp X. By Proposition 4.9 (3), the subobject L := Ker η_{M} of M belongs to X, and Im η_{M} is an essential subobject of GF (M). By Proposition 3.12 (2), β ∈ AAss(Im η_{M}) = AAss(M/L). Hence there exists a subobject L′ of M with L ⊂ L′ such that L′/L is a monoform object representing β. Since β ∈ ASupp L′, by Setting 8.3 (2), there exists α ∈ AAss L′ with α ≤ β. Since β ∉ ASupp X, it holds that α ∉ ASupp L by Proposition 3.17. By Proposition 3.10 and Proposition 3.12 (1), α ∈ AAss(L′/L) = {β}. Therefore β = α ∈ AAss L′ ⊂ AAss M. □

We show two lemmas as parts of the proof of Theorem 8.8. It is useful to determine whether an object belongs to a given prelocalizing subcategory.

Lemma 8.6. Let λ ∈ Λ, and let Y′ be a prelocalizing subcategory of U_{λ}. Let U′ be a uniform object in U_{λ} with AAss U′ = {α}. If U′_{α} belongs to Y′_{α}, then U′ belongs to Y′.

Proof. There exists an object N′ in U_{λ} which belongs to Y′ such that U′_{α} ≅ N′_{α}. By Setting 8.3 (3), the object G_{α}^{λ}F_{α}^{λ}(U′) ≅ G_{α}^{λ}F_{α}^{λ}(N′) belongs to Y′. Let η: 1_{U_{λ}} → G_{α}^{λ}F_{α}^{λ} be the unit morphism. Then by Proposition 4.9 (3), α ∉ ASupp(Ker η_{U′}). If Ker η_{U′} ≠ 0, then by Proposition 3.12 (2), α ∈ AAss(Ker η_{U′}) ⊂ ASupp(Ker η_{U′}). This is a contradiction. Hence η_{U′} is a monomorphism. The object U′ belongs to Y′. □

Lemma 8.7. Let Y be a prelocalizing subcategory of A, and let U be a uniform object in A with AAss U = {α}. If U_{α} belongs to Y_{α}, then U belongs to Y.

Proof. Let L be the largest subobject of U which belongs to Y. Assume that L ⊊ U. Then by Proposition 6.7, there exists β ∈ AAss(U/L). By Setting 8.3 (2), α ≤ β. By Setting 8.3 (1), there exists λ ∈ Λ such that β ∈ ASpec U_{λ} = ASpec A \ ASupp X_{λ}. Then by Proposition 3.17, we also have α ∈ ASpec U_{λ}. By a similar argument to that in the proof of Lemma 8.6, the canonical morphism U → G_{λ}F_{λ}(U) is a monomorphism, and U is X_{λ}-torsionfree. By Proposition 4.12 (2), the object F_{λ}(U) is uniform, and AAss F_{λ}(U) = {α} by Proposition 3.12 (1). Since F_{λ}(U)_{α} = U_{α} belongs to Y_{α} = F_{λ}(Y)_{α}, by Lemma 8.6, the object F_{λ}(U) belongs to F_{λ}(Y). We obtain an object N in A which belongs to Y such that F_{λ}(N) ≅ F_{λ}(U). Let V be the image of the composite of the canonical morphism N → G_{λ}F_{λ}(N) and G_{λ}F_{λ}(N) \underrightarrow{∼} G_{λ}F_{λ}(U). By Proposition 4.9 (3), the object G_{λ}F_{λ}(U)/V belongs to X_{λ}. Hence

\frac{G_{λ}F_{λ}(U)}{U ∩ V} ↪ \frac{G_{λ}F_{λ}(U)}{U} ⊕ \frac{G_{λ}F_{λ}(U)}{V} ∈ X_{λ}.

Since U ∩ V belongs to Y, we have U ∩ V ⊂ L by the maximality of L. Hence U/L also belongs to X_{λ}, and β ∈ ASupp(U/L) ⊂ ASupp X_{λ}. This is a contradiction. Therefore L = U. □

Theorem 8.8. Assume Setting 8.3. Let Y be a prelocalizing subcategory of A, and let M be an object in A. If M_{α} belongs to Y_{α} for every α ∈ AAss M, then M belongs to Y.

Proof. Since A has enough atoms, there exists a family {α_{ω}}_{ω∈Ω} of elements of ASpec A such that

E(M) ≅ ⨁_{ω∈Ω} E(α_{ω}).

Let Z = ⟨M⟩_{preloc}. For each ω ∈ Ω, let L_{ω} be the largest subobject of E(α_{ω}) which belongs to Z. Then by Proposition 4.5, M ⊂ ⨁_{ω∈Ω} L_{ω}. Since L_{ω} is uniform for each ω ∈ Ω, by Proposition 3.12,

{α_{ω}} = AAss L_{ω} ⊂ AAss E(M) = AAss M.

By Proposition 4.13 (2), it is straightforward to show that Z_{α_{ω}} = ⟨M_{α_{ω}}⟩_{preloc}. Hence by the assumption, Z_{α_{ω}} ⊂ Y_{α_{ω}}. Since L_{ω} belongs to Z, the object (L_{ω})_{α_{ω}} belongs to Y_{α_{ω}}. By Lemma 8.7, we deduce that L_{ω} belongs to Y. Therefore the subobject M of ⨁_{ω∈Ω} L_{ω} also belongs to Y. □

The following results are consequences of Theorem 8.8.

Proposition 8.9. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A. Then for every object M in A, the object GF (M) belongs to ⟨M⟩_{preloc}.

Proof. Let η: 1_{A} → GF be the unit morphism. Let α ∈ AAss GF (M). By Proposition 5.6 (1),

α ∈ AAss GF (M) = AAss F (M) ⊂ ASpec \frac{A}{X} = ASpec A \ ASupp X .

By Proposition 4.9 (3), the objects Ker η_{M} and Cok η_{M} belong to X. By applying (−)_{α} to the exact sequence

0 → Ker η_{M} → M → GF (M) → Cok η_{M} → 0,

we obtain the isomorphism M_{α} \underrightarrow{∼} GF (M)_{α}. Hence GF (M)_{α} belongs to (⟨M⟩_{preloc})_{α}. By Theorem 8.8, we deduce that GF (M) belongs to ⟨M⟩_{preloc}.

□

Proposition 8.10. Let Y be a prelocalizing subcategory of A and α ∈ ASpec A. Then α ∈ ASupp Y if and only if H(α) belongs to Y.

Proof. If H(α) belongs to Y, then α = \overline{H(α)} ∈ ASupp Y. Assume α ∈ ASupp Y. Then there exists a monoform object H in A with \overline{H} = α such that H belongs to Y. By Proposition 8.9, the object G_{α}F_{α}(H) belongs to Y. By the proof of Proposition 6.10 (2), the object G_{α}F_{α}(H) is isomorphic to H(α). □

We show the main result in this section.

Theorem 8.11. Assume Setting 8.3. Then there exist bijections between the following sets.

(1) The set of prelocalizing subcategories of A.

(2) The set of families {Y_{λ} ⊂ U_{λ}}_{λ∈Λ} of prelocalizing subcategories such that F_{μ}^{λ}(Y_{λ}) = Y_{μ} for each λ, μ ∈ Λ with ASpec U_{μ} ⊂ ASpec U_{λ}.

(3) The set of families {Y(α) ⊂ A_{α}}_{α∈ASpec A} of prelocalizing subcategories such that Y(β)_{α} = Y(α) for each α, β ∈ ASpec A with α ≤ β.

The correspondences are given as follows.

(1) Y ↦ \begin{cases}(2) \{F_{λ}(Y)\}_{λ∈Λ} \\ (3) \{Y_{α}\}_{α∈ASpec A}, \end{cases}

(2) \{Y_{λ}\}_{λ∈Λ} ↦ \begin{cases}(1) ⋂_{λ∈Λ} F_{λ}^{−1}(Y_{λ}) \\ (3) \{Y(α)\}_{α∈ASpec A}, \\ where Y(α) = (Y_{λ})_{α} for λ ∈ Λ with α ∈ ASpec U_{λ}, \end{cases}

(3) \{Y(α)\}_{α∈ASpec A} ↦ \begin{cases}(1) ⋂_{α∈ASpec A} F_{α}^{−1}(Y(α)) \\ (2) \{ ⋂_{α∈ASpec U_{λ}}(F_{α}^{λ})^{−1}(Y(α))\}_{λ∈Λ}. \end{cases}

Proof. ((1)↔(2)) Let Y be a prelocalizing subcategory of A. It is obvious that Y ⊂ ⋂_{λ∈Λ} F_{λ}^{−1}F_{λ}(Y). Let M be an object in A which belongs to ⋂_{λ∈Λ} F_{λ}^{−1}F_{λ}(Y). For each α ∈ AAss M, by Setting 8.3 (1), there exists λ ∈ Λ such that α ∈ ASpec U_{λ}. Then

M_{α} = F_{λ}(M)_{α} ∈ F_{λ}(Y)_{α} = Y_{α}.

By Theorem 8.8, the object M belongs to Y. We obtain Y = ⋂_{λ∈Λ} F_{λ}^{−1}F_{λ}(Y). Let {Y_{λ}}_{λ∈Λ} be an element of (2) and Y := ⋂_{λ∈Λ} F_{λ}^{−1}(Y_{λ}). It is obvious that F_{λ}(Y) ⊂ Y_{λ}. Let M′ be an object in U_{λ} which belongs to Y_{λ}. We show that F_{λ′}G_{λ}(M′) belongs to Y_{λ′} for each λ′ ∈ Λ. For each β ∈ AAss F_{λ′}G_{λ}(M′), by Lemma 8.5 and Proposition 5.6 (1),

β ∈ AAss M′ ∩ ASpec U_{λ′} ⊂ ASpec U_{λ} ∩ ASpec U_{λ′}.

Hence there exists μ ∈ Λ such that

β ∈ ASpec U_{μ} ⊂ ASpec U_{λ} ∩ ASpec U_{λ′}.

Since the object

F_{λ′}G_{λ}(M′)_{β} = G_{λ}(M′)_{β} = F_{λ}G_{λ}(M′)_{β} = M′_{β}

belongs to

(Y_{λ})_{β} = F_{μ}^{λ}(Y_{λ})_{β} = (Y_{μ})_{β} = F_{μ}^{λ′}(Y_{λ′})_{β} = (Y_{λ′})_{β},

by Proposition 8.4 and Theorem 8.8, the object F_{λ′}G_{λ}(M′) belongs to Y_{λ′}. Hence G_{λ}(M′) belongs to Y, and M′ ≅ F_{λ}G_{λ}(M′) belongs to F_{λ}(Y). We obtain F_{λ}(Y) = Y_{λ}. (Well-definedness of (2)→(3)) Let {Y_{λ}}_{λ∈Λ} be an element of (2) and α ∈ ASpec A. Let λ_{1}, λ_{2} ∈ Λ such that α ∈ ASpec U_{λ_{i}} for each i = 1, 2. Then by Setting 8.3 (1), there exists μ ∈ Λ such that

α ∈ ASpec U_{μ} ⊂ ASpec U_{λ_{1}} ∩ ASpec U_{λ_{2}}.

