FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRAHIROKI MATSUI AND RYO TAKAHASHI
Abstract. Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studiedin Matsui–Nam–Takahashi–Tri–Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R.
In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), andexplore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrationsof subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due
to Gabriel, Krause, Neeman, Takahashi and Angeleri Hügel–Marks–Šťovı́ček–Takahashi–Vitória.

Contents
1. Introduction
2. Preliminaries
3. Basic properties of n-wide subcategories and n-coherent subsets
4. Classification of n-wide (E, ⊕)-closed subcategories of Mod R
5. Classification of n-uniform localizing subcategories of D(Mod R)
6. Several classifications of subcategories by restriction
7. A remark on cofinite modules
References

1
2
3
6
8
11
14
16

1. Introduction
A localizing subcategory of an abelian category is defined to be a full subcategory closed under coproducts, extensions,subobjects and quotient objects. This notion was introduced by Gabriel [20] to study the reconstruction problem of anoetherian scheme from the category of quasi-coherent sheaves. Since then, localizing subcategories of an abelian category
have been explored by a lot of authors to classify them and study abelian categories geometrically; see [22, 23, 24, 27, 30, 48].
A wide subcategory of an abelian category is by definition a full subcategory closed under extensions, kernels and cokernels.
In recent years, wide subcategories have actively been studied in representation theory of algebras; see [3, 12, 32, 35, 47, 49].
A localizing subcategory of a triangulated category is defined as a full triangulated subcategory closed under coproducts.
As with localizing subcategories of an abelian category, one of the main topics in the study of localizing subcategories
of a triangulated category is to classify them. The first classification theorem has been obtained by Neeman [42] forthe unbounded derived category of a commutative noetherian ring. Nowadays, localizing subcategories of a triangulated
category are widely and deeply investigated in many areas of mathematics; see [4, 6, 14, 15, 28] for instance.
Let R be a commutative noetherian ring. Gabriel [20] gives a complete classification of the localizing subcategories ofthe module category Mod R by specialization-closed subsets of Spec R. Krause [32] introduces the notion of a coherent
subset of Spec R to classify the wide subcategories of Mod R closed under direct sums, which extends Gabriel’s classification
theorem. Takahashi [48] classifies the E-stable1 subcategories of Mod R closed under direct sums and summands, extending
Krause’s classification theorem.
In [36] the notion of an n-wide subcategory of Mod R is introduced for each n ∈ N. In this paper we extend this ton = ∞. Also, we introduce the notions of an n-uniform subcategory of the derived category D(Mod R) of Mod R, andan n-coherent subset of Spec R for each n ∈ N ∪ {∞}. The (small) support in the sense of [18], denoted by supp, is ourfundamental tool. We prove the following theorem; the notation and terminology are explained in Convention 2.1 andDefinitions 2.2, 2.4, 3.1, 3.5, 5.1, 5.3, 6.4.
2010 Mathematics Subject Classification. 13C60, 13D09, 13D45.
Key words and phrases. bilocalizing subcategory, bireflective Giraud subcategory, bismashing subcategory, derived category, E-closed subcategory, s E-closed subcategory, generalization-closed subset, local cohomology, localizing subcategory, module category, n-coherent subset,
n-consistent subcategory, n-uniform subcategory, n-wide subcategory, specialization-closed subset, (small) support.
Matsui was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158. Takahashi was partly supported by JSPS Grants-in-Aid for
Scientific Research 16K05098 and JSPS Fund for the Promotion of Joint International Research 16KK0099.
1This means s E-closed in our sense.
1

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HIROKI MATSUI AND RYO TAKAHASHI

Theorem 1.1 (Theorems 4.10, 5.5, 6.3 and Corollaries 4.11, 4.13, 5.10, 6.6). Let R be a commutative noetherian ring,and let n ∈ N ∪ {∞}. Define the sets
E = {(s E, ⊕, ⊖)-closed subcategories of Mod R},
S = {subsets of Spec R},
L = {localizing subcategories of D(Mod R)},E′ = {(s E, ⊕, ⊖, Π)-closed subcategories of Mod R},
S′ = {generalization-closed subsets of Spec R},
L′ = {bilocalizing subcategories of D(Mod R)},

Then there is a commutative diagram! " ··· "E
E∞ "supp ∼=!
S#

supp ∼=

L

supp ∼=!
S# ∞ "! " En "! " ··· "supp ∼=! " ··· "supp ∼=

L∞ "En = {n-wide (E, ⊕)-closed subcategories of Mod R},
Sn = {n-coherent subsets of Spec R},
Ln = {n-uniform localizing subcategories of D(Mod R)},E′n = {n-wide (E, ⊕, Π)-closed subcategories of Mod R},
S′n = {n-coherent generalization-closed subsets of Spec R},
L′n = {n-uniform bilocalizing subcategories of D(Mod R)}.

!
! " Sn "#! " ··· "supp ∼=! " ··· "! " Ln "! " ··· "! " E2 "! " E1 "! " E0supp ∼=

supp ∼=

supp ∼=

supp ∼=

supp ∼=

supp ∼=! " L2 "! " L1 "! " L0!
! " S2 "#!
! " S1 "#!
! " S0#!
of sets, where the horizontal arrows are inclusion maps and the vertical arrows given by X &→ supp X = X∈X supp X arebijections. This diagram restricts to the one where all the E∗ , S∗ , L∗ are replaced with E′∗ , S′∗ , L′∗ respectively. One has
E1 = {⊕-closed wide subcategories of Mod R},
S1 = {coherent subsets of Spec R},
L1 = {cohomology-closed localizing subcategories of D(Mod R)},E′1 = {bireflective Giraud subcategories of Mod R},
S′1 = {generalization-closed coherent subsets of Spec R},
L′1 = {cohomology-closed bilocalizing subcategories of D(Mod R)},E0 = {localizing subcategories of Mod R},
S0 = {specialization-closed subsets of Spec R},
L0 = {smashing subcategories of D(Mod R)},E′0 = {bilocalizing subcategories of Mod R},
S′0 = {clopen subsets of Spec R},
L′0 = {bismashing subcategories of D(Mod R)}.

The bijections E0 → S0 , E → S, S0 ← L0 , S ← L, E1 → S1 ← L1 , and E′1 → S′1 ← L′1 in Theorem 1.1 arerespectively the same as the ones given by Gabriel [20], Takahashi [48], Neeman [42], Neeman [42], Krause [32], andAngeleri Hügel, Marks, Šťovı́ček, Takahashi and Vitória [2]. By restricting the diagram in the theorem in other ways, we
also give classifications of certain thick subcategories of Db (Mod R), D+ (Mod R) and D(Mod R)fid , the last one of which
denotes the full subcategory of D(Mod R) consisting of complexes of finite injective dimension.
The organization of this paper is as follows. Section 2 is devoted to recalling basic facts on minimal injective resolutions
and supports of modules/complexes. In Section 3, we introduce the notions of n-wide subcategories of Mod R, n-coherent
subsets of Spec R for n ∈ N ∪ {∞} and investigate fundamental properties of them. In Section 4, we give a classificationof n-wide subcategories of Mod R via n-coherent subsets of Spec R, which contains results in [20, 32, 48]. In Section
5, we introduce the notions of n-uniform and n-consistent subcategories of D(Mod R) and classify some of them, whichcontains results in [32, 42]. In Section 6, we consider two kinds of restrictions of classifications obtained in the previous
two sections. The first one is the restriction to classifications of Π-closed subcategories and hence the diagram in Theorem
1.1 is completed here. The second one concerns classifications of n-uniform subcategories of D(Mod R)fid , Db (Mod R)
and D+ (Mod R). As an application we obtain a higher-dimensional analogue of Brüning’s classification theorem [10]. In
Section 7, we apply the classification of n-wide subcategories to a problem presented by Hartshorne [26]. We consider aweakened version of the notion of cofinite modules and study wideness of the subcategory of those modules.
2. Preliminaries
In this section, we recall several definitions which play important roles throughout this paper.
Convention 2.1. Throughout this paper, we use the following convention. We assume that all rings are commutativeand noetherian and all subcategories are full. We set N := Z!0 = {0, 1, 2, . . . }. Let R be a ring. We denote by Mod R
the category of (all) R-modules and by D(Mod R) the (unbounded) derived category of the abelian category Mod R. We
denote by ER (M ) the injective hull of an R-module M and set κ(p) = Rp /pRp for each p ∈ Spec R. We set n + ∞ = ∞and n − ∞ = −∞ for all n ∈ Z. We may simply say that a subcategory is ⊕-closed (resp. ⊖-closed, Π-closed) to mean that
it is closed under (existing) direct sums (resp. direct summands, (existing) direct products)2. We may omit subscripts
and superscripts as long as there is no danger of confusion. We may tacitly use statements given in Remarks.
2In this paper, we say that a subcategory X of an additive category C is closed under direct sums (resp. direct products) provided that if{Xλ }λ∈Λ is a family of objects in X such that the direct sum Y =belongs to X .

!