Hence

(Y_{λ_{1}})_{α} = F_{μ}^{λ_{1}}(Y_{λ_{1}})_{α} = (Y_{μ})_{α} = F_{μ}^{λ_{2}}(Y_{λ_{2}})_{α} = (Y_{λ_{2}})_{α}.

((2)↔(3)) Let {Y_{λ}}_{λ∈Λ} be an element of (2). For each λ ∈ Λ, let

Ỹ_{λ} := ⋂_{α∈ASpec U_{λ}}(F_{α}^{λ})^{−1}F_{α}^{λ}(Y_{λ}).

Then Y_{λ} ⊂ Ỹ_{λ}. Let M′ be an object in U_{λ} which belongs to Ỹ_{λ}. For each α ∈ AAss M′, we have M′_{α} ∈ (Y_{λ})_{α}. Hence by Proposition 8.4 and Theorem 8.8, the object M′ belongs to Y_{λ}, and we obtain Y_{λ} = Ỹ_{λ}. Let {Y(α)}_{α∈ASpec A} be an element of (3). For each λ ∈ Λ, let

Y_{λ} := ⋂_{α∈ASpec U_{λ}}(F_{α}^{λ})^{−1}(Y(α)).

For each α ∈ ASpec A, by Setting 8.3 (1), there exists μ ∈ Λ such that α ∈ ASpec U_{μ}. It is obvious that F_{α}^{μ}(Y_{λ}) ⊂ Y(α). Let M′′ be an object in A_{α} which belongs to Y(α). We show that F_{β}^{λ}G_{α}^{λ}(M′′) belongs to Y(β) for each β ∈ ASpec U_{λ}. For each γ ∈ AAss F_{β}^{λ}G_{α}^{λ}(M′′), by Lemma 8.5 and Proposition 5.6 (1),

γ ∈ AAss M′′ ∩ Λ(β) ⊂ Λ(α) ∩ Λ(β).

Since the object

F_{β}^{λ}G_{α}^{λ}(M′′)_{γ} = G_{α}^{λ}(M′′)_{γ} = F_{α}^{λ}G_{α}^{λ}(M′′)_{γ} = M′′_{γ}

belongs to

Y(α)_{γ} = Y(γ) = Y(β)_{γ},

the object F_{β}^{λ}G_{α}^{λ}(M′′) belongs to Y(β). Hence G_{α}^{λ}(M′′) belongs to Y_{λ}, and M′′ ≅ F_{α}^{λ}G_{α}^{λ}(M′′) ∈ F_{α}^{λ}(Y_{λ}). We obtain F_{β}^{μ}(Y_{λ}) = Y(β). □

For a family {Y^{ω}}_{ω∈Ω} of prelocalizing subcategories of A, we can consider the smallest prelocalizing subcategory ⟨⋃_{ω∈Ω} Y^{ω}⟩_{preloc} containing Y^{ω} for every ω ∈ Ω and the intersection ⋂_{ω∈Ω} Y^{ω}. These are described in terms of prelocalizing subcategories of quotient categories in the following ways.

Proposition 8.12. Assume that the following elements correspond to each other by the bijections in Theorem 8.11 for each ω ∈ Ω.

(1) Y^{ω}.

(2) {Y_{λ}^{ω}}_{λ∈Λ}.
(3) {Y^{ω}(α)}_{α∈ASpec A}.

Then the following elements correspond to each other by the bijections. (1) ⟨⋃_{ω∈Ω} Y^{ω}⟩_{preloc}.

(2) {⟨⋃_{ω∈Ω} Y_{λ}^{ω}⟩_{preloc}}_{λ∈Λ}.
(3) {⟨⋃_{ω∈Ω} Y^{ω}(α)⟩_{preloc}}_{α∈ASpec A}.

Proof. For each λ ∈ Λ,

F_{λ}(⟨ ⋃_{ω∈Ω} Y^{ω}⟩_{preloc}) = ⟨F_{λ}( ⋃_{ω∈Ω} Y^{ω})⟩_{preloc}

= ⟨ ⋃_{ω∈Ω} F_{λ}(Y^{ω})⟩_{preloc}

= ⟨ ⋃_{ω∈Ω} Y_{λ}^{ω}⟩_{preloc}.

It is shown similarly that (⟨⋃_{ω∈Ω} Y^{ω}⟩_{preloc})_{α} = ⟨⋃_{ω∈Ω} Y^{ω}(α)⟩_{preloc} for each α ∈ ASpec A. □

Proposition 8.13. Assume that the following elements correspond to each other by the bijections in Theorem 8.11 for each ω ∈ Ω.

(1) Y^{ω}.

(2) {Y_{λ}^{ω}}_{λ∈Λ}.
(3) {Y^{ω}(α)}_{α∈ASpec A}.

Then the following elements correspond to each other by the bijections.

(1) ⋂_{ω∈Ω} Y^{ω}.
(2) {⋂_{ω∈Ω} Y_{λ}^{ω}}_{λ∈Λ}.
(3) {⋂_{ω∈Ω} Y^{ω}(α)}_{α∈ASpec A}.

Proof. Let λ ∈ Λ. It is obvious that F_{λ}(⋂_{ω∈Ω} Y^{ω}) ⊂ ⋂_{ω∈Ω} F_{λ}(Y^{ω}) = ⋂_{ω∈Ω} Y_{λ}^{ω}. Let M′ be an object in U_{λ} which belongs to ⋂_{ω∈Ω} Y_{λ}^{ω}. Then for each ω ∈ Ω, there exists an object M_{ω} in A which belongs to Y^{ω} such that F_{λ}(M_{ω}) ≅ M′. By Proposition 8.9, the object G_{λ}(M′) ≅ G_{λ}F_{λ}(M_{ω}) belongs to ⟨M_{ω}⟩_{preloc}. Hence G_{λ}(M′) belongs to ⋂_{ω∈Ω} Y^{ω}, and M′ ≅ F_{λ}G_{λ}(M′) ∈ F_{λ}(⋂_{ω∈Ω} Y^{ω}). This shows that F_{λ}(⋂_{ω∈Ω} Y^{ω}) = ⋂_{ω∈Ω} Y_{λ}^{ω}. It is shown similarly that (⋂_{ω∈Ω} Y^{ω})_{α} = ⋂_{ω∈Ω} Y^{ω}(α) for each α ∈ ASpec A.

□

Families in Theorem 8.11 (3) have the following characterization.

Proposition 8.14. For each family {Y(α) ⊂ A_{α}}_{α∈ASpec A} of prelocalizing subcategories, the following assertions are equivalent.

(1) There exists a prelocalizing subcategory Y of A satisfying Y_{α} = Y(α) for each α ∈ ASpec A.

(2) For each α ∈ ASpec A, there exist λ ∈ Λ with α ∈ ASpec U_{λ} and a prelocalizing subcategory Y′ of U_{λ} satisfying Y′_{β} = Y(β) for each β ∈ ASpec U_{λ}.

(3) For each α, β ∈ ASpec A with α ≤ β, it holds that Y(β)_{α} = Y(α).

Proof. This can be shown straightforwardly by using Theorem 8.11. □

In order to investigate the localizing subcategories of QCoh X, we improve Proposition 4.13 under the assumptions of Setting 8.3.

Proposition 8.15. Let X be a localizing subcategory of A. Denote the canonical functor by F: A → A/X and its right adjoint by G: A/X → A.

(1) Let Y be a prelocalizing subcategory of A. Then Y ∗ X ⊂ X ∗ Y.

(2) Let Y_{1} and Y_{2} be prelocalizing subcategories of A. Then

F (Y_{1} ∗ Y_{2}) = F (Y_{1}) ∗ F (Y_{2}).

(3) Let Y be a localizing subcategory of A. Then F (Y) is a localizing subcategory of A/X.

Proof. (1) Let M be an object in A which belongs to Y ∗ X. Then there exists an exact sequence

0 → L → M → N → 0

in A, where L belongs to Y, and N belongs to X. Since F (L) ≅ F (M), the object GF (M) ≅ GF (L) belongs to Y by Proposition 8.9. Let η: 1_{A} → GF be the unit morphism. There is an exact sequence

0 → Ker η_{M} → M → Im η_{M} → 0.

By Proposition 4.9 (3), the object Ker η_{M} belongs to X. The subobject Im η_{M} of GF (M) belongs to Y. Therefore M belongs to X ∗ Y.

(2) By Proposition 4.13 (3),

F (Y_{1} ∗ Y_{2}) ⊂ F (Y_{1} ∗ X ∗ Y_{2}) = F (Y_{1}) ∗ F (Y_{2}).

By (1),

F (Y_{1} ∗ X ∗ Y_{2}) ⊂ F (X ∗ Y_{1} ∗ Y_{2}) ⊂ F (X) ∗ F (Y_{1} ∗ Y_{2}) = F (Y_{1} ∗ Y_{2}).

(3) By (2),

F (Y) ∗ F (Y) = F (Y ∗ Y) = F (Y).  □

Theorem 8.16. Assume that the following elements correspond to each other by the bijections in Theorem 8.11 for each i = 1, 2.

(1) Y^{i}.

(2) {Y_{λ}^{i}}_{λ∈Λ}.
(3) {Y^{i}(α)}_{α∈ASpec A}.

Then the following elements correspond to each other by the bijections. (1) Y^{1} ∗ Y^{2}.

(2) {Y_{λ}^{1} ∗ Y_{λ}^{2}}_{λ∈Λ}.
(3) {Y^{1}(α) ∗ Y^{2}(α)}_{α∈ASpec A}.

Proof. This follows from Proposition 8.15. □

Corollary 8.17. The bijections in Theorem 8.11 restrict to bijections between following sets.

(1) The set of localizing subcategories of A.

(2) The set of families {Y_{λ} ⊂ U_{λ}}_{λ∈Λ} of localizing subcategories such that F_{μ}^{λ}(Y_{λ}) = Y_{μ} for each λ, μ ∈ Λ with ASpec U_{μ} ⊂ ASpec U_{λ}.

(3) The set of families {Y(α) ⊂ A_{α}}_{α∈ASpec A} of localizing subcategories such that Y(β)_{α} = Y(α) for each α, β ∈ ASpec A with α ≤ β.

Proof. This follows from Theorem 8.16. □

Prime localizing subcategories of A are characterized as follows.

Theorem 8.18. Assume Setting 8.3, and let X be a localizing subcategory of A. Then the following assertions are equivalent.

(1) X is a prime localizing subcategory of A.

(2) There exists α ∈ ASpec A such that X = X (α).

(3) For each family {X_{ω}}_{ω∈Ω} of localizing subcategories of A satisfying X = ⋂_{ω∈Ω} X_{ω}, there exists ω ∈ Ω such that X = X_{ω}.

(4) For each family {Y_{ω}}_{ω∈Ω} of prelocalizing subcategories of A satisfying X = ⋂_{ω∈Ω} Y_{ω}, there exists ω ∈ Ω such that X = Y_{ω}.

Proof. The equivalence (1)⇔(2) follows from Theorem 5.14. Let {Y_{ω}}_{ω∈Ω} be a family of prelocalizing subcategories of A satisfying X (α) = ⋂_{ω∈Ω} Y_{ω}. Since H(α) does not belong to X (α), there exists ω ∈ Ω such that H(α) does not belong to Y_{ω}. By Proposition 8.10, α ∉ ASupp Y_{ω}, and hence Y_{ω} ⊂ X (α). This shows (2)⇒(4).

The implication (4)⇒(3) is obvious. The implication (3)⇒(2) follows from Corollary 6.9. □

9. Classification of prelocalizing subcategories

Let X be a locally noetherian scheme with the structure sheaf O_{X}. In this section, we classify the prelocalizing subcategories of QCoh X. Let {U_{λ}}_{λ∈Λ} be an open affine basis of X. Let i_{λ}: U_{λ} ↪ X be the immersion for each λ ∈ Λ, and let i_{λ,μ}: U_{μ} ↪ U_{λ} be the immersion for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}. We recall the notion of a filter. This is an essential tool to classify prelocalizing subcategories.

Definition 9.1. Let A be a Grothendieck category, and let M be an object in A.

(1) A filter of subobjects of M in A is a set F of subobjects of M satisfying the following conditions.

(a) M ∈ F.

(b) If L ⊂ L′ are subobjects of M with L ∈ F, then L′ ∈ F.

(c) If L_{1}, L_{2} ∈ F, then L_{1} ∩ L_{2} ∈ F.

If there is no danger of confusion, we simply say that F is a filter of M.

(2) For each subobject L of M, denote by F(L) the filter consisting of all subobjects L′ of M with L ⊂ L′. A filter of the form F(L) is called a principal filter.

Remark 9.2. In Definition 9.1 (2), the principal filter F(L) is closed under arbitrary intersection. Conversely, if a filter F of M is closed under arbitrary intersection, then F = F(L), where L is the smallest element of F. It is obvious that the map

{ subobjects of M } → { principal filters of M }

given by L ↦ F(L) is bijective.

For a ring Λ, we say that a filter F of right ideals of Λ is prelocalizing if for each L ∈ F and a ∈ Λ, the right ideal

a^{−1}L = { b ∈ Λ | ab ∈ L }

of Λ belongs to F. For a ring Λ, Gabriel [Gab62] gave a classification of the prelocalizing subcategories of Mod Λ.

Theorem 9.3 ([Gab62, Lemma V.2.1]). Let Λ be a ring. There is a bijection

{ prelocalizing subcategories of Mod Λ }

→ { prelocalizing filters of right ideals of Λ }

given by

Y ↦ \{ L ⊂ Λ in Mod Λ | \frac{Λ}{L} ∈ Y \}.

Its inverse is given by

F ↦ { M ∈ Mod Λ | Ann_{Λ}(x) ∈ F for every x ∈ M }

= ⟨ \frac{Λ}{L} ∈ Mod Λ | L ∈ F ⟩_{preloc}.

Proof. [Pop73, Theorem 4.9.1]. □

For a commutative ring R, every filter F of R is prelocalizing. Indeed, for L ∈ F and a ∈ R, we have L ⊂ a^{−1}L, and hence a^{−1}L ∈ F. Therefore the following assertion holds.

Corollary 9.4. Let R be a commutative ring. There is a bijection

{ prelocalizing subcategories of Mod R } → { filters of ideals of R }

given by

Y ↦ \{ I ⊂ R in Mod R | \frac{R}{I} ∈ Y \}.

Its inverse is given by

F ↦ { M ∈ Mod R | Ann_{R}(x) ∈ F for every x ∈ M }

= ⟨ \frac{R}{I} ∈ Mod R | I ∈ F ⟩_{preloc}.

Proof. This is immediate from Theorem 9.3. □

In the case of a locally noetherian scheme X, we need to use the notion of a local filter instead of a filter (see Theorem 9.14 and Example 12.13).

Definition 9.5. Let X be a locally noetherian scheme. We say that a filter F of subobjects of O_{X} in QCoh X is a local filter of O_{X} if it satisfies the following condition: let I be a subobject of O_{X}, and assume that for each x ∈ X, there exist an open affine neighborhood U of x in X and I′ ∈ F such that I′|_{U} ⊂ I|_{U} as a subobject of O_{U}. Then I ∈ F.

Proposition 9.6. Every principal filter of O_{X} is a local filter.

Proof. For every subobject I of O_{X}, we show that F(I) is a local filter. Let I′ be a subobject of O_{X} such that for each x ∈ X, there exist an open affine neighborhood U(x) of x in X and J(x) ∈ F(I) such that J(x)|_{U(x)} ⊂ I′|_{U(x)}. Let J := ⋂_{x∈X} J(x) in QCoh X. Then J ∈ F(I), and J|_{U(x)} ⊂ I′|_{U(x)} for each x ∈ X. For each open subset U of X,

J(U) = { s ∈ O_{X}(U) | s|_{U(x)∩U} ∈ J(U(x) ∩ U) for each x ∈ X }
⊂ { s ∈ O_{X}(U) | s|_{U(x)∩U} ∈ I′(U(x) ∩ U) for each x ∈ X }
= I′(U),

and hence J ⊂ I′ follows. This implies that I′ ∈ F. □

The next result shows that the local filters of O_{X} are exactly the same as the filters of O_{X} in the case where X is quasi-compact. This is the reason that we do not need to consider a local filter in the case of a commutative ring.

Proposition 9.7. If X is a noetherian scheme, then every filter of O_{X} is a local filter.

Proof. Let F be a filter of O_{X}. Let I be a subobject of O_{X}, and assume that for each x ∈ X, there exist an open affine neighborhood U(x) of x in X and I′(x) ∈ F such that I′(x)|_{U(x)} ⊂ I|_{U(x)}. Since X is quasi-compact, there exists x_{1}, …, x_{n} ∈ X such that X = ⋃_{j=1}^{n} U(x_{j}). Let I′ := ⋂_{j=1}^{n} I′(x_{j}). Then I′ belongs to F. Since I′|_{U(x_{j})} ⊂ I|_{U(x_{j})} for each j = 1, …, n, we have I′ ⊂ I, and hence I also belongs to F. □

The following result describes the local filter generated by a set of subobjects of O_{X}.

Proposition 9.8. Let S be a set of subobjects of O_{X}. Let F be the set consisting of all subobjects I of O_{X} satisfying the following condition: for each x ∈ X, there exist an open affine neighborhood U of x in X and n ∈ ℤ_{≥1} and I_{1}, …, I_{n} ∈ S such that

(I_{1} ∩ ⋯ ∩ I_{n})|_{U} ⊂ I|_{U}.

Then F is the smallest local filter of O_{X} including S.

Proof. It is obvious that F satisfies the conditions (a) and (b) in Definition 9.1 (1). We show that (c) is satisfied. Let I^{(1)}, I^{(2)} ∈ F. Then for each j = 1, 2 and x ∈ X, there exist an open affine neighborhood U^{(j)} of x in X and n_{j} ∈ ℤ_{≥1} and I_{1}^{(j)}, …, I_{n_{j}}^{(j)} ∈ S such that

(I_{1}^{(j)} ∩ ⋯ ∩ I_{n_{j}}^{(j)})|_{U^{(j)}} ⊂ I^{(j)}|_{U^{(j)}}.

Then

(I_{1}^{(1)} ∩ ⋯ ∩ I_{n_{1}}^{(1)} ∩ I_{1}^{(2)} ∩ ⋯ ∩ I_{n_{2}}^{(2)})|_{U^{(1)}∩U^{(2)}} ⊂ (I^{(1)} ∩ I^{(2)})|_{U^{(1)}∩U^{(2)}}.

This shows I^{(1)} ∩ I^{(2)} ∈ F. Hence F is a filter of O_{X}. Let I be a subobject of O_{X} such that for each x ∈ X, there exist an open affine neighborhood U of x in X and I′ ∈ F such that I′|_{U} ⊂ I|_{U}. Let x ∈ X, and take such U and I′. Then there exists an open affine neighborhood U′ of x in X and n ∈ ℤ_{≥1} and I′_{1}, …, I′_{n} ∈ S such that

(I′_{1} ∩ ⋯ ∩ I′_{n})|_{U′} ⊂ I′|_{U′}.

Since

(I′_{1} ∩ ⋯ ∩ I′_{n})|_{U∩U′} ⊂ I′|_{U∩U′} ⊂ I|_{U∩U′},

it holds that I ∈ F. This shows that F is a local filter. It is obvious that F is the smallest local filter of O_{X} including S. □

In the setting of Proposition 9.8, the local filter F is denoted by ⟨S⟩_{locfilt}. We investigate the restriction of a filter to an open affine subscheme and the localization at a point.

Proposition 9.9. Let F be a filter of O_{X}.

(1) For every λ ∈ Λ, the set

F|_{U_{λ}} := { I|_{U_{λ}} ⊂ O_{U_{λ}} in QCoh U_{λ} | I ∈ F }

is a filter of O_{U_{λ}}.

(2) For every x ∈ X, the set

F_{x} := { I_{x} ⊂ O_{X,x} in Mod O_{X,x} | I ∈ F }

is a filter of O_{X,x}.

Proof. (1) Since O_{X} ∈ F, we have O_{U_{λ}} ∈ F|_{U_{λ}}. Let Ĩ ⊂ Ĩ′ be subobjects of O_{U_{λ}} with Ĩ ∈ F|_{U_{λ}}. By Proposition 4.10, there exists a largest subobject I (resp. I′) of O_{X} satisfying I|_{U_{λ}} ⊂ Ĩ (resp. I′|_{U_{λ}} ⊂ Ĩ′), and it holds that I|_{U_{λ}} = Ĩ (resp. I′|_{U_{λ}} = Ĩ′). Then I ∈ F and I ⊂ I′ imply I′ ∈ F. We deduce that Ĩ′ = I′|_{U_{λ}} ∈ F|_{U_{λ}}. Let Ĩ_{1}, Ĩ_{2} ∈ F|_{U_{λ}}. Then for each i = 1, 2, there exists I_{i} ∈ F such that I_{i}|_{U_{λ}} = Ĩ_{i}. It holds that I_{1} ∩ I_{2} ∈ F. Since (−)|_{U_{λ}}: QCoh X → QCoh U_{λ} is an exact functor, I_{1}|_{U_{λ}} ∩ I_{2}|_{U_{λ}} = (I_{1} ∩ I_{2})|_{U_{λ}} ∈ F|_{U_{λ}}.