λ∈Λ Xλ (resp.

the direct product Y = Πλ∈Λ Xλ ) exists in C, then YFILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

3

Definition 2.2. (1) Let X be a subcategory of Mod R. We say that X is Serre if it is closed under extensions, submodules
and quotient modules. We say that X is localizing if it is Serre and closed under direct sums, and that X is bilocalizing
if it is localizing and closed under direct products. We say that X is wide if it is closed under extensions, kernels andcokernels.
(2) Let X be a subcategory of D(Mod R). We say that X is thick if it is closed under extensions, shifts and direct
summands. We say that X is localizing if it is thick and closed under direct sums, and that X is bilocalizing if it islocalizing and closed under direct products.
Remark 2.3. We define a strictly localizing subcategory of an abelian category A (resp. a triangulated category T ) as a
Serre (resp. thick) subcategory X such that the quotient functor A → A/X (resp. T → T /X ) has a right adjoint, see [31].
In the situation where A = Mod R (resp. T = D(Mod R)) with R being a commutative noetherian ring, the localizing
subcategories and the strictly localizing subcategories are the same. Indeed, since A = Mod R is a Grothendieck category,
this follows by [20, Corollaire 1, 375 page]. For the category T = D(Mod R), it is shown in [42, Theorem 2.8] that every
localizing subcategory is generated by a set of objects and hence every localizing subcategory is strictly localizing by [5,Lemma 2.1].
Definition 2.4. (1) For X ∈ D(Mod R) we define the (small) support of X by
∼ 0}.
supp X = {p ∈ Spec R | X ⊗L κ(p) ∕=R

R(2) Let X be a complex of R-modules with Hi X = 0 for i ≪ 0. Then one can take a minimal injective resolution
∂ i−1

∂i

∂ i+1

ER (X) = (0 → · · · −−−→ EiR (X) −→ Ei+1−−→ · · · )R (X) −of X, that is, a bounded below complex of injective R-modules quasi-isomorphic to X such that EiR (X) is the injective
hull of the kernel of the map ∂ i for all i.
(3) Let M be an R-module. For an integer i > 0, we denote by ✵i M the ith cosyzygy of M , that is, the image of the(i − 1)st diﬀerential map in the minimal injective resolution of M . We set ✵0 M = M and ✵M = ✵1 M .
(4) We say that a subcategory X of Mod R is E-closed if Ei (M ) ∈ X for all M ∈ X and i ! 0. This is equivalent tosaying that each object M ∈ X admits an injective resolution I with I i ∈ X for all i ! 0. If in addition X satisfies
M ∈ X whenever Ei (M ) ∈ X for all i ! 0, we say that X is s E-closed. In a similar fashion, s E-closed and E-closed
subcategories of D+ (Mod R) are defined by using minimal injective resolutions of complexes.
Notation 2.5. Let X be a subcategory of Mod R and Φ a subset of Spec R. For ∆ ∈ {supp, Ass} we denote by ∆−1X (Φ)
the subcategory of X consisting of objects X with ∆(X) ⊆ Φ.
Remark 2.6. [36, Proposition 2.3 and Corollary 2.5][37, Theorem 18.7]!
(1) Let X ∈ D(Mod R). Let I be a complex of injective R-modules quasi-isomorphic to X. Then supp X ⊆ i∈Z Ass I i .
The equality holds if H≪0 (X) = 0 and I = E(X). In particular, for each R-module M there is an inclusion Ass M ⊆supp M , whose equality holds if M is injective.
(2) Let Φ be a subset of Spec R. Then supp−1
Mod R (Φ) is closed under direct sums, direct summands and extensions. One−1−1
also has suppMod R (Φ) ⊆ AssMod R (Φ).
(3) Let M ∈ Mod R, p ∈ Spec R and i ∈ Z. The number µi (p, M ) = dimκ(p) ExtiRp (κ(p), Mp ) is equal to the cardinality of
the number of direct summands
ER (R/p) of Ei (M ). Thus, there is a direct sum decomposition into indecomposable"i
injective modules E (M ) ∼= p∈Spec R ER (R/p)⊕µi (p,M ) .
3. Basic properties of n-wide subcategories and n-coherent subsets

In this section, we introduce the notion of n-coherent subsets of Spec R with n ∈ N ∪ {∞}, which form a filtrationof classes of subsets of Spec R. The class of specialization-closed subsets and that of coherent subsets introduced in [32]appear among this filtration. As a categorical counterpart of this filtration, we introduce the notion of n-wide subcategoriesof Mod R with n ∈ N ∪ {∞}, which has been introduced in [36] for n ∈ N. Likewise, two important classes of subcategories,that is, those of Serre subcategories and wide subcategories, appear in this filtration.
First of all, we recall here the definition of an n-wide subcategory and extend it to n = ∞.
Definition 3.1. Let n be a nonnegative integer (resp. n = ∞). A subcategory X of Mod R is said to be closed undern-kernels3 if for every exact sequence 0 → M → X 0 → X 1 → X 2 → · · · in Mod R with X i ∈ X for all 0 " i " n(resp. all i ! 0) the module M is in X . Dually, X is said to be closed under n-cokernels if for every exact sequence· · · → X2 → X1 → X0 → M → 0 in Mod R with Xi ∈ X for all 0 " i " n (resp. all i ! 0) the module M is in X . We saythat a subcategory X of Mod R is n-wide if it is closed under extensions, n-kernels and n-cokernels.
3Our definitions of n-kernels and n-cokernels are diﬀerent from those in [29, Definition 2.2].

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HIROKI MATSUI AND RYO TAKAHASHI

Remark 3.2. Let X be a subcategory of Mod R.
(1) Let n ∈ N∪{∞}. If X is closed under n-kernels (resp. n-cokernels), then X is closed under (n+1)-kernels (resp. (n+1)cokernels) and ∞-kernels (resp. ∞-cokernels). In particular, any n-wide subcategory of Mod R is both (n + 1)-wide
and ∞-wide. In general, there is an (n + 1)-wide subcategory which is not n-wide, see the example below.
(2) X is closed under 0-kernels (resp. 0-cokernels) if and only if X is closed under submodules (resp. quotient modules).
In particular, X is 0-wide and ⊕-closed if and only if it is localizing.
(3) X is closed under 1-kernels (resp. 1-cokernels) if and only if X is closed under kernels (resp. cokernels). In particular,
X is 1-wide if and only if it is wide.
(4) Let n ∈ N ∪ {∞}. If X is closed under n-kernels (resp. n-cokernels) e.g., if X is n-wide, then X is closed under
direct summands and kernels of epimorphisms (resp. cokernels of monomorphisms). Indeed, for any R-modules M, N
( 0 01 )( 1 00 )( 0 01 )the sequence · · · −−0−−→ M ⊕ N −−0−−→ M ⊕ N −−0−−→ · · · is exact, whose cocycles are M, N . This shows that X is
closed under direct summands. For a short exact sequence 0 → L → M → N → 0, one has the long exact sequence
0 → L → M → N → 0 → · · · → 0 → 0. Hence X is closed under kernels of epimorphisms provided X is closed undern-kernels and similarly for X being closed under n-cokernels.
(5) If X is closed under ∞-kernels and E-closed, then X is s E-closed. In particular, every n-wide E-closed subcategory of
Mod R with n ∈ N ∪ {∞} is s E-closed.
(6) In [44], they use the notation Presn (X ) for the subcategory of R-modules M admitting a presentation of length n by
objects of X i.e., an exact sequence Xn → Xn−1 → . . . → X0 → M → 0 with Xi ∈ X for all i = 0, 1, . . . , n. Using
this notation, we have that X is closed under n-cokernels if and only if Presn (X ) ⊆ X . Dually, using the subcategory
Copresn (X ) consisting of R-modules admitting copresentation of length n by objects of X , we have that X is closed
under n-kernels if and only if Copresn (X ) ⊆ X .
Example 3.3. (1) Fix an R-module C. Then we define the C-grade of an R-module M by C-gradeR (M ) := inf{i ! 0 |ExtiR (C, M ) ∕= 0}. One can easily check that the subcategory of Mod R consisting of R-modules with C-gradeR (M ) " n
is closed under n-kernels (cf. [11, Proposition 1.2.9]). Dually, the C-cograde of M is defined by C-cogradeR (M ) :=inf{i ! 0 | TorR
i (C, M ) ∕= 0}, see [45, Corollary 3.11]. Then the subcategory of Mod R consisting of R-modules with
C-cogradeR (M ) " n is closed under n-cokernels.
(2) ([36, Theorem 4.3]) Let a be an ideal of R with cohomological dimension at most n, i.e., H>na (M ) = 0 for all M . Then−1supp−1(D(a))
isn-wide.
For
example,ifaisgeneratedbyn-elements,
then
supp
(D(a))
is n-wide. Moreover,
Mod R
Mod R
this subcategory is not (n − 1)-wide provided a is generated by a regular sequence of length n.
For a subset Φ of Spec R and a prime ideal p of R, we set
Φp := {P ∈ Spec Rp | P ∩ R ∈ Φ}.