(2) This is shown similarly to (1). □

We give a characterization of a local filter.

Proposition 9.10. Let F be a filter of O_{X}. Then the following assertions are equivalent.

(1) F is a local filter.

(2) Let I be a subobject of O_{X} such that for each x ∈ X, there exists an open affine neighborhood U of x in X satisfying I|_{U} ∈ F|_{U}. Then I ∈ F.

Proof. It is obvious that (1) implies (2). Assume (2). Let I be a subobject of O_{X} such that for each x ∈ X, there exist an open affine neighborhood U of x in X and I′ ∈ F satisfying I′|_{U} ⊂ I|_{U}. Since F|_{U} is a filter of O_{U} by Proposition 9.9 (1), we have I|_{U} ∈ F|_{U}. Hence I ∈ F. This shows (1). □

The following lemmas show that the bijection in Corollary 9.4 commutes with the restriction to an open affine subscheme and the localization at a point.

Lemma 9.11. Let λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}. Let Y be a prelocalizing subcategory of QCoh U_{λ}, and let F be the corresponding filter of O_{U_{λ}} by the bijection in Corollary 9.4. Then the filter F|_{U_{μ}} of O_{U_{μ}} corresponds to the prelocalizing subcategory Y|_{U_{μ}} of QCoh U_{μ} by the bijection.

Proof. Let F′ be the filter of O_{U_{μ}} corresponding to Y|_{U_{μ}}, that is,

F′ = \{ Ĩ ⊂ O_{U_{μ}} in QCoh U_{μ} | \frac{O_{U_{μ}}}{Ĩ} ∈ Y|_{U_{μ}} \}.

It is obvious that F|_{U_{μ}} ⊂ F′. Let Ĩ ∈ F′. Then there exists an object M in QCoh X which belongs to Y such that O_{U_{μ}}/Ĩ ≅ M|_{U_{μ}}. By Proposition 4.10, there exists a subobject I of O_{U_{λ}} such that I|_{U_{μ}} = Ĩ, and O_{U_{λ}}/I is X -torsionfree, where

X = { M′ ∈ QCoh U_{λ} | M′|_{U_{μ}} = 0 }.

By Proposition 4.9 (3), the canonical morphism O_{U_{λ}}/I → (i_{λ,μ})_{∗}i_{λ,μ}^{∗}(O_{U_{λ}}/I) is a monomorphism. By Proposition 8.9, the object

(i_{λ,μ})_{∗}i_{λ,μ}^{∗}(\frac{O_{U_{λ}}}{I} ) ≅ (i_{λ,μ})_{∗}(\frac{O_{U_{μ}}}{Ĩ} ) ≅ (i_{λ,μ})_{∗}i_{λ,μ}^{∗}M

belongs to Y. Hence O_{U_{λ}}/I also belongs to Y. This shows that I ∈ F and that Ĩ = I|_{U_{μ}} ∈ F|_{U_{μ}}. Therefore F|_{U_{μ}} = F′. □

Lemma 9.12. Let x, y ∈ X with y ∈ \overline{\{x\}}. Let Y be a prelocalizing subcategory of Mod O_{X,y}, and let F be the corresponding filter of O_{X,y} by the bijection in Corollary 9.4. Then the filter F_{x} of O_{X,x} corresponds to the prelocalizing subcategory Y_{x} of Mod O_{X,x} by the bijection.

Proof. This is shown similarly to Lemma 9.11. □

We show a lemma to glue filters on open affine basis to a local filter of O_{X}.

Lemma 9.13.

(1) For every local filter F of O_{X},

F = { I ⊂ O_{X} in QCoh X | I|_{U_{λ}} ∈ F|_{U_{λ}} for each λ ∈ Λ }.

(2) Let F_{λ} be a filter of O_{U_{λ}} for each λ ∈ Λ, and assume that F_{λ}|_{U_{μ}} = F_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}. Then there exists a unique local filter F of O_{X} satisfying F|_{U_{λ}} = F_{λ} for each λ ∈ Λ.

Proof. (1) This follows from Proposition 9.10.

(2) The uniqueness follows from (1). Let

F := { I ⊂ O_{X} in QCoh X | I|_{U_{λ}} ∈ F_{λ} for each λ ∈ Λ }.

It is straightforward to show that F is a filter of O_{X} satisfying F|_{U_{λ}} ⊂ F_{λ} for each λ ∈ Λ. Let I be a subobject of O_{X} such that for each x ∈ X, there exists an open affine neighborhood U of x in X satisfying I|_{U} ∈ F|_{U}. For each λ ∈ Λ and y ∈ U_{λ}, there exists an open affine neighborhood U′ of y in X satisfying I|_{U′} ∈ F|_{U′}. Take μ ∈ Λ satisfying y ∈ U_{μ} ⊂ U_{λ} ∩ U′. Then (I|_{U_{λ}})|_{U_{μ}} = (I|_{U′})|_{U_{μ}} ∈ (F|_{U′})|_{U_{μ}} = (F|_{U_{λ}})|_{U_{μ}}. Since F|_{U_{λ}} is a local filter by Proposition 9.9 (1) and Proposition 9.7, we have I|_{U_{λ}} ∈ F|_{U_{λ}} ⊂ F_{λ}. This shows that I ∈ F. By Proposition 9.10, the filter F is a local filter. We show that F_{λ} ⊂ F|_{U_{λ}}. Let J̃ ∈ F_{λ}. By Proposition 4.10, there exists a subobject J of O_{X} such that J|_{U_{λ}} = J̃, and O_{X}/J is X_{U_{λ}}-torsionfree (see Proposition 7.2). It suffices to show that J ∈ F, that is, J|_{U_{μ}} ∈ F_{μ} for each μ ∈ Λ. Denote by Y_{λ} and Y_{μ} the prelocalizing subcategories of QCoh U_{λ} and QCoh U_{μ} corresponding to F_{λ} and F_{μ} by Corollary 9.4, respectively. We show that the object O_{U_{μ}}/J|_{U_{μ}} belongs to Y_{μ}. Let x ∈ Ass_{U_{μ}}(O_{U_{μ}}/J|_{U_{μ}}). By Lemma 8.5,

x ∈ Ass_{U_{μ}} \frac{O_{X}}{J} |_{U_{μ}} = Ass_{X} \frac{O_{X}}{J} ∩ U_{μ} ⊂ U_{λ} ∩ U_{μ}.

Hence

(\frac{O_{U_{μ}}}{J|_{U_{μ}}} )_{x} = \frac{O_{X,x}}{J_{x}} = (\frac{O_{U_{λ}}}{J|_{U_{λ}}} )_{x} = (\frac{O_{U_{λ}}}{J̃} )_{x} ∈ (Y_{λ})_{x}.

Take ν ∈ Λ such that x ∈ U_{ν} ⊂ U_{λ} ∩ U_{μ}. Then (Y_{λ})_{x} = (Y_{λ}|_{U_{ν}})_{x} = (Y_{ν})_{x} = (Y_{μ}|_{U_{ν}})_{x} = (Y_{μ})_{x}. Hence by Theorem 8.8, the object O_{U_{μ}}/J|_{U_{μ}} belongs to Y_{μ}. This shows that J|_{U_{μ}} ∈ F_{μ}. □

The following theorem is the main result in this section, which gives a classification of the prelocalizing subcategory of QCoh X.

Theorem 9.14. Let X be a locally noetherian scheme, and let {U_{λ}}_{λ∈Λ} be an open affine basis of X. Then there exist bijections between the following sets.

(1) The set of prelocalizing subcategories of QCoh X.

(2) The set of families {Y_{λ} ⊂ QCoh U_{λ}}_{λ∈Λ} of prelocalizing subcategories such that Y_{λ}|_{U_{μ}} = Y_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(3) The set of families {Y(x) ⊂ Mod O_{X,x}}_{x∈X} of prelocalizing subcategories such that Y(y)_{x} = Y(x) for each x, y ∈ X with y ∈ \overline{\{x\}}.

(4) The set of local filters of O_{X}.

(5) The set of families {F_{λ}}_{λ∈Λ}, where F_{λ} is a filter of O_{U_{λ}} for each λ ∈ Λ, such that F_{λ}|_{U_{μ}} = F_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(6) The set of families {F(x)}_{x∈X}, where F(x) is a filter of O_{X,x} for each x ∈ X, such that F(y)_{x} = F(x) for each x, y ∈ X with y ∈ \overline{\{x\}}.

The correspondences are given as follows.

(1) Y ↦ \begin{cases}(4) \{ I ⊂ O_{X} in QCoh X | \frac{O_{X}}{I} ∈ Y \} \\ (2) \{Y|_{U_{λ}}\}_{λ∈Λ} \\ (3) \{Y_{x}\}_{x∈X} \end{cases}

(4) F ↦ \begin{cases}(1) ⟨ \frac{O_{X}}{I} | I ∈ F ⟩_{preloc}. \\ (5) \{F|_{U_{λ}}\}_{λ∈Λ} \\ (6) \{F_{x}\}_{x∈X} \end{cases}

(2) {Y_{λ}}_{λ∈Λ} ↦ (1) { M ∈ QCoh X | M|_{U_{λ}} ∈ Y_{λ} for each λ ∈ Λ }

(3) {Y(x)}_{x∈X} ↦ (1) { M ∈ QCoh X | M_{x} ∈ Y(x) for each x ∈ X }

(5) {F_{λ}}_{λ∈Λ} ↦ (4) { I ⊂ O_{X} in QCoh X | I|_{U_{λ}} ∈ F_{λ} for each λ ∈ Λ }

(6) {F(x)}_{x∈X} ↦ (4) { I ⊂ O_{X} in QCoh X | I_{x} ∈ F(x) for each x ∈ X } Proof. Theorem 8.11 gives bijections between (1), (2), and (3). Corollary 9.4 and Lemma 9.11 (resp. Lemma 9.12) give a bijection between (2) and (5) (resp.

(3) and (6)). Lemma 9.13 gives a bijection between (4) and (5). □

For a family of prelocalizing subcategories of QCoh X, the supremum and the intersection are described in terms of local filters as follows.

Proposition 9.15. Assume that the following elements correspond to each other by the bijections in Theorem 9.14 for each ω ∈ Ω.

(1) Y^{ω}. ; (2) {Y_{λ}^{ω}}_{λ∈Λ}. ; (3) {Y^{ω}(x)}_{x∈X}.  (4) F^{ω}. ; (5) {F_{λ}^{ω}}_{λ∈Λ}. ; (6) {F^{ω}(x)}_{x∈X}.

Then the following elements correspond to each other by the bijections.

(1) ⟨⋃_{ω∈Ω} Y^{ω}⟩_{preloc}.
(2) {⟨⋃_{ω∈Ω} Y_{λ}^{ω}⟩_{preloc}}_{λ∈Λ}.
(3) {⟨⋃_{ω∈Ω} Y^{ω}(x)⟩_{preloc}}_{x∈X}.

(4) ⟨⋃_{ω∈Ω} F^{ω}⟩_{locfilt}.
(5) {⟨⋃_{ω∈Ω} F_{λ}^{ω}⟩_{locfilt}}_{λ∈Λ}.
(6) {⟨⋃_{ω∈Ω} F^{ω}(x)⟩_{locfilt}}_{x∈X}.

Proof. This follows from Proposition 8.12. □

Proposition 9.16. Assume that the following elements correspond to each other by the bijections in Theorem 9.14 for each ω ∈ Ω.

(1) Y^{ω}. ; (2) {Y_{λ}^{ω}}_{λ∈Λ}. ; (3) {Y^{ω}(x)}_{x∈X}.  (4) F^{ω}. ; (5) {F_{λ}^{ω}}_{λ∈Λ}. ; (6) {F^{ω}(x)}_{x∈X}.

Then the following elements correspond to each other by the bijections.

(1) ⋂_{ω∈Ω} Y^{ω}. ; (2) {⋂_{ω∈Ω} Y_{λ}^{ω}}_{λ∈Λ}. ; (3) {⋂_{ω∈Ω} Y^{ω}(x)}_{x∈X}.  (4) ⋂_{ω∈Ω} F^{ω}. ; (5) {⋂_{ω∈Ω} F_{λ}^{ω}}_{λ∈Λ}. ; (6) {⋂_{ω∈Ω} F^{ω}(x)}_{x∈X}.

Proof. This follows from Proposition 8.13. □

We demonstrate a calculation of the prelocalizing subcategories by using Theorem 9.14.

Example 9.17. Let k be an algebraically closed field, and consider the polynomial ring k[x] with a variable x. For each a ∈ k, let 𝔭_{a} := (x − a) ⊂ k[x] and 𝔪_{a} := 𝔭_{a}k[x]_{𝔭_{a}}. Then

Spec k[x] = { 𝔭_{a} | a ∈ k } ∪ {0}.

Since k[x]_{𝔭_{a}} is a discrete valuation ring, the set of ideals of k[x]_{𝔭_{a}} is

{ 𝔪_{a}^{i} | i ∈ ℤ_{≥0} } ∪ {0},

where 𝔪_{a}^{0} = k[x]_{𝔭_{a}}. For each n ∈ ℤ_{≥0}, define the filter F_{a}^{n} of k[x]_{𝔭_{a}} by

F_{a}^{n} = { 𝔪_{a}^{i} | 0 ≤ i ≤ n },

and let

F_{a}^{∞} := { 𝔪_{a}^{i} | i ∈ ℤ_{≥0} }, F_{a} := { 𝔪_{a}^{i} | i ∈ ℤ_{≥0} } ∪ {0}.

Then the set of filters of k[x]_{𝔭_{a}} is

{ F_{a}^{n} | n ∈ ℤ_{≥0} ∪ {∞} } ∪ {F_{a}}.

Since k[x]_{0} = k(x) is a field, the set of the filters of k(x) consists of F^{∞} = {k(x)} and F = {0, k(x)}. For each a ∈ k and n ∈ ℤ_{≥0} ∪ {∞}, (F_{a}^{n})_{0} = F^{∞}, and (F_{a})_{0} = F. Hence the set

{ ({F_{a}^{r(a)}}_{a∈k}, F^{∞}) | r = {r(a)}_{a∈k} ∈ ∏_{a∈k}(ℤ_{≥0} ∪ {∞}) } ∪ {({F_{a}}_{a∈k}, F)}

is the set of families of filters which are compatible with localizations. By Theorem 9.14, the set of prelocalizing subcategories of Mod k[x] is

{ Y_{r} | r ∈ ∏_{a∈k}(ℤ_{≥0} ∪ {∞}) } ∪ {Mod k[x]},

where

Y_{r} = { M ∈ Mod k[x] | M_{𝔭_{a}}𝔪_{a}^{r(a)} = 0 for each a ∈ k with r(a) ≠ ∞ }

for each r ∈ ∏_{a∈k}(ℤ_{≥0} ∪ {∞}).

10. Classification of localizing subcategories

In this section, we investigate extensions of prelocalizing subcategories (Definition 4.1 (1)) in terms of local filters and classify the localizing subcategories of QCoh X for a locally noetherian scheme X. The classification is given as a restriction of Theorem 9.14. We start with recalling Gabriel’s classification of the localizing subcategories of Mod Λ for a ring Λ.

Definition 10.1. Let Λ be a ring.

(1) For prelocalizing filters F_{1} and F_{2} of right ideals of Λ, define the product F_{1} ∗ F_{2} as follows: L ∈ F_{1} ∗ F_{2} if and only if there exists L_{1} ∈ F_{1} satisfying a^{−1}L ∈ F_{2} for every a ∈ L_{1}.

(2) A prelocalizing filter F of right ideals of Λ is called a Gabriel filter if F ∗ F ⊂ F holds.

Proposition 10.2. Let Λ be a ring. If F_{1} and F_{2} are prelocalizing filters of right ideals of Λ, then F_{1} ⊂ F_{1} ∗ F_{2}, and F_{2} ⊂ F_{1} ∗ F_{2}.

Proof. Let L_{1} ∈ F_{1}. Then a^{−1}L_{1} = Λ ∈ F_{2} for each a ∈ L_{1}. This shows that F_{1} ⊂ F_{1} ∗ F_{2}. Let L_{2} ∈ F_{2}. Then Λ ∈ F_{1}, and a^{−1}L_{2} ∈ F_{2} for each a ∈ Λ. This shows that F_{2} ⊂ F_{1} ∗ F_{2}. □

Theorem 10.3 ([Gab62, p. 412]). Let Λ be a ring.

(1) For each i = 1, 2, let Y_{i} be a prelocalizing subcategory of Mod Λ, and let F_{i} be the prelocalizing filter of right ideals of Λ corresponding to F_{i} by the bijection in Theorem 9.3. Then Y_{1} ∗ Y_{2} corresponds to F_{2} ∗ F_{1} by the bijection.

(2) The bijection in Theorem 9.3 restricts to a bijection

{ localizing subcategories of Mod Λ } → { Gabriel filters of right ideals of Λ }.

Proof. [Ste75, Theorem VI.5.1]. □

For a commutative ring R, we say that a filter F of R is closed under products if I_{1}, I_{2} ∈ F implies I_{1}I_{2} ∈ F. In the case of a commutative noetherian ring, products of filters and Gabriel filters are characterized as follows.

Proposition 10.4. Let R be a commutative noetherian ring.

(1) Let F_{1} and F_{2} be filters of R. Then

F_{1} ∗ F_{2} = { I ⊂ R in Mod R | I_{1}I_{2} ⊂ I for some I_{1} ∈ F_{1}, I_{2} ∈ F_{2} }.

(2) Let F be a filter of R. Then F is a Gabriel filter if and only if F is closed under products.

Proof. (1) Let I ∈ F_{1} ∗ F_{2}. Then there exists I_{1} ∈ F_{1} such that a^{−1}I ∈ F_{2} for each a ∈ I_{1}. Since R is noetherian, there exist b_{1}, …, b_{n} ∈ R such that I_{1} = b_{1}R + ⋯ + b_{n}R. Let I_{2} := b_{1}^{−1}I ∩ ⋯ ∩ b_{n}^{−1}I. Then I_{2} ∈ F_{2}, and

I_{1}I_{2} = b_{1}I_{2} + ⋯ + b_{n}I_{2} ⊂ b_{1}(b_{1}^{−1}I) + ⋯ + b_{n}(b_{n}^{−1}I) ⊂ I.

Conversely, let J_{1} ∈ F_{1} and J_{2} ∈ F_{2}. For each a ∈ J_{1}, we have J_{2} ⊂ a^{−1}J_{1}J_{2}, and hence a^{−1}J_{1}J_{2} ∈ F_{2}. This implies that J_{1}J_{2} ∈ F_{1} ∗ F_{2}.

(2) This follows from (1). □

For a commutative noetherian ring R, the classification of the localizing subcategories of Mod R is stated as follows.

Corollary 10.5. Let R be a commutative noetherian ring. Then the bijection in Corollary 9.4 restricts to a bijection

{ localizing subcategories of Mod R } → { filters of R closed under products }.

Proof. This follows from Theorem 10.3 (2) and Proposition 10.4 (2). □

In the rest of this section, let X be a locally noetherian scheme, and let {U_{λ}}_{λ∈Λ} be an open affine basis of X. For an object M in QCoh X and a subobject I of O_{X}, the subobject MI of M is defined as the image of the canonical morphism M ⊗_{O_{X}} I → M in QCoh X.

Definition 10.6.

(1) Let F_{1} and F_{2} be local filters of O_{X}. We define the product F_{1} ∗ F_{2} by

F_{1} ∗ F_{2} = ⟨ I_{1}I_{2} ⊂ O_{X} in QCoh X | I_{i} ∈ F_{i} for each i = 1, 2 ⟩_{locfilt}.

(2) We say that a local filter F is closed under products if F ∗F ⊂ F holds.

Note that a local filter F is closed under products if and only if I_{1}, I_{2} ∈ F implies I_{1}I_{2} ∈ F. Products of local filters of O_{X} commute with the restriction to an open affine subscheme and the localization at a point.

Lemma 10.7. Let F_{i} be a local filter of O_{X} for each i = 1, 2.

(1) For every λ ∈ Λ,

(F_{1} ∗ F_{2})|_{U_{λ}} = F_{1}|_{U_{λ}} ∗ F_{2}|_{U_{λ}}.

(2) For every x ∈ X,

(F_{1} ∗ F_{2})_{x} = (F_{1})_{x} ∗ (F_{2})_{x}.