Here, P ∩ R means the pullback of P along the canonical ring homomorphism R → Rp . Then Φp is homeomorphic to thegeneralization-closure {q ∈ Φ | q ⊆ p} of {p} in Φ by the assignment P &→ P ∩ R. The following fact is frequently used inthis paper.
Proposition 3.4. Let n be either a nonnegative integer or ∞. Let Φ be a subset of Spec R. Then supp−1
Mod R (Φ) is n-wide−1
if and only if suppMod Rp (Φp ) is n-wide for each p ∈ Spec R.
Proof. Note that for a multiplicatively closed subset S of R and an R-module M one has suppRS MS = {P ∈ Spec RS |−1
P ∩ R ∈ suppR M }. Hence M ∈ supp−1
Mod R (Φ) if and only if Mp ∈ suppMod Rp (Φp ) for all p ∈ Spec R. The “if” part of the
proposition is deduced from this. We show the “only if” part from now on. Let n ∈ N (resp. n = ∞), fix a prime ideal pof R, and take an exact sequence 0 → M → X 0 → X 1 → · · · in Mod Rp with X i ∈ supp−1
Mod Rp (Φp ) for all 0 " i " n (resp.
−1ifor all i ! 0). Then we see that X ∈ suppMod R (Φ) for all 0 " i " n (resp. for all i ! 0). The n-wideness of supp−1
Mod R (Φ)−1yields M ∈ supp−1
(Φ),
that
is,
supp
M
⊆
Φ.
We
then
observe
that
supp
M
⊆
Φ
,
namely,
M
∈
supp
p
R
Rp
Mod R
Mod Rp (Φp ).
−1
This shows that suppMod Rp (Φp ) is closed under n-kernels. Closure under n-cokernels is shown similarly.
#
Next, we introduce the notion of n-coherent subsets of Spec R. Denote by Inj R the subcategory of Mod R consisting of
injective R-modules.
Definition 3.5. Let n be a nonnegative integer (resp. n = ∞). A subset Φ of Spec R is called n-coherent if for an exact
sequence · · · → I2 → I1 → I0 → C → 0 in Mod R with Ii ∈ Inj R and Ass Ii ⊆ Φ for all 0 " i " n (resp. all i ! 0) themodule C is embedded in some J ∈ Inj R with Ass J ⊆ Φ.
Remark 3.6. (1) Let n ∈ N (resp. n = ∞). A subset Φ of Spec R is n-coherent if and only if for an exact sequence· · · → I1 → I0 → C → 0 in Mod R with Ii ∈ Ass−1
Inj R (Φ) for all 0 " i " n (resp. all i ! 0) one has AssR C ⊆ Φ.

FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

5(2) Let n ∈ N∪{∞}. Then n-coherence implies (n+1)-coherence and ∞-coherence. In general, there is an (n+1)-coherent
subset which is not n-coherent.
(3) The 1-coherent subsets are the same as the coherent subsets in the sense of [32].
(4) Let n ∈ N ∪ {∞}. If supp−1
Mod R (Φ) is closed under n-cokernels, then Φ is n-coherent.
Example 3.7. ([36, Example 4.11]) Let x := x1 , . . . , xn be a sequence of elements of R. Then D(x) := Spec R \ V(x) is
n-coherent which is not (n − 1)-coherent if x is an R-regular sequence.
The latter assertion of the following proposition includes [32, Proposition 4.1(2)].
Proposition 3.8. The 0-coherent subsets are the same as the specialization-closed subsets. In particular, specializationclosed subsets are coherent.
Proof. The second assertion follows from the first one and Remark 3.6(2)(3). To show the first assertion, let Φ be asubset of Spec R. Suppose that Φ is specialization-closed. Then by [36, Theorem 1.1] the subcategory Ass−1
Mod R (Φ) is−1closed under quotient modules. Let I ↠ C be a surjective homomorphism with I ∈ Ass−1
(Φ).
Then
I
is
in
Ass
Inj R
Mod R (Φ),−1and so is C. Hence E(C) ∈ AssInj R (Φ), which shows that Φ is 0-coherent. Conversely, assume that Φ is 0-coherent. Let−100
M ∈ Ass−1
Mod R (Φ). Then E (M ) = E(M ) ∈ AssInj R (Φ) and the surjection E (M ) ↠ ✵M shows that ✵M is embedded in−1−1
a module in AssInj R (Φ), which implies E1 (M ) = ER (✵M ) ∈ AssInj R (Φ). Iterating this procedure, we see that Ei (M ) ∈
−1
−1
−1
Ass−1
Inj R (Φ) for all i ! 0, and M ∈ suppMod R (Φ). Thus AssMod R (Φ) = suppMod R (Φ), and Φ is specialization-closed by [36,Theorem 1.1].
#
The following extends [32, Proposition 4.1(1)(4)] on coherent subsets to n-coherent ones.
Proposition 3.9. Let n be either a nonnegative integer or ∞.
#(1) Let {Φλ }λ∈Λ be a family of n-coherent subsets of Spec R. Then λ∈Λ Φλ is n-coherent.
(2) A subset Φ of Spec R is n-coherent if and only if Φp is n-coherent for each p ∈ Spec R.

Proof. Let n be a nonnegative integer (resp. n = ∞).
#(1) Let · · · → I2 → I1 → I0 → C → 0 be an exact sequence with Ii ∈ Ass−1
Inj R ( λ∈Λ Φλ ) for all 0 " i " n (resp. all−1
i ! 0). Then by assumption, C is embedded in some Iλ ∈ Ass−1
Inj R (Φλ ) for each λ ∈ Λ. Hence ER (C) ∈ AssInj R (Φλ ) for all−1 #λ ∈ Λ, that is, ER (C) ∈ AssInj R ( λ∈Λ Φλ ).
(2) First of all, recall the fundamental fact that for a multiplicatively closed subset S of R one has (Inj R)S ⊆ Inj(RS ) ⊆Inj R and AssRS (MS ) = {P ∈ Spec RS | P ∩ R ∈ AssR M } for M ∈ Mod R (see [37, Theorem 6.2]).
The “only if” part: Fix p ∈ Spec R. Let · · · → I2 → I1 → I0 → C → 0 be an exact sequence of Rp -modules with−1
Ii ∈ Ass−1
Inj Rp (Φp ) for all 0 " i " n (resp. all i ! 0). Then Ii ∈ AssInj R (Φ) for all 0 " i " n (resp. all i ! 0), and the−1n-coherence of Φ implies that C is embedded in some J ∈ Ass−1
Inj R (Φ). Hence C = Cp is embedded in Jp ∈ AssInj Rp (Φp ).
The “if” part: Let · · · → I2 → I1 → I0 → C → 0 be an exact sequence of R-modules with Ii ∈ Ass−1
Inj R (Φ) for all
0 " i " n (resp. all i ! 0). For each p ∈ Spec R the exact sequence · · · → (I2 )p → (I1 )p → (I0 )p → Cp → 0 of Rp -modules
is induced, and (Ii )p ∈ Ass−1
Inj Rp (Φp ) for all 0 " i " n (resp. all i ! 0). Since Φp is n-coherent, Cp is embedded in some−1−1J(p) ∈ AssInj Rp (Φp ). Hence ERp (Cp ) ∈ Ass−1#
Inj Rp (Φp ) for all p ∈ Spec R. It is easy to see that ER (C) ∈ AssInj R (Φ).
Let Φ be a specialization-closed subset of Spec R, i an integer and M an R-module. We denote by HiΦ (M ) the ith local
cohomology module of M with respect to Φ. The details of local cohomology with respect to a specialization-closed subset
are found in [36, §3] for instance. Here we relate n-coherence to local cohomology.

Proposition 3.10. Let Φ be a specialization-closed subset of Spec R, and let n ! 0 be an integer. Then Φ∁ is n-coherent#n
if and only if H>n
Φ (M ) = 0 for all R-modules M with HΦ (M ) = 0.
f0

f1

fn

f n+1

∁
Proof. To show the “if” part, take an exact sequence 0 → K −→ I 0 −→ · · · −−→ I n −−−→ C → 0 with I i ∈ Ass−1
Inj R (Φ ) fori
all 0 " i " n. It follows from [36, Proposition 4.5] that HiΦ (K) = 0 for 0 " i " n. By assumption Hn+1
Φ (K) = 0. Let K
i
i
i
i+1
be the image of f for each 0 " i " n + 1. There is an exact sequence 0 → K → I → K→ 0 for each 0 " i " n. We
0nget H1Φ (K n ) ∼
(K
)
=
0.
Combining
this
with
the
equalityΓ(I)= 0 from [36, Proposition= H2Φ (K n−1 ) ∼= ··· ∼
= Hn+1ΦΦ∁
3.2(1)], we see that ΓΦ (K n+1 ) = 0. The module C = K n+1 is embedded in ER (K n+1 ), which belongs to Ass−1
Inj R (Φ ).
#n
We show the “only if” part. By Remark 3.6(2), it suﬃces to verify Hn+1
Φ (M ) = 0 for an R-module M with HΦ (M ) = 0.
−1
0
n
i
∁
By [36, Proposition 4.5] we get an exact sequence 0 → M → I → · · · → I with I ∈ AssInj R (Φ ) for all i " n. As Φ is∁n-coherent, this is extended to an exact sequence 0 → M → I 0 → · · · → I n → I n+1 with I n+1 ∈ Ass−1
Inj R (Φ ). Using [36,n+1Proposition 4.5] again, we obtain HΦ (M ) = 0.
#

6

HIROKI MATSUI AND RYO TAKAHASHI

Remark 3.11. For the case n = 1, this proposition recovers [2, Theorem 4.9 (2) ⇔ (4)]. Indeed, for an R-module M , there
is an R-module DΦ (M ) such that HiΦ (DΦ (M )) = 0 for i = 0, 1 and HiΦ (M ) ∼= HiΦ (DΦ (M )) for i > 1, see [36, Proposition
3.9] for details. Therefore, by the above proposition, Φ∁ is coherent if and only if H>1
Φ (M ) = 0 for all R-modules M .
4. Classification of n-wide (E, ⊕)-closed subcategories of Mod R
In the previous section, we recalled/introduced the notions of an n-wide subcategory of Mod R and an n-coherent subsetof Spec R and studied their basic properties. The aim of this section is to explore the relationship between them.
We introduce a new series of subcategories of modules, and investigate some properties of them.
nNotation 4.1. Let X be a subcategory of Mod R. For n ∈ N we denote by CX
the subcategory of Mod R consisting of−1
01
nmodules M admitting an exact sequence 0 → M → I → I → · · · → I with I i ∈ Inj R ∩ X for all i. Set CX= Mod R.
−1nn
When we consider X := AssMod R (Φ) for a subset Φ of Spec R, we set CΦ := CX for each n.