Proof. (1) Let J ∈ (F_{1} ∗ F_{2})|_{U_{λ}}. Then there exists I ∈ F_{1} ∗ F_{2} such that I|_{U_{λ}} = J. For each x ∈ U_{λ}, there exist an open affine neighborhood U of x in X and I_{1} ∈ F_{1} and I_{2} ∈ F_{2} such that (I_{1}I_{2})|_{U} ⊂ I|_{U}. Hence

(I_{1}|_{U_{λ}}I_{2}|_{U_{λ}})|_{U_{λ}∩U} = (I_{1}I_{2})|_{U_{λ}∩U} ⊂ I|_{U_{λ}∩U} = J|_{U_{λ}∩U}.

This shows that J ∈ F_{1}|_{U_{λ}} ∗ F_{2}|_{U_{λ}}. Conversely, assume J ∈ F_{1}|_{U_{λ}} ∗ F_{2}|_{U_{λ}}. Then for each x ∈ U_{λ}, there exist an open affine neighborhood V of x in U_{λ} and J_{1} ∈ F_{1}|_{U_{λ}} and J_{2} ∈ F_{2}|_{U_{λ}} such that (J_{1}J_{2})|_{V} ⊂ J|_{V}. For each i = 1, 2, there exists I_{i} ∈ F_{i} such that I_{i}|_{U_{λ}} = J_{i}. Then (I_{1}I_{2})|_{U_{λ}} ∈ (F_{1} ∗ F_{2})|_{U_{λ}}, and

((I_{1}I_{2})|_{U_{λ}})|_{V} = (J_{1}J_{2})|_{V} ⊂ J|_{V} .

Since (F_{1} ∗ F_{2})|_{U_{λ}} is a local filter by Proposition 9.9 (1) and Proposition 9.7, we obtain J ∈ (F_{1} ∗ F_{2})|_{U_{λ}}.

(2) This can be shown similarly to (1). □

We describe extensions of prelocalizing subcategories of QCoh X in terms of products of local filters.

Theorem 10.8. Assume that the following elements correspond to each other by the bijections in Theorem 9.14 for each i = 1, 2.

(1) Y^{i}. ; (2) {Y_{λ}^{i}}_{λ∈Λ}. ; (3) {Y^{i}(x)}_{x∈X}.  (4) F^{i}. ; (5) {F_{λ}^{i}}_{λ∈Λ}. ; (6) {F^{i}(x)}_{x∈X}.

Then the following elements correspond to each other by the bijections.

(1) Y^{1} ∗ Y^{2}. ; (2) {Y_{λ}^{1} ∗ Y_{λ}^{2}}_{λ∈Λ}. ; (3) {Y^{1}(x) ∗ Y^{2}(x)}_{x∈X}.  (4) F^{1} ∗ F^{2}. ; (5) {F_{λ}^{1} ∗ F_{λ}^{2}}_{λ∈Λ}. ; (6) {F^{1}(x) ∗ F^{2}(x)}_{x∈X}.

Proof. This follows from Theorem 8.16, Theorem 10.3 (1), and Lemma 10.7.

□

Corollary 10.9. The bijections in Theorem 9.14 restrict to bijections between following sets.

(1) The set of localizing subcategories of QCoh X.

(2) The set of families {X_{λ} ⊂ QCoh U_{λ}}_{λ∈Λ} of localizing subcategories such that X_{λ}|_{U_{μ}} = X_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(3) The set of families {X (x) ⊂ Mod O_{X,x}}_{x∈X} of localizing subcategories such that X (y)_{x} = X (x) for each x, y ∈ X with y ∈ \overline{\{x\}}.

(4) The set of local filters of O_{X} closed under products.

(5) The set of families {F_{λ}}_{λ∈Λ}, where F_{λ} is a filter of O_{U_{λ}} closed under products for each λ ∈ Λ, such that F_{λ}|_{U_{μ}} = F_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(6) The set of families {F(x)}_{x∈X}, where F(x) is a filter of O_{X,x} closed under products for each x ∈ X, such that F(y)_{x} = F(x) for each x, y ∈ X with y ∈ \overline{\{x\}}.

Proof. This follows from Theorem 10.8. □

We apply Corollary 10.9 to Example 9.17.

Example 10.10. In the setting of Example 9.17,

F_{a}^{m} ∗ F_{a}^{n} = F_{a}^{m+n}

for each a ∈ k and m, n ∈ ℤ_{≥0} ∪ {∞}. Hence by Corollary 10.9, the set of localizing subcategories of Mod k[x] is

{ Y_{r} | r ∈ ∏_{a∈k}{0, ∞} } ∪ {Mod k[x]}.

In Theorem 7.8, we showed that there exists a bijection between the localizing subcategories of QCoh X and the specialization-closed subsets of X. For a local filter F of O_{X} closed under products, the corresponding specialization-closed subset of X is { x ∈ X | F_{x} ≠ {O_{X,x}} }. Prime localizing subcategories of QCoh X are characterized in terms of local filters as follows.

Theorem 10.11. Let F be a local filter of O_{X} closed under products. Then the following assertions are equivalent.

(1) By the bijection in Theorem 9.14, the local filter F corresponds to a prime localizing subcategory of QCoh X.

(2) There exists x ∈ X such that

F = { I ⊂ O_{X} in QCoh X | I_{x} = O_{X,x} }.

(3) For each family {F_{ω}}_{ω∈Ω} of local filters of O_{X} closed under products satisfying F = ⋂_{ω∈Ω} F_{ω}, there exists ω ∈ Ω such that F = F_{ω}.

(4) For each family {F_{ω}}_{ω∈Ω} of local filters of O_{X} satisfying F = ⋂_{ω∈Ω} F_{ω}, there exists ω ∈ Ω such that F = F_{ω}.

Proof. This follows from Theorem 8.18. □

11. Classification of closed subcategories

In this section, we investigate the closed subcategories of QCoh X for a locally noetherian scheme X, whose definition is as follows.

Definition 11.1. Let A be a Grothendieck category. A prelocalizing subcategory X of A is called a closed subcategory of A if X is closed under arbitrary direct products.

Note that every Grothendieck category has arbitrary direct products ([Pop73, Corollary 3.7.10]). Closed subcategories are characterized by Proposition 4.3 and the following result.

Proposition 11.2. Let A be a Grothendieck category (or more generally, an abelian category admitting arbitrary direct products), and let Y be a full subcategory of A closed under subobjects and quotient objects. Then the following assertions are equivalent.

(1) Y is closed under arbitrary direct products.

(2) The inclusion functor Y ↪ A has a left adjoint.

(3) For each object M in A, there exists a smallest subobject L of M satisfying M/L ∈ Y.

Proof. Apply Proposition 4.3 to the opposite category of A. □

Note that for a Grothendieck category, every full subcategory which is closed under subobjects and arbitrary direct products is also closed under arbitrary direct sums. For a ring Λ, Rosenberg [Ros95] showed that there exists a bijection between the closed subcategories of Mod Λ and the two-sided ideals of Λ. This result can be unified into Theorem 9.3 as follows.

Theorem 11.3 (Gabriel [Gab62, Lemma V.2.1] and Rosenberg [Ros95, Proposition III.6.4.1]). Let Λ be a ring. Then there exist bijections between the following sets.

(1) The set of closed subcategories of Mod Λ.

(2) The set of principal prelocalizing filters of right ideals of Λ.

(3) The set of two-sided ideals of Λ. The bijection between (1) and (2) is induced by the bijection in Theorem 9.3. The bijection between (1) and (3) is given by

(1) → (3) : Y ↦ ⋂_{M∈Y} Ann_{Λ}(M),

(3) → (1) : I ↦ \{ M ∈ Mod Λ | MI = 0 \} = ⟨\frac{Λ}{I}⟩_{preloc}.

Proof. We show that for each right ideal L of Λ, the principal filter F(L) of right ideals of Λ is prelocalizing if and only if L is a two-sided ideal of Λ. Assume that F(L) is prelocalizing. Then for each a ∈ Λ, we have a^{−1}L ∈ F(L). This implies L ⊂ a^{−1}L, and hence aL ⊂ L. Therefore L is a two-sided ideal of Λ. The converse is obvious. The bijection between (2) and (3) follows from Remark 9.2. Let Y be a prelocalizing subcategory of A, and let F be the corresponding prelocalizing filter of right ideals of Λ. If Y is a closed subcategory of A, then by Proposition 11.2, there exists a smallest element of F. Hence F is principal. Conversely, assume that F is principal. Then F = F(I) for some two-sided ideal I of Λ. Since

Y = { M ∈ Mod Λ | I ⊂ Ann_{Λ}(x) for each x ∈ M }
= { M ∈ Mod Λ | MI = 0 },

the prelocalizing subcategory Y of A is also closed under arbitrary direct products. □

The aim of this section is to generalize Theorem 11.3 to a locally noetherian scheme X. Let {U_{λ}}_{λ∈Λ} be an open affine basis of X. We show a lemma on gluing of subobjects on open affine subschemes.

Lemma 11.4. Let M be an object in QCoh X, and let L_{λ} be a subobject of M|_{U_{λ}} for each λ ∈ Λ. Assume that L_{λ}|_{U_{μ}} = L_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}. Then there exists a unique subobject L of M such that L|_{U_{λ}} = L_{λ} for each λ ∈ Λ.

Proof. (Existence) Define a subsheaf L of M by

L(U) = { s ∈ M(U) | s|_{U_{λ}} ∈ L_{λ}(U_{λ}) for each λ ∈ Λ with U_{λ} ⊂ U }

for each open subset U of X. It is straightforward to show that L is a subsheaf of M satisfying L|_{U_{λ}} = L_{λ} for each λ ∈ Λ. In particular, the sheaf L is quasi-coherent. (Uniqueness) Let L′ be a subobject of M in QCoh X such that L′|_{U_{λ}} = L_{λ} for each λ ∈ Λ. Then

L′(U) = { s ∈ M(U) | s|_{U_{λ}} ∈ L′(U_{λ}) for each λ ∈ Λ with U_{λ} ⊂ U }
= { s ∈ M(U) | s|_{U_{λ}} ∈ L_{λ}(U_{λ}) for each λ ∈ Λ with U_{λ} ⊂ U }

for each open subset U of X. □

The following lemma shows that for a principal filter of O_{X}, its restriction to an open affine subscheme and its localization at a point are also principal filters.

Lemma 11.5. Let I be a subobject of O_{X}.

(1) For every λ ∈ Λ, we have F(I)|_{U_{λ}} = F(I|_{U_{λ}}).

(2) For every x ∈ X, we have F(I)_{x} = F(I_{x}).

Proof. (1) For each J′ ∈ F(I)|_{U_{λ}}, there exists J ∈ F(I) such that J|_{U_{λ}} = J′. Since I ⊂ J, it holds that I|_{U_{λ}} ⊂ J|_{U_{λ}} = J′. This shows that F(I)|_{U_{λ}} ⊂ F(I|_{U_{λ}}). It is follows from I ∈ F(I) that I|_{U_{λ}} ∈ F(I)|_{U_{λ}}. Since F(I)|_{U_{λ}} is a filter of O_{U_{λ}} by Proposition 9.9 (1), we have F(I)|_{U_{λ}} ⊃ F(I|_{U_{λ}}).

(2) This is shown similarly by using Proposition 9.9 (2). □

Conversely, if the restriction of a local filter of O_{X} to each open affine subscheme U_{λ} is principal, then the local filter is principal.

Lemma 11.6. Let F be a local filter of O_{X}. Then F is a principal filter if and only if the filter F|_{U_{λ}} of O_{U_{λ}} is principal for every λ ∈ Λ.

Proof. If F is a principal filter, then F|_{U_{λ}} is a principal filter for every λ ∈ Λ by Lemma 11.5 (1). Assume that there exists a subobject I_{λ} of O_{U_{λ}} such that F|_{U_{λ}} = F(I_{λ}) for each λ ∈ Λ. For each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ},

F(I_{λ}|_{U_{μ}}) = F(I_{λ})|_{U_{μ}} = (F|_{U_{λ}})|_{U_{μ}} = F|_{U_{μ}} = F(I_{μ}).

Hence I_{λ}|_{U_{μ}} = I_{μ}. By Lemma 11.4, there exists a subobject I of O_{X} such that I|_{U_{λ}} = I_{λ} for each λ ∈ Λ. Since F(I)|_{U_{λ}} = F(I|_{U_{λ}}) = F(I_{λ}) = F_{λ} for each λ ∈ Λ, it follows from Lemma 9.13 (2) that F(I) = F. □

Remark 11.7. Let F be a local filter of O_{X}. Even if F_{x} is a principal filter of O_{X,x} for each x ∈ X, the local filter F is not necessarily a principal filter. A counter-example is given in Example 11.12.

We characterize closed subcategories of QCoh X in terms of local filters.

Lemma 11.8. Let Y be a prelocalizing subcategory of QCoh X, and let F be the corresponding local filter of O_{X} by the bijection in Theorem 9.14. Then Y is a closed subcategory of QCoh X if and only if F is a principal filter. If F = F(I) for a subobject I of O_{X}, then I is the smallest subobject of O_{X} satisfying O_{X}/I ∈ Y, and

Y = { M ∈ QCoh X | MI = 0 }.

Proof. Assume that Y is a closed subcategory of QCoh X. Then by Proposition 11.2, there exists a smallest subobject I of O_{X} satisfying O_{X}/I ∈ Y. Hence F = F(I). Conversely, assume that F = F(I) for some subobject I of O_{X}. Then for each λ ∈ Λ, we have F|_{U_{λ}} = F(I|_{U_{λ}}) by Lemma 11.5 (1), and hence

Y|_{U_{λ}} = { M′ ∈ QCoh U_{λ} | M′(I|_{U_{λ}}) = 0 }

by Theorem 11.3. By Theorem 9.14,

Y = { M ∈ QCoh X | M|_{U_{λ}} ∈ Y|_{U_{λ}} for every λ ∈ Λ }
= { M ∈ QCoh X | M|_{U_{λ}}I|_{U_{λ}} = 0 for every λ ∈ Λ }
= { M ∈ QCoh X | MI = 0 }.

For each object M in QCoh X, the subobject MI of M is the smallest among the subobjects L of M satisfying (M/L)I = 0. Therefore Y is a closed subcategory of QCoh X. □

As in Remark 11.7, the same type of theorem as Corollary 10.9 does not hold for the closed subcategories. For this reason, we use the characterization in Proposition 8.14 in order to obtain a generalization to the closed subcategories.

Theorem 11.9. Let X be a locally noetherian scheme, and let {U_{λ}}_{λ∈Λ} be an open affine basis of X. Then there exist bijections between the following sets.

(1) The set of closed subcategories of QCoh X.

(2) The set of families {Y_{λ} ⊂ QCoh U_{λ}}_{λ∈Λ} of closed subcategories such that Y_{λ}|_{U_{μ}} = Y_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(3) The set of families {Y(x) ⊂ Mod O_{X,x}}_{x∈X} of closed subcategories such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and a closed subcategory Y′ of QCoh U_{λ} satisfying Y′_{y} = Y(y) for each y ∈ U_{λ}.

(4) The set of principal filters of O_{X}.

(5) The set of families {F_{λ}}_{λ∈Λ}, where F_{λ} is a principal filter of O_{U_{λ}} for each λ ∈ Λ, such that F_{λ}|_{U_{μ}} = F_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(6) The set of families {F(x)}_{x∈X}, where F(x) is a principal filter of O_{X,x} for each x ∈ X, such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and a principal filter F′ of O_{U_{λ}} satisfying F′_{y} = F(y) for each y ∈ U_{λ}.

(7) The set of subobjects of O_{X}.

(8) The set of families {I_{λ}}_{λ∈Λ}, where I_{λ} is a subobject of O_{U_{λ}} for each λ ∈ Λ, such that I_{λ}|_{U_{μ}} = I_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(9) The set of families {I(x)}_{x∈X}, where I(x) is an ideal of O_{X,x} for each x ∈ X, such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and a subobject I′ of O_{U_{λ}} satisfying I′_{y} = I(y) for each y ∈ U_{λ}.

The bijections between the sets (1), …, (6) are induced by Theorem 9.14. The bijections (4)↔(7), (5)↔(8), and (6)↔(9) are defined by the bijection L ↦ F(L) in Remark 9.2.

Proof. This follows from Theorem 9.14, Theorem 11.3, Lemma 11.6, and Lemma 11.8. □

We establish a bijection between the closed subcategories of QCoh X and the closed subschemes of X by using the following fact.

Proposition 11.10. There is a bijection

{ subobjects of O_{X} } → { closed subschemes of X }

given by I ↦ (Supp(O_{X}/I), i^{−1}(O_{X}/I)), where i: Supp(O_{X}/I) ↪ X is the immersion. For each closed subscheme Y of X, the corresponding subobject I of O_{X} is given by the exact sequence

0 → I → O_{X} → i_{∗}O_{Y} → 0,

where i: Y ↪ X is the immersion, and O_{X} → i_{∗}O_{Y} is the canonical morphism.

Proof. [Har77, Proposition II.5.9]. □

Theorem 11.11. Let X be a locally noetherian scheme. Then there exists a bijection between (1) The set of closed subcategories of QCoh X.

(2) The set of closed subschemes of X. For each closed subscheme Y of X with the immersion i: Y ↪ X, the functor i_{∗}: QCoh Y → QCoh X is fully faithful and induces an equivalence between QCoh Y and the closed subcategory of QCoh X corresponding to Y.

Proof. The bijection is obtained by Theorem 11.9 and Proposition 11.10. By [Gro60, 0.5.1.4], [Gro60, Proposition I.5.5.1 (i)], and [Gro60, Corollary I.9.2.2 (a)], we have the functor i^{∗}: QCoh X → QCoh Y and its right adjoint i_{∗}: QCoh Y → QCoh X. It is straightforward to show that the counit morphism i^{∗}i_{∗} → 1_{QCoh Y} is an isomorphism. Hence i_{∗} is fully faithful. An object M in QCoh X is isomorphic to the image of an object in QCoh Y by i_{∗} if and only if the canonical morphism M → i_{∗}i^{∗}M is an isomorphism. Let I be the subobject of O_{X} corresponding to Y. Since we have the exact sequence

0 → MI → M → i_{∗}i^{∗}M → 0,

M → i_{∗}i^{∗}M is an isomorphism if and only if MI = 0. Therefore the claim follows. □

Example 11.12. We follow the notations in Example 9.17 and Example 10.10. Each nonzero proper ideal I of k[x] is generated by an element of the form (x − a_{1})^{r_{1}} ⋯ (x − a_{l})^{r_{l}}, where l ∈ ℤ_{≥1}, a_{1}, …, a_{l} are distinct elements of k, and r_{1}, …, r_{l} ∈ ℤ_{≥1}. We have

I_{𝔭_{a}} = \begin{cases}𝔪_{a_{i}}^{r_{i}} & if a = a_{i} for some i ∈ \{1, …, l\} \\ k[x]_{𝔭_{a}} & if a ∈ k \ \{a_{1}, …, a_{l}\} \end{cases}.

For each r ∈ ∏_{a∈k}(ℤ_{≥0} ∪ {∞}), the object k[x]/I belongs to Y_{r} if and only if r_{i} ≤ r(a_{i}) for every i = 1, …, l. Hence the corresponding filter of k[x] to Y_{r} is

\{(x − a_{1})^{r_{1}} ⋯ (x − a_{l})^{r_{l}} ⊂ k[x] | \begin{matrix}l ∈ ℤ_{≥1}, a_{1}, …, a_{l} ∈ k (distinct) \\ r_{1}, …, r_{l} ∈ ℤ_{≥1} \\ r_{i} ≤ r(a_{i}) for each i = 1, …, l \end{matrix}\} ∪ \{k[x]\}.

This is equal to

⟨ (x − a)^{r} ⊂ k[x] | a ∈ k, r ∈ ℤ_{≥1}, r ≤ r(a) ⟩_{locfilt},

and we have the description

Y_{r} = ⟨ \frac{k[x]}{(x − a)^{r}} | a ∈ k, r ∈ ℤ_{≥1}, r ≤ r(a) ⟩_{preloc}.

By Theorem 11.9, the set of closed subcategories of Mod k[x] is

{ Y_{r} | r ∈ ⨁_{a∈k} ℤ_{≥0} } ∪ {Mod k[x]}.

Let r ∈ (∏_{a∈k} ℤ_{≥0}) \ (⨁_{a∈k} ℤ_{≥0}). Then for every 𝔭 ∈ Spec k[x], the prelocalizing subcategory (Y_{r})_{𝔭} of Mod k[x]_{𝔭} is a closed subcategory, and F_{𝔭} is a principal filter of k[x]_{𝔭}. However, the prelocalizing subcategory Y_{r} of Mod k[x] is not a closed subcategory, and the corresponding local filter of k[x] is not a principal filter.

12. Classification of bilocalizing subcategories

Let X be a locally noetherian scheme. We investigate extensions of closed subcategories. The following lemma shows that products of principal filters are also principal.

Lemma 12.1. Let I_{1} and I_{2} be subobjects of O_{X}. Then F(I_{1})∗F(I_{2}) = F(I_{1}I_{2}).

Proof. This follows from Proposition 9.6. □

Extensions of closed subcategories are described in terms of products of principal filters.

Theorem 12.2. Assume that the following elements correspond to each other by the bijections in Theorem 11.9 for each i = 1, 2.