Here, we list basic properties of the above series of subcategories, which easily follow from the definition.
Remark 4.2. Let n ! 0 be an integer, X a subcategory of Mod R and Φ a subset of Spec R.
ni(1) The subset Φ is n-coherent if and only if CΦ= CΦ
for all i ! n.
#
#
−1
0
1
i
i
(2) There is a filtration Mod R = CX ⊇ CX
⊇ CX
⊇ · · · ⊇ i!0 CX
of subcategories, and the equality i!0 CΦ
=
−1
−1
n
suppMod R (Φ) holds. In particular,# Φi is n-coherents if and only if suppMod R (Φ) = CΦ .
(3) If X is E-closed and contains i!0 CX, then X is E-closed.
(4) Suppose that X is closed under direct summands. Then the following hold.
n(a) An R-module M is in CX
if and only if Ei (M ) ∈ X#for all 0 " i " n.
#ii(b) The subcategory X is E-closed if and only if X ⊆ i!0 CX. Hence, X is s E-closed if and only if X = i!0 CX.
fg

Lemma 4.3. Let X be a subcategory of Mod R closed under finite direct sums. Let 0 → L −→M −→ N → 0 be an exact
sequence of R-modules. Let n ! 0 be an integer. Then the first two of the following implications hold true, and so is thethird if X is closed under direct summands.
nnn(i) L ∈ CX
, N ∈ CX
⇒ M ∈ CX,n−1nn(ii) M ∈ CX , N ∈ CX ⇒ L ∈ CX
,
n−1
n−1
n
(iii) L ∈ CX , M ∈ CX ⇒ N ∈ CX.
Proof. The first implication directly follows from the horseshoe lemma. To show the second, take a lift I → J of g to
injective resolutions of M and N with I i , J i ∈ X for all 0 " i " n and 0 " j " n − 1. Taking the mapping cone, we getan injective resolution (0 → I 0 → J 0 ⊕ I 1 → · · · → J n−1 ⊕ I n → · · · ) of the module L, whose first n + 1 terms are in X .
nThis shows L ∈ CX
. The third implication is similarly shown: taking a lift I → J of f to injective resolutions of L and
M and the mapping cone, we get an injective resolution (0 → (J 0 ⊕ I 1 )/I 0 → J 1 ⊕ I 2 → · · · → J n−1 ⊕ I n → · · · ) of N
whose first n terms are in X , if X is closed under direct summands. Here, the injectivity of (J 0 ⊕ I 1 )/I 0 follows since I 0
is injective and I 0 → J 0 ⊕ I 1 is a monomorphism.
#nRemark 4.4. This lemma generalizes the depth lemma. Indeed, if we consider Φ := D(a) with a an ideal of R, then CΦ
is nothing but the class of R-modules M with grade(a, M ) > n in the sense of [11, Definition 9.1.1] by [19, Theorem 2.1]
and [36, Proposition 4.5].

The following result gives a way to construct an n-wide subcategory from a given n-coherent subset.
Proposition 4.5. Let n ! 0 be an integer, and let Φ be a subset of Spec R.
n(1) The subcategory CΦ
of Mod R is closed under direct sums, extensions and n-kernels.
ns(2) If Φ is n-coherent, then supp−1
Mod R (Φ) = CΦ is an n-wide ( E, ⊕)-closed subcategory of Mod R.
f0

f1

fn

n
Proof. (1) Let 0 → M −→ X0 −→ · · · −→ Xn be an exact sequence with Xi ∈ CΦfor all i. Let Ui be the image of fi . The
0
inclusions Un ↩→ Xn ↩→ ER (Xn ) show Un ∈ CΦ . Using the second implication in Lemma 4.3 inductively, we are done.
(2) The equality follows from Remark 4.2(2). By (1) it suﬃces to show that supp−1
Mod R (Φ) is closed under n-cokernels.
Consider an exact sequence 0 → M → X0 → X1 → · · · → Xn → N → 0 with Xi ∈ supp−1
Mod R (Φ) for all i. Applying thethird implication in Lemma 4.3 repeatedly and using Remark 4.2(2), we observe that N belongs to supp−1#
Mod R (Φ).

For an R-module M we denote by dimR M the (Krull) dimension of M , i.e., dimR M = sup{dim R/p | p ∈ SuppR M }.
The next proposition gives a suﬃcient condition for a subset of a given n-coherent subset to be again n-coherent.
Proposition 4.6. Let n ! 0 be an integer. Let Φ, Ψ be subsets of Spec R, and assume that Ψ is n-coherent. Suppose
either that (i) ht p " n for all p ∈ Φ or that (ii) dim R/p " n for all p ∈ Ψ. Then Ψ \ Φ is n-coherent.

FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

7

Proof. Let 0 → M → I 0 → · · · → I n → N → 0 be an exact sequence with I i ∈ Ass−1
Inj R (Ψ \ Φ) for all 0 " i " n. We wantto prove that Ass N ⊆ Ψ \ Φ. Since Ψ is n-coherent, we have Ass N ⊆ Ψ, and it is enough to show that Ass N ⊆ Φ∁ . Takeany p ∈ Φ. Then p ∈/ Ass I i for all 0 " i " n. Hence ΓpRp (Ipi ) = 0, and therefore HipRp (Mp ) = 0 for all 0 " i " n.
We claim that dimRp Mp " n. Indeed, in case (i), the statement holds since dim Rp = ht p " n. In case (ii), we have
SuppR M ⊆ SuppR I 0 ⊆ cl(Ψ), where cl(Ψ) stands for the set of prime ideals p such that p ⊇ q for some q ∈ Ψ. It is easy
to see that dimR M " n, and hence dimRp Mp " n.
iIt follows from Grothendieck’s vanishing theorem [9, 6.1.2] that H>npRp (Mp ) = 0. Therefore, we get HpRp (Mp ) = 0 for all∁
integers i, and Mp ∈ supp−1/ suppRp Mp , which implies p ∈/ suppR M .
Mod Rp ({pRp } ) by [36, Remark 3.8(1)]. We have pRp ∈
∁
n+1
n
∼
Now we conclude that supp M ⊆ Φ . Note that N = ✵M ⊕ J for some direct summand J of I , which implies

Ass N = Ass ✵n+1 M ∪ Ass J. We have Ass ✵n+1 M = Ass En+1 (M ) ⊆ supp M ⊆ Φ∁ , while Ass J ⊆ Ass I n ⊆ Φ∁ . Hence
Ass N ⊆ Φ∁ , which is what we have wanted to deduce.
#
The following result refines [32, Corollary A.5] to assert that in the case dim R ! 2 there exists a coherent subset whichis generalization-closed. Also, for n = 1 this theorem contains [32, Corollary 4.3].
Corollary 4.7. The following are equivalent for an integer n ! 0.
(1) Every subset of Spec R is n-coherent.
(2) Every generalization-closed subset of Spec R is n-coherent.
(3) The set (Max R)∁ of nonmaximal prime ideals of R is n-coherent.
(4) One has H>nΦ (−) = 0 for every specialization-closed subset Φ of Spec R.
(5) The inequality dim R " n holds.
Proof. The equivalences (3) ⇔ (4) ⇔ (5) follow from [36, Theorem 4.13], Proposition 4.5(2) and Remark 3.6(4), while
the implications (1) ⇒ (2) ⇒ (3) are obvious. Let Ξ be a subset of Spec R. Applying Proposition 4.6(i) to Φ := Ξ∁ andΨ := Spec R, we see that (5) implies (1).
#Remark 4.8. Using the above result, we see that Proposition 4.6 fails without (i) or (ii). In fact, let Φ = Max R andΨ = Spec R. Then for any integer n ! 0 the subset Ψ is n-coherent, but Ψ \ Φ is not if dim R > n by Corollary 4.7.
We here record a remarkable statement.
Corollary 4.9. Every subset Φ of Spec R is ∞-coherent.
Proof. Thanks to Proposition 3.9(2), we may reduce to the case where R is a local ring. Then every subset of Spec R is(dim R)-coherent by Corollary 4.7, and hence is ∞-coherent by Remark 3.6(2).
#Now we prove the following theorem. In view of Remarks 3.2, 3.6, Proposition 3.8 and [32, Lemma 3.5], one observes
that this theorem for n = 0 (resp. n = 1) yields a one-to-one correspondence between the localizing (resp. wide and⊕-closed) subcategories of Mod R and the specialization-closed (resp. coherent) subsets of Spec R, which is nothing but
the classification theorem of Gabriel [20, Proposition VI.4] (resp. Krause [32, Theorem 3.1]).
Theorem 4.10. Let n be a nonnegative integer or ∞. The assignments X &→ supp X and supp−1
Mod R (Φ) ←! Φ give abijective correspondence between the n-wide (E, ⊕)-closed subcategories of Mod R, and the n-coherent subsets of Spec R.
Proof. Let n ∈ N (resp. n = ∞). We prove that the map X &→ supp X is well-defined. Let X be an n-wide (E, ⊕)-closed
subcategory of Mod R. Let · · · → I1 → I0 → C → 0 be an exact sequence with Ii ∈ Ass−1
Inj R (supp X ) for all 0 " i " n(resp. all i ! 0). Then we have Ass Ii ⊆ supp X and find Xi ∈ X with Ass
I
⊆
supp
X
since X is closed under direct
ii"sums. We observe that Ii is a direct summand of a direct sum of copies of j!0 Ej (Xi ). As X is (E, ⊕)-closed and it isalso closed under direct summands by Remark 3.2(4), the module Ii belongs to X for all 0 " i " n (resp. all i ! 0). The
n-wideness of X shows C ∈ X , which implies ER (C) ∈ Ass−1
Inj R (supp X ). We conclude that supp X is n-coherent.
In view of [48, Theorem 2.3] and Remark 3.2(5), it is enough to prove that the map supp−1
Mod R (Φ) ←! Φ is well-defined,
which follows from Proposition 4.5 in the case n ∈ N. Let Φ be an ∞-coherent subset of Spec R. For any prime ideal pof R, we see from Corollary 4.7 and Proposition 4.5(2) that supp−1
Mod Rp (Φp ) is (ht p)-wide, and it is ∞-wide by Remark
3.2(1). Proposition 3.4 deduces that supp−1(Φ)
is∞-wide.
#Mod R

Corollary 4.11. The assignments X &→ supp X and supp−1
Mod R (Φ) ←! Φ give a bijective correspondence between thelocalizing (resp. ⊕-closed wide) subcategories of Mod R and the specialization-closed (resp. coherent) subsets of Spec R.
Proof. It follows from Remark 3.2(2)(3) that the 0-wide (resp. 1-wide) subcategories are the Serre (resp. wide) subcategories of Mod R and it follows from Proposition 3.8 and Remark 3.6(3) that the 0-coherent (resp. 1-coherent) subsets
are the specialization-closed (resp. coherent) subsets of Spec R. Therefore, for n = 0 (resp. n = 1), Theorem 4.10 is

8

HIROKI MATSUI AND RYO TAKAHASHI

directly translated as the bijective correspondence between E-closed localizing (resp. (E, ⊕)-closed wide) subcategoriesof Mod R and specialization-closed (resp. coherent) subsets of Spec R. Since [32, Lemma 3.5] shows every ⊕-closed wide
subcategory is E-closed, this bijection is nothing but the one that we want.
#
Question 4.12. As we have seen above, ⊕-closed n-wide subcategories of Mod R are automatically E-closed for n = 0
and 1 by [32, Lemma 3.5]. However, its proof breaks down for n > 1 and we do not have any examples of ⊕-closed
n-wide subcategories of Mod R which are not E-closed. Can we show that all ⊕-closed n-wide subcategories of Mod R are
E-closed?
Takahashi [48, Theorem 2.3] gives a classification of the (s E, ⊕, ⊖)-closed subcategories of Mod R. As an application ofTheorem 4.10, we recover this classification by making a connection with ∞-wide subcategories.

Corollary 4.13. Let X be an (E, ⊕)-closed subcategory of Mod R. Then X is (s E, ⊖)-closed if and only if X is ∞-wide.
In particular, there are one-to-one correspondences among the (s E, ⊕, ⊖)-closed subcategories of Mod R, the (E, ⊕)-closed∞-wide subcategory of Mod R, and the subsets of Spec R.
#i
Proof. Suppose that X is (s E, ⊖)-closed. Then the equality X = i!0 CX
holds by Remark 4.2(4), and this means that−1
X = suppMod R (Ass(Inj R ∩ X )) by [32, Lemma 3.3]. It follows from Corollary 4.9 and Theorem 4.10 that X is ∞-wide.
Conversely, suppose that X is ∞-wide. Then X is ⊖-closed by Remark 3.2(4). For an R-module M with Ei (M ) ∈ X for
all i ! 0, one has M ∈ X since X is closed under ∞-kernels. This shows that X is s E-closed. Thus the proof of the first
assertion is completed. The second assertion follows from the first one and Theorem 4.10.
#
We state another corollary of Theorem 4.10. In general, it is diﬃcult to check whether a given subset is n-coherent or
not. The first assertion of the corollary gives a necessary condition of n-coherence. We mentioned in Remark 3.6(2) thatn-coherence implies (n + 1)-coherence for each n ! 0. The second assertion of the corollary shows that this implication
can be strict for an arbitrary n.
Corollary 4.14. (1) Let a be an ideal of R and n ! 0 an integer. Let M be a finitely generated R-module with aM ∕= M .
If D(a) is an n-coherent subset of Spec R, then one has grade(a, M ) " n.
(2) Let x = x1 , . . . , xn be a sequence of elements of R. Then D(x) is an n-coherent subset of Spec R. If the sequence x
is R-regular, then D(x) is not (n − 1)-coherent.
(3) Let Φ be a specialization-closed subset of Spec R. Then the following equivalences hold.
−1
∁
∁
H>0
Φ (M ) = 0 for all M ∈ Mod R ⇔ suppMod R (Φ ) is localizing ⇔ Φ is specialization-closed ⇔ Φ is clopen.

Proof. Assertions (1) and (2) immediately follow from Theorem 4.10 and [36, Proposition 4.10 and Example 4.11]. As for
assertion (3), the first and second equivalences are direct consequences of [36, Theorem 4.9(1)] and Theorem 4.10.
√
Assume Φ is clopen and write Φ = V(a) and Φ∁ = V(b) with radical ideals a, b. Then a + b = R and a ∩ b = 0. Hence
>0
>0
H!0
a+b (M ) = Ha∩b (M ) = 0 for any M ∈ Mod R. The Mayer–Vietoris sequence [9, 3.2.3] implies Ha (M ) = 0.
∁
If Φ is specialization-closed, then Φ is both specialization-closed and generalization-closed
and so is Φ∁ . By symmetry,!
it is enough to show that Φ is closed. As Φ is specialization-closed, we can write Φ = p∈min Φ V(p), where min Φ stands
for the set of minimal elements of Φ with respect to the inclusion relation. As Φ is generalization-closed, min Φ consists
only of minimal prime ideals and hence is a finite set. We see that Φ is closed.
#
5. Classification of n-uniform localizing subcategories of D(Mod R)
In the last section, we gave a correspondence between the n-coherent subsets of Spec R and the (E, ⊕)-closed n-wide
subcategories of Mod R. In this section, we consider the derived category analogue of this correspondence. To this end,
we introduce two kinds of subcategories of D(Mod R).
Definition 5.1. Let n ∈ N ∪ {∞} .
(1) We say that a subcategory X of D(Mod R) is n-uniform provided that if X ∈ X and i ∈ Z satisfy Hj X = 0 for alli ∕= j ∈ (i − n, i + n) and X j ∈ X for all integers j ∈ [i − n, i + n], then Zi X, X i /Bi X ∈ X .
(2) We say that a subcategory X of D(Mod R) is n-consistent provided that if X ∈ X and i ∈ Z satisfy Hj X = 0 for alli ∕= j ∈ (i − n, i + n), then Hi X ∈ X .

Remark 5.2. (1) A subcategory X of D(Mod R) is 0-uniform if and only if for each X ∈ X and i ∈ Z with X i ∈ X one
has Zi X, X i /Bi X ∈ X .
(2) A subcategory of D(Mod R) is 0-consistent, if and only if it is 1-consistent, if and only if it is closed under cohomologies.
(3) Let n be a nonnegative integer. Then any n-uniform (resp. n-consistent) subcategory of D(Mod R) is both (n + 1)uniform and ∞-uniform (resp. (n + 1)-consistent and ∞-consistent).

FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

9(4) Every thick subcategory of D(Mod R) is ∞-consistent. In fact, let X ∈ X , i ∈ Z and Hj X = 0 for all i ∕= j ∈ (−∞, ∞).
Then the complex X is isomorphic to the stalk complex Hi X[−i] in D(Mod R). Since X ∈ X , we have Hi X ∈ X .
The condition in the definition of n-consistency is closed under quasi-isomorphisms, but the condition in the definitionof n-uniformity is not. Also, n-consistency looks simpler than n-uniformity. We shall show in Theorem 5.9 that, whenevern ∕= 0, these two notions are equivalent for localizing subcategories4. We first investigate 0-uniform localizing subcategories
in relation to smashing subcategories.
Definition 5.3. A thick subcategory X of D(Mod R) is called smashing if the inclusion functor X ↩→ D(Mod R) admits
a right adjoint which preserves direct sums. In particular, it is localizing; see Remark 2.3.
Proposition 5.4. The 0-uniform localizing subcategories of D(Mod R) are the smashing subcategories of D(Mod R).
Proof. Let X be a localizing subcategory of D(Mod R). Then X = supp−1
D(Mod R) (Φ) for some subset Φ of Spec R by [42,Theorem 2.8]. If X is smashing, then Φ is specialization-closed by [42, Theorem 3.3], and X ∩ Mod R = supp−1
Mod R (Φ) is
localizing by [20, page 425] (see also [32, Corollary 3.6]). Conversely, if X ∩ Mod R is localizing, then by [20, page 425]
again supp X = supp(X ∩ Mod R) = Φ is specialization-closed, and X is smashing by [42, Theorem 3.3] again. Thus it
suﬃces to show that X is 0-uniform if and only if X ∩ Mod R is localizing.
Suppose that the subcategory X is 0-uniform. The localizing property of X implies that X ∩ Mod R is closed under
direct sums. Let 0 → L → M → N → 0 be an exact sequence of R-modules. Then an exact triangle L → M → N → L[1]in D(Mod R) is induced. If L, N ∈ X , then M ∈ X since X is localizing. Now, assume M ∈ X . Then consider the complex
X = (0 → L → M → N → 0) with M being in degree 0. This complex is exact, so that X ∼= 0 in D(Mod R). As X
contains 0, it also contains X. Since X 0 = M belongs to X , the modules L = Z0 X and N = X 0 /B0 X belong to X as
well. We conclude that X ∩ Mod R is localizing.
Conversely, suppose that X ∩ Mod R is a localizing subcategory of Mod R. Let X ∈ X , i ∈ Z and X i ∈ X . Then X ibelongs to X ∩ Mod R. The natural injection Zi X ↩→ X i and surjection X i ↠ X i /Bi X show that Zi X and X i /Bi X arein X ∩ Mod R. It follows that X is 0-uniform.
#
The following theorem is one of the main results in this section, which gives a derived category analogue of thecorrespondence 4.10 using n-uniform subcategories.
Theorem 5.5. Let n ∈ N ∪ {∞}. The assignments X &→ supp X and supp−1
D(Mod R) (Φ) ←! Φ give a bijective correspondence
between the n-uniform localizing subcategories of D(Mod R) and the n-coherent subsets of Spec R.
Proof. In view of [42, Theorem 2.8], it suﬃces to show that (a) supp X is n-coherent for an n-uniform localizing subcategory
X of D(Mod R) and that (b) supp−1
D(Mod R) (Φ) is n-uniform for an n-coherent subset Φ of Spec R.
(a) The case n = 0 is settled since the 0-uniform subcategories are the localizing subcategories of D(Mod R) whose
supports are specialization-closed by Remark 5.4 and [42, Theorem 3.3], while Corollary 4.9 shows the case n = ∞. So,−1
let 0 < n < ∞. Put Φ = supp X . We have X = supp−1
D(Mod R) (Φ) by [42, Theorem 2.8], and X ∩ Mod R = suppMod R (Φ).
By [48, Theorem 2.3] we get supp(X ∩ Mod R) = Φ. In view of Theorem 4.10, it suﬃces to show that X ∩ Mod R is
n-wide and (E, ⊕)-closed. It follows from [48, Theorem 2.3] again that X ∩ Mod R is (E, ⊕)-closed. As X is localizing, weeasily see that X ∩ Mod R is closed under extensions. It remains to verify that X ∩ Mod R is closed under n-kernels andn-cokernels. Let 0 → K → X 0 → · · · → X n → C → 0 be an exact sequence of R-modules with X i ∈ X for all 0 " i " n.
Then consider the complex X = (· · · → 0 → X 0 → · · · → X n → 0 → · · · ). For each 1 " i " n there is an exact triangle
X i [−i] → Ci → Ci−1 → X i [−i + 1] with Cn = X and C0 = X 0 . We inductively see that X belongs to X . Since Hj X = 0
for all 0 ∕= j ∈ (−n, n) and n ∕= j ∈ (0, 2n), the modules K = Z0 X and C = X n /Bn X belong to X . Therefore, X ∩ Mod R
is closed under n-kernels and n-cokernels.
jj(b) Put X = supp−1
D(Mod R) (Φ). Let X ∈ X , i ∈ Z, H X = 0 for all integers j with i ∕= j ∈ (i − n, i + n), and X ∈ X
for all integers j ∈ [i − n, i + n]. Let n ∈ N (resp. n = ∞). There are two exact sequences 0 → Zi X → X i → · · · → X i+n
and X i−n → · · · → X i → X i /Bi X → 0 (resp. 0 → Zi X → X i → X i+1 → · · · and · · · → X i−1 → X i → X i /Bi X → 0).
Theorem 4.10 says that X ∩ Mod R = supp−1
Mod R (Φ) is n-wide. In particular, it is closed under n-kernels and n-cokernels.
The exact sequences show that Zi X and X i /Bi X are in X ∩Mod R, and hence they belong to X . Thus, X is n-uniform. #Corollary 4.9, Theorem 5.5 and [42, Theorem 2.8] imply the following.
Corollary 5.6. Every localizing subcategory of D(Mod R) is ∞-uniform.
4Thus the reader may wonder why we introduce the notion of n-uniform subcategories. A benefit from introducing this notion is that
Theorem 5.5 can be shown by combination of classical results. More precisely, it can be done without Lemma 5.7 which comes from the
Nakamura–Yoshino theory, and so even the reader who does not know this theory can understand it.

10

HIROKI MATSUI AND RYO TAKAHASHI

Next, we consider classifying n-consistent subcategories of D(Mod R). We shall do it by showing that n-uniformity andn-consistency are equivalent for localizing subcategories. To show that n-uniformity implies n-consistency, we use thelemma below. This follows from the dual version of [41, Corollary 7.14], whose establishment is guaranteed by [41, thethree lines at the end of Section 7].
Lemma 5.7 (Nakamura–Yoshino). Suppose dim R < ∞. Let Φ be a subset of Spec R, and set X = supp−1
D(Mod R) (Φ).
Then for every complex X ∈ X there exists a complex Y ∈ D(Mod R) such that Y ∼= X and Y i ∈ X for all i ∈ Z.
Sketch of Proof. One can verify this lemma inductively via [41, Definition 7.6] by combining [40, Theorem 3.12], [41,Lemma 9.1(2)], [34, Lemma 3.5.1] and [41, Theorem 3.22] (or [40, Theorem 3.13]), see also [40, Corollary 3.4].
Concretely, we can construct such a complex Y as follows. Let W = {Wi }ni=1 be a system of slices of Φ in the sense of[41, Definition 7.6] (for example, we can take Wi := {p ∈ Φ | dim R/p = i}). We then replace the functor λthe complex of functors LW in [41] with their categorical duals$
γ (i ,...,i ) := γ W · · · γ W ,γ W :=
ΓV(p) HomR (Rp , −)
m

and

%

0

RW := &0 → γ (d,...,0) →im$

i0 <···<id−1

i0

γ (i

d−1 ,...,i0 )(im ,...,i0 )

and

i

p∈Wi

→ ··· →$

i0 <i1γ (i ,i ) → · · · →
1

0

$
i

'

γ (i) → 0(

respectively. Here the diﬀerentials of RW are induced from the canonical natural transformations γ W → idMod R . Thisi)
complex can be constructed inductively as follows. Let Wr := {Wi }ri=0 and denote by R
Wr the augmentation of RWr .
))!
Then, we see that R
R
Wr+1 is the mapping cone of γ W
Wr → RWr and hence its de-augmentation is RWr+1 . Therefore,
r+1
by induction, one can show that there is a natural isomorphism γΦ (M ) ∼= RW (M ) in D(Mod R) for any injective R-module
M . Here, the induction basis follows from [40, Theorem 3.12]. For a complex X ∈ D(Mod R) with a semi-injective
resolution I, we get isomorphisms X ∼= γΦ (I) ∼= Tot(RW (I)) =: Y in D(Mod R) such that each component of Y belongs−1
to suppD(Mod R) (Φ), see [34, Lemma 3.5.1] and [41, Theorem 3.22].
#
Proposition 5.8. Let n ∈ N ∪ {∞}.
(1) Assume n ∕= 0. For an n-consistent localizing subcategory X of D(Mod R), the subset supp X of Spec R is n-coherent.
(2) For an n-coherent subset Φ of Spec R, the subcategory supp−1
D(Mod R) (Φ) of D(Mod R) is n-consistent.
Proof. (1) The assertion is shown by the same arguments as in (a) in the proof of Theorem 5.5 for n ∕= 0, where using
n-uniformity at the end is just replaced with using n-consistency.
j(2) Set X = supp−1
D(Mod R) (Φ). Let X ∈ X and i ∈ Z be such that H X = 0 for all i ∕= j ∈ (i − n, i + n). We provei
H X ∈ X in two steps: (i) the case where dim R < ∞ and (ii) the general case.
(i) By Lemma 5.7, we may assume X j ∈ X for all j ∈ Z. Since X is n-uniform by Theorem 5.5, we have X i /Bi X ∈ X .
Note by Theorem 4.10 that X ∩ Mod R = supp−1
Mod R (Φ) is n-wide, and in particular, it is closed under n-kernels. Fromthe exact sequence 0 → Hi X → X i /Bi X → X i+1 → · · · → X i+n , we obtain Hi X ∈ X ∩ Mod R ⊆ X .
(ii) Fix a prime ideal p of R. Since Rp has finite Krull dimension, it follows from (i) and Proposition 3.9(2) that
LL∼
Y := supp−1
D(Mod Rp ) (Φp ) is n-consistent. As κ(qRp ) ⊗Rp Xp = κ(q) ⊗R X for q ∈ Spec R with q ⊆ p, we observe that Xp
belongs to Y. Also, Hj (Xp ) = (Hj X)p = 0 for all integers j with i ∕= j ∈ (i − n, i + n). Hence Hi (Xp ) ∈ Y, and therefore
suppRp (Hi X)p ⊆ Φp for all p ∈ Spec R. It is seen that Hi X ∈ X .
#As we have promised, we give the following classification theorem of n-consistent localizing subcategories.
Theorem 5.9. Let 0 < n ∈ N ∪ {∞}. Then a localizing subcategory of D(Mod R) is n-uniform if and only if it is nconsistent. In particular, the assignments X &→ supp X and supp−1
D(Mod R) (Φ) ←! Φ give a bijective correspondence between
the n-consistent localizing subcategories of D(Mod R) and the n-coherent subsets of Spec R.
Proof. We begin with showing the first assertion. According to Theorem 5.5 and Proposition 5.8(2), it is enough to provethat every n-consistent subcategory X of D(Mod R) is n-uniform. Proposition 5.8(1) implies that supp X is n-coherent,−1and hence supp−1
D(Mod R) (supp X ) is n-uniform by Theorem 5.5. We have X = suppD(Mod R) (supp X ) by [42, Theorem 2.8].
It follows that X is n-uniform, as desired. The second assertion follows from the first and Theorem 5.5.
#
Letting n = 0 in Theorem 5.5 and n = 1, ∞ in Theorem 5.9 respectively and using Proposition 5.4 and Remark 5.2(2),we have the following corollary. These bijections are given in [42, Theorem 3.3] and [32, Theorem 5.2] (see also [48, Main
Theorem]).

FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

11Corollary 5.10. The assignments X &→ supp X and supp−1
D(Mod R) (Φ) ←! Φ give one-to-one correspondences:• {localizing subcategories of D(Mod R)} ↔ {subsets of Spec R},• {cohomology-closed localizing subcategories of D(Mod R)} ↔ {coherent subsets of Spec R}.
• {smashing subcategories of D(Mod R)} ↔ {specialization-closed subsets of Spec R},

Remark 5.11. It is clarified by [13, Remark 2.3] that [32, Proposition 5.1] does not necessarily hold true, namely, for an
unbounded complex X with minimal semi-injective resolution I, the equality*
supp X =Ass I ii∈Z

may fail5. However, Corollary 5.10 guarantees that the assertion of [32, Theorem 5.2] is correct. Indeed, as a directconsequence of Corollary 5.10, for a subset Φ of Spec R, supp−1
D(Mod R) (Φ) is closed under taking cohomologies if and only
if Φ is coherent. Here we give a proof of [48, Proposition 2.3(2)], i.e., the equalityisupp−1
D(Mod R) (supp M) = M := {X ∈ D(Mod R) | H (X) ∈ M (∀i ∈ Z)}

for a ⊕-closed wide subcategory M of Mod R without using [32, Proposition 5.1].
Fix such a subcategory M of Mod R. We note that the proof of the inclusion M ⊆ supp−1
D(Mod R) (supp M) in [48,Proposition 2.3(2)] just uses [32, Theorem 5.2]. By contrast, the converse inclusion uses [32, Proposition 5.1]. Thus,+
we need to check the inclusion M ⊇ supp−1
D(Mod R) (supp M). Let M be the smallest localizing subcategory of D(Mod R)
containing M. Then we have−1++supp−1
D(Mod R) (supp M) ⊆ suppD(Mod R) (supp M) = M ⊆ M.

+ contains M and the equality follows from [42, Theorem 2.8]. It is easy to see
The first inclusion is trivial since M
that M is localizing, and contains M. The last inclusion follows from this. Therefore, we obtain the inclusion M ⊇supp−1
D(Mod R) (supp M). Thus the proof of [48, Proposition 2.3(2)] is completed.
6. Several classifications of subcategories by restriction
This section concerns some restrictions of classifications obtained in the previous two sections. First of all, we consider
restriction to Π-closed subcategories of Mod R and D(Mod R).
Proposition 6.1. The following are equivalent for a subset Φ of Spec R.
(1) The subset Φ is generalization-closed.
(2) The subcategory supp−1
Mod R (Φ) is closed under direct products.
(3) The subcategory supp−1
Mod R (Φ) is closed under countable direct products.
(4) The subcategory supp−1
D(Mod R) (Φ) is closed under direct products.
(5) The subcategory supp−1
D(Mod R) (Φ) is closed under countable direct products.
Proof. We prove that (1) implies (2). Let {Mλ }λ∈Λ be a family of modules in supp−1
Mod R (Φ). The product Πλ∈Λ E(Mλ )of complexes is an injective resolution of the module Πλ∈Λ Mλ . We may assume that each Mλ is injective to proveAss(Πλ∈Λ Mλ ) ⊆ Φ. Take p ∈ Ass(Πλ∈Λ Mλ ). Then p = ann(mλ )λ∈Λ for some 0 ∕= (mλ )λ∈Λ ∈ Πλ∈Λ Mλ . There is λ ∈ Λwith mλ ∕= 0, and there exists q ∈ Ass Mλ such that ann mλ ⊆ q (see [37, Theorem 6.1]). Note that p ⊆ ann mλ and
Ass Mλ ⊆ Φ. As Φ is generalization-closed, it follows that p ∈ Φ.
We prove that (3) implies (1). Take an inclusion p ⊆ q of prime ideals with q ∈ Φ. There is a sequence(6.1.1)

R/p ↩→ Rq /pRq ↩→ Πr!1 Rq /(qr + p)Rq ↩→ Πr!1 ER (R/(qr + p))qof injective maps, where the second injection follows from Krull’s intersection theorem. Since the ideal (qr + p)Rq is
qRq -primary, we have AssRq (ER (Rq /(qr + p))q ) = {qRq }, and hence AssR (ER (R/(qr + p))q ) = {q} by [37, Theorem 6.2].
Note that ER (R/(qr + p))q is injective as an R-module. The module ER (R/(qr + p))q is in supp−1
Mod R (Φ). By assumption,r
the product Πr!1 ER (R/(qr + p))q is also in supp−1
(Φ),
and
hence
Ass
(Π
E
(R/(q
+
p))
R
r!1
R
q ) ⊆ Φ. It follows from
Mod R
(6.1.1) that p ∈ Φ, and thus Φ is generalization-closed.
It is obvious that the implications (3) ⇐ (2) ⇐ (4) ⇒ (5) ⇒ (3) hold. The proof of the proposition is done once weverify that (1) implies (4), which is a consequence of [43, Lemma 4.5].
#
5This does not aﬀect [32, Lemma 3.3], [32, Lemma 3.5] or [48, Theorem 2.3]

12

HIROKI MATSUI AND RYO TAKAHASHI

Remark 6.2. The above proposition gives an alternative proof of the smashing conjecture [42, Theorem 3.3] for D(Mod R),
that is, if the inclusion X ↩→ D(Mod R) from a localizing subcategory of D(Mod R) admits a coproduct preserving rightadjoint, then X is generated by compact objects. Indeed, let X be such a subcategory of D(Mod R). Let us show that
supp X is specialization-closed. Let Y be the kernel of F . Then Y is closed under direct products as F is a right adjoint
functor, while Y is closed under direct sums as F preserves direct sums. It is easy to check that the equaity ⊥ Y = X holds.
−1
By [42, Theorem 2.8] and Proposition 6.1, we have X = supp−1
D(Mod R) (Φ) and Y = suppD(Mod R) (Ψ) for some subset Φ andsome generalization-closed subset Ψ of Spec R. Here, recall that HomR (ER (R/p), ER (E/q)) = 0 if and only if p ∕⊆ q forp, q ∈ Spec R; see [17, Theorem 3.3.8(5)]. This fact and the equality ⊥ Y = X show Φ = Ψ∁ , which is specialization-closed.
Now, we give classifications of Π-closed subcategories of Mod R and D(Mod R). The second classification can beconsidered as a dual of [42, Theorem 3.3].
Theorem 6.3. Let n ∈ N.
(1) The assignments X &→ supp X and Φ &→ supp−1
Mod R (Φ) give a one-to-one correspondence{(s E, ⊕, ⊖, Π)-closed subcategories of Mod R} ↔ {generalization-closed subsets of Spec R},

which restricts to a one-to-one correspondence{n-wide (E, ⊕, Π)-closed subcategories of Mod R} ↔ {n-coherent generalization-closed subsets of Spec R}.

(2) The assignments X &→ supp X and Φ &→ supp−1
D(Mod R) (Φ) give a one-to-one correspondence{bilocalizing subcategories of D(Mod R)} ↔ {generalization-closed subsets of Spec R},

which restricts to a one-to-one correspondence{n-uniform bilocalizing subcategories of D(Mod R)} ↔ {n-coherent generalization-closed subsets of Spec R}.
Proof. The combination of Theorems 4.10, 5.5 and Proposition 6.1 shows the two statements of the theorem.

#

We study more about the above classification in the cases n = 0, 1. Here, let us recall two notions from category theory.
Definition 6.4. (1) Define a bismashing subcategory X of D(Mod R) as a bilocalizing subcategory such that the inclusion
functor X ↩→ D(Mod R) has a right adjoint preserving direct sums and a left adjoint preserving direct products.
(2) Recall that a subcategory X of Mod R is called bireflective (resp. Giraud) if the inclusion functor X ↩→ Mod R admits
left and right adjoints (resp. admits an exact left adjoint).
Remark 6.5. A wide (⊕, Π)-closed subcategory of Mod R is nothing but a bireflective Giraud subcategory of Mod R. In
fact, it follows from [25, Proposition 3.8], [21, Theorem 1.2] and [46, Theorem 4.8] (see also [1, Definition 5.2 and Theorem5.3]) that the wide (⊕, Π)-closed subcategories X of Mod R are the bireflective extension-closed subcategories X of Mod R,
which bijectively correspond to the epiclasses of ring epimorphisms R → S with TorR
1 (S, S) = 0 such that X is the imageof the induced fully faithful functor Mod S ↩→ Mod R, which are the epiclasses of flat ring epimorphisms R → S by [2,Proposition 5.4]. Let φ : R → S be a ring epimorphism and θ : Mod S ↩→ Mod R the induced functor. If φ is flat, thenthe functor − ⊗R S : Mod R → Mod S is an exact left adjoint to θ. Conversely, if θ has an exact left adjoint, then theuniqueness of (left) adjoints implies θ ∼= − ⊗R S, and hence φ is flat.
Now we can state and prove the following corollary. The first and second assertions come from the cases n = 0 and
n = 1 in Theorem 6.3, respectively. The second assertion is (essentially) shown in [2, Corollary 4.10].
Corollary 6.6. There are one-to-one correspondences
,
,
,
supp
supp
bilocalizing
bismashing subcategories$ clopen subsets "
"
$
(1)
.
subcategories of Mod R
of Spec Rof D(Mod R)
supp−1supp−1Mod R
D(Mod R)
,
,
,
supp
supp
bireflective Giraud
generalization-closed
cohomology-closed bilocalizing
$
"
"
$
(2)
.
subcategories of Mod R
coherent subsets of Spec R supp−1
subcategories of D(Mod R)
supp−1Mod R