(1) Y^{i}. ; (2) {Y_{λ}^{i}}_{λ∈Λ}. ; (3) {Y^{i}(x)}_{x∈X}. ; (4) F^{i}. ; (5) {F_{λ}^{i}}_{λ∈Λ}. ; (6) {F^{i}(x)}_{x∈X}.  (7) I^{i}. ; (8) {I_{λ}^{i}}_{λ∈Λ}. ; (9) {I^{i}(x)}_{x∈X}.

Then the following elements correspond to each other by the bijections.

(1) Y^{1} ∗ Y^{2}. ; (2) {Y_{λ}^{1} ∗ Y_{λ}^{2}}_{λ∈Λ}. ; (3) {Y^{1}(x) ∗ Y^{2}(x)}_{x∈X}. ; (4) F^{1} ∗ F^{2}. ; (5) {F_{λ}^{1} ∗ F_{λ}^{2}}_{λ∈Λ}. ; (6) {F^{1}(x) ∗ F^{2}(x)}_{x∈X}.  (7) I^{1}I^{2}. ; (8) {I_{λ}^{1}I_{λ}^{2}}_{λ∈Λ}. ; (9) {I^{1}(x)I^{2}(x)}_{x∈X}.

Proof. This follows from Theorem 10.8 and Lemma 12.1. □

As a corollary of Theorem 12.2, we obtain a classification of the bilocalizing subcategories of QCoh X. They are defined as follows.

Definition 12.3. Let A be a Grothendieck category. A prelocalizing subcategory X of A is called a bilocalizing subcategory of A if X is both localizing and closed.

Bilocalizing subcategories have the following characterization.

Proposition 12.4. Let A be a Grothendieck category, and let X be a localizing subcategory of A. Then X is a bilocalizing subcategory of A if and only if the canonical functor A → A/X has a left adjoint.

Proof. [Pop73, Theorem 4.21.1]. □

For a ring Λ, the bilocalizing subcategories of Mod Λ are classified by the idempotent two-sided ideals of Λ.

Definition 12.5. Let Λ be a ring. A two-sided ideal I of Λ is called idempotent if I^{2} = I holds.

Proposition 12.6. Let Λ be a ring.

(1) For each i = 1, 2, let Y_{i} be a closed subcategory of Mod Λ, and let I_{i} be the corresponding two-sided ideal of Λ by the bijection in Theorem 11.3. Then Y_{1} ∗ Y_{2} corresponds to I_{2}I_{1} by the bijection.

(2) The bijection in Theorem 11.3 restricts to a bijection

{ bilocalizing subcategories of Mod Λ } → { idempotent two-sided ideals of Λ }.

Proof. (1) For two-sided ideals I_{1} and I_{2} of Λ, it is straightforward to show that F(I_{1}) ∗ F(I_{2}) = F(I_{1}I_{2}). Therefore the claim follows from Theorem 10.3 (1).

(2) This follows from (1). □

A subobject I of O_{X} is called idempotent if I^{2} = I holds. We classify the bilocalizing subcategories of QCoh X as follows.

Corollary 12.7. The bijections in Theorem 11.9 restrict to bijections between following sets.

(1) The set of bilocalizing subcategories of QCoh X.

(2) The set of families {Y_{λ} ⊂ QCoh U_{λ}}_{λ∈Λ} of bilocalizing subcategories such that Y_{λ}|_{U_{μ}} = Y_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(3) The set of families {Y(x) ⊂ Mod O_{X,x}}_{x∈X} of bilocalizing subcategories such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and a bilocalizing subcategory Y′ of QCoh U_{λ} satisfying Y′_{y} = Y(y) for each y ∈ U_{λ}.

(4) The set of principal filters of O_{X} closed under products.

(5) The set of families {F_{λ}}_{λ∈Λ}, where F_{λ} is a principal filter of O_{U_{λ}} closed under products for each λ ∈ Λ, such that F_{λ}|_{U_{μ}} = F_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(6) The set of families {F(x)}_{x∈X}, where F(x) is a principal filter of O_{X,x} closed under products for each x ∈ X, such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and a principal filter of subobjects F′ of O_{U_{λ}} which is closed under products and satisfies F′_{y} = F(y) for each y ∈ U_{λ}.

(7) The set of idempotent subobjects of O_{X}.

(8) The set of families {I_{λ}}_{λ∈Λ}, where I_{λ} is an idempotent subobjects of O_{U_{λ}} for each λ ∈ Λ, such that I_{λ}|_{U_{μ}} = I_{μ} for each λ, μ ∈ Λ with U_{μ} ⊂ U_{λ}.

(9) The set of families {I(x)}_{x∈X}, where I(x) is an idempotent ideal of O_{X,x} for each x ∈ X, such that for each x ∈ X, there exist λ ∈ Λ with x ∈ U_{λ} and an idempotent subobject I′ of O_{U_{λ}} satisfying I′_{y} = I(y) for each y ∈ U_{λ}.

Proof. This follows from Theorem 12.2. □

Example 12.8. In the setting of Example 11.12, the set of bilocalizing subcategories of Mod k[x] is

{ Y_{r} | r = {0}_{a∈k} } ∪ {Mod k[x]} = {0, Mod k[x]}.

We show that the sets in Corollary 12.7 also bijectively correspond to the set of open closed subsets of X. We start with the following well-known fact on a commutative noetherian ring.

Lemma 12.9. Let R be a commutative noetherian ring, and let I be an idempotent ideal of R. Then there exists an ideal J of R such that R = I ⊕ J in Mod R. In particular, the subset Supp(R/I) of Spec R is open and closed.

Proof. By Nakayama’s lemma ([Mat89, Theorem 2.2]), there exists a ∈ R such that aI = 0 and 1 − a ∈ I. Then a^{2} = a and (1 − a)R = I. By letting J = aR, we obtain R = I ⊕ J, and Spec R is the disjoint union of the closed subsets V (I) and V (J) determined by I and J, respectively. □

The idempotence of a subobject of O_{X} is characterized in terms of the corresponding closed subscheme.

Lemma 12.10. Let X be a locally noetherian scheme. Let I be a subobject of O_{X}, and let Y be the corresponding closed subscheme of X by the bijection in Proposition 11.10. Then I is idempotent if and only if Y is also an open subscheme of X.

Proof. Assume that I is idempotent. For each open affine subscheme U of X, the subobject I|_{U} of O_{U} is idempotent. By Lemma 12.9, the subset Supp(O_{U}/I|_{U}) of U is open and closed. Since

Supp \frac{O_{U}}{I|_{U}} = U ∩ Supp \frac{O_{X}}{I} ,

the underlying space Supp(O_{X}/I) of Y is an open subset of X. For each y ∈ Y, the ideal I_{y} of O_{X,y} is idempotent, and (O_{X}/I)_{y} ≠ 0. Hence I_{y} = 0. It follows that O_{Y} = (O_{X}/I)|_{Y} = O_{X}|_{Y}. Conversely, assume that Y is also an open subscheme. Let i: Y ↪ X be the immersion. There is an exact sequence

0 → I → O_{X} → i_{∗}(O_{X}|_{Y}) → 0.

For each x ∈ X,

I_{x} = \begin{cases}0 & if x ∈ Y \\ O_{X,x} & if x ∉ Y \end{cases} ,

and hence I_{x} is idempotent. It follows that I is idempotent. □

Corollary 12.11. Let X be a locally noetherian scheme. Then there exist bijections between the following sets.

(1) The set of bilocalizing subcategories of QCoh X.

(2) The set of idempotent subobjects of O_{X}.

(3) The set of closed subschemes of X which are also open subschemes.

(4) The set of subsets of X which are open and closed. The bijection (1)↔(2) is in Corollary 12.7. The bijection (2)↔(3) is induced by the bijection in Proposition 11.10. For each element Y of (3), the corresponding element of (4) is the underlying space of Y.

Proof. This follows from Corollary 12.7 and Lemma 12.10. □

By using the classification of the prelocalizing (resp. localizing, closed) subcategories of Mod k[x], we can obtain a classification of the prelocalizing (resp. localizing, closed) subcategories for the projective line.

Example 12.12. Let k be an algebraically closed field, and consider the projective line X = ℙ_{k}^{1}. Denote by Φ the set of closed points in X. For each r ∈ ∏_{x∈Φ}(ℤ_{≥0} ∪ {∞}), we define a prelocalizing subcategory Y_{r} of QCoh X by

Y_{r} = { M ∈ QCoh X | M_{x}𝔪_{x}^{r(x)} = 0 for each x ∈ Φ with r(x) ≠ ∞ }.

Then by the main results (Theorem 9.14, Corollary 10.9, Theorem 11.9, and Corollary 12.7) and the examples on Spec k[x] (Example 9.17, Example 10.10, Example 11.12, and Example 12.8), the set of prelocalizing subcategories of QCoh X is

{ Y_{r} | r ∈ ∏_{x∈Φ}(ℤ_{≥0} ∪ {∞}) } ∪ {QCoh X},

the set of localizing subcategories of QCoh X is

{ Y_{r} | r ∈ ∏_{x∈Φ}{0, ∞} } ∪ {QCoh X},

the set of closed subcategories of QCoh X is

{ Y_{r} | r ∈ ⨁_{x∈Φ} ℤ_{≥0} } ∪ {QCoh X},

and the set of bilocalizing subcategories of QCoh X is

{ Y_{r} | r = {0}_{x∈Φ} } ∪ {QCoh X} = {0, QCoh X}.

Example 12.13. For each i ∈ ℤ, let k_{i} be a field, and let U_{i} := Spec k_{i}. Consider the disjoint union X := ∐_{i∈ℤ} U_{i}. For each subset B of ℤ, define a prelocalizing subcategory Y_{B} of QCoh X by

Y_{B} = { M ∈ QCoh X | M|_{U_{i}} = 0 for each i ∈ ℤ \ B }.

Then by Theorem 9.14, Corollary 10.9, Theorem 11.9, and Corollary 12.7, the set

{ Y_{B} | B ⊂ ℤ }

is the set of prelocalizing subcategories of QCoh X, and every prelocalizing subcategory of QCoh X is bilocalizing. Therefore every local filter of O_{X} is a principal filter. For each subset B of ℤ, let I_{B} be the idempotent subobject of O_{X} corresponding to the bilocalizing subcategory Y_{B}. Then the filter

F = { I_{B} | ℤ \ B is a finite set }

of O_{X} is not a local filter since F is not a principal filter.

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Ryo Kanda

Graduate School of Mathematics Nagoya University Furo-cho, Chikusa-ku Nagoya-shi, Aichi-ken 464-8602, Japan ryo.kanda.math@gmail.com 1466

Documenta Mathematica 20 (2015)