D(Mod R)

Proof. (1) Letting n = 0, we have that the bijection in Theorem 6.3(1) is nothing but the first one-to-one correspondence.
Indeed, Remark 3.2(2), Propositions 3.8, and [32, Lemma 3.5] show that 0-wide (E, ⊕, Π)-closed subcategories of Mod R
are the same things as bilocalizing subcategories of Mod R. By Proposition 3.8, the 0-coherent generalization closed subsetsof Spec R are the clopen subsets of Spec R. Using Proposition 5.4, we see that the bijection in Theorem 6.3(2) gives a
one-to-one correspondence between the set of clopen subsets of Spec R and the set of smashing bilocalizing subcategoriesof D(Mod R). Thus, it suﬃces to check that every smashing bilocalizing subcategory X of D(Mod R) is bismashing, i.e.,
the inclusion functor X ↩→ D(Mod R) has a left adjoint which preserves direct products. There is a clopen subset ΦFILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

13

−1
∁
of Spec R with X = supp−1
D(Mod R) (Φ). Set Y = suppD(Mod R) (Φ ). By [33, Proposition 4.9.1] and [43, Lemma 4.5], theinclusion functor X = Y ⊥ ↩→ D(Mod R) has a left adjoint D(Mod R) → D(Mod R)/Y ∼= X . Using Proposition 6.1, we see∁that Y = supp−1(Φ)
isclosedunder
directproducts,
and
hence
this
left
adjoint
preserves direct products.
D(Mod R)
(2) The assertion follows by Remarks 3.2(3), 3.6(3), 5.2(2), 6.5, [32, Lemma 3.5] and letting n = 1 in Theorem 6.3.
Indeed, Remark 3.2(3) and [32, Lemma 3.5] show that the 1-wide (E, ⊕, Π)-closed subcategories are the wide (⊕, Π)-closed
subcategories which are by Remark 6.5 the bireflective Giraud subcategories of Mod R. By Remark 5.2(2), the 1-uniform
bilocalizing subcategories are the cohomology-closed bilocalizing subcategories of D(Mod R) and by Remark 3.6(3) the
1-coherent generalization-closed subsets are the generalization-closed coherent subsets of Spec R.
#

Next, we consider classifying certain thick subcategories of D(Mod R)fid , Db (Mod R) and D+ (Mod R). Here, D(Mod R)fid
stands for the subcategory of D(Mod R) consisting of complexes of finite injective dimension. Recall that ⊕-closedness(resp. Π-closedness) means closed under existing coproducts (resp. products) for a subcategory of D(Mod R)fid , Db (Mod R)
or D+ (Mod R).
Theorem 6.7. Let n ∈ N ∪ {∞}.

(1) The assignments X &→ supp X and supp−1
D+ (Mod R) (Φ) ←! Φ give a bijective correspondence between the n-uniform(E, ⊕, Π)-closed thick subcategories of D+ (Mod R) and the n-coherent generalization-closed subsets of Spec R.
(2) The assignments X &→ supp X and supp−1
D(Mod R)fid (Φ) ←! Φ give a bijective correspondence between the n-uniform(E, ⊕)-closed thick subcategories of D(Mod R)fid and the n-coherent subsets of Spec R.
Proof. We begin with establishing two claims.
Claim 1. It suﬃces to show the assertion for n = ∞.
Proof of Claim. Assume that this claim is shown to hold. We prove that (1) and (2) in the theorem hold for n ∈ N.
(1) Let X be an n-uniform (E, ⊕, Π)-closed thick subcategory of D+ (Mod R) and Φ an n-coherent generalizationclosed subset of Spec R. Then X is ∞-uniform and Φ is ∞-coherent. By assumption, supp X is generalization-closed,
and supp−1
D+ (Mod R) (Φ) is thick and (E, ⊕, Π)-closed. The proof of Theorem 5.5 shows that supp X is n-coherent andsupp−1
D+ (Mod R) (Φ) is n-uniform. Thus the assignments are well-defined and bijective.
(2) This is shown in a similar way to the proof of (1).

□Claim 2. Let X be an (E, Π)-closed thick subcategory of D+ (Mod R) or an E-closed thick subcategory of D(Mod R)fid .
Then X is s E-closed.
Proof of Claim. The second case is obvious as every complex X ∈ D(Mod R)fid is isomorphic to the bounded complex
E(X). Let X be an (E, Π)-closed thick subcategory of D+ (Mod R), and take a complex X ∈ D+ (Mod R) with Ei (X) ∈X for all i ∈ Z. Let E(X)#i := (· · · → Ei−1 (X) → Ei (X) → 0) be a (hard) truncation of E(X) for each i. As
E(X)#i is a bounded complex of objects in X , it is in X . The canonical morphisms X ∼= E(X) → E(X)#i for i ∈ Z#i+∼induce an isomorphism X = holimi∈Z E(X) in D (Mod R); see [8, Remark 2.3]. Thus there is an exact triangle
X → Πi∈Z E(X)#i → Πi∈Z E(X)#i → X[1] in D+ (Mod R). As E(X)#i ∈ X and X is a Π-closed thick subcategory, it isseen that X ∈ X .
□−1
Using Remark 2.6(1), we get supp(supp−1
D+ (Mod R) (Φ)) = Φ and supp(suppD(Mod R)fid (Φ)) = Φ for any subset Φ of+Spec R. Let X be an (E, ⊕, Π)-closed thick subcategory of D (Mod R) (resp. an (E, ⊕)-closed thick subcategory of−1iD(Mod R)fid ). Let X be a complex in supp−1
D+ (Mod R) (supp X ) (resp. suppD(Mod R)fid (supp X )). Then E (X) ∈ X for alli ∈ Z by Remark 2.6 and the proof of Theorem 4.10. Claim 2 shows X ∈ X . Hence supp−1
D+ (Mod R) (supp X ) = X (resp.

supp−1
D(Mod R)fid (supp X ) = X ), and the proof of the theorem for n = ∞ is completed. Combining this with Claim 1, weare done.
#
Remark 6.8. By tracing the argument in Section 5, the above classifications show that if 0 ∕= n ∈ N ∪ {∞}, thenn-uniformity and n-consistency are equivalent for (E, ⊕, Π)-closed thick subcategories of D+ (Mod R) and for (E, ⊕)-closed
thick subcategories of D(Mod R)fid ; see the proofs of Proposition 5.8 and Theorem 5.9.
As an application of Theorem 6.7 together with Corollary 4.7, we obtain a higher-dimensional analogue of Brüning’sclassification theorem [10, Theorem 5.1] for the module category.
Corollary 6.9. Let n ∈ N. Assume that gldim R " n. Then any (E, ⊕)-closed thick subcategory of Db (Mod R) is nuniform. Hence, there are one-to-one correspondences among the (E, ⊕)-closed thick subcategories of Db (Mod R), the(E, ⊕)-closed n-wide subcategories of Mod R, and the subsets of Spec R.

14

HIROKI MATSUI AND RYO TAKAHASHI

Proof. Since gldim R " n, the ring R has Krull dimension at most n and D(Mod R)fid ∼= Db (Mod R). Thus, the firststatement follows from Corollary 4.7 and Theorem 6.7(2). The second statement is a consequence of the combination of
the first statement and Theorems 6.7(2), 4.10.
#
Taking n = 1 in the above corollary, we obtain the following.
Corollary 6.10. Assume that R is hereditary, i.e., gldim R " 1. Then, there are one-to-one correspondences among the⊕-closed thick subcategories of Db (Mod R), the ⊕-closed wide subcategories of Mod R, and the subsets of Spec R.
Proof. In view of Corollaries 6.9 and 4.11, it is enough to show that every ⊕-closed (1-uniform) thick subcategory X of
Db (Mod R) is E-closed. The proof is almost the same as [32, Lemma 3.5]. We need to prove that for all X ∈ X andp ∈ supp X it holds that E(R/p) ∈ X . By the definition of a small support, TorRt (X, κ(p)) ∕= 0 for some t ∈ Z. Since R
is hereditary, κ(p) has a projective R-resolution of length at most one. As X is thick, X ⊗L
R κ(p) belongs to X . Since X
is 1-uniform, it is closed under cohomologies. Hence TorR
(X,
κ(p))
is
in
X
,
and
so
is
its
directsummand κ(p). Note thatt
E(R/p) is a directed union of direct sums of copies of κ(p). It follows that E(R/p) ∈ X .
#Let X be an R-complex. For each integer n, we define (soft) truncations
τ #n X = (· · · → X n−2 → X n−1 → Zn X → 0),

τ !n X = (0 → Bn X → X n → X n+1 → · · · )of X, where Zn X, Bn X are placed in degree n, n−1 respectively. Note that there is a natural exact sequence 0 → τ #n X →X → τ !n+1 X → 0 of complexes. The following result is worth comparing with Corollary 6.9.
Proposition 6.11. Let n ∈ N. When gldim R " n, every thick subcategory of D(Mod R) is n-consistent.

Proof. First of all, we claim that for an integer k and a complex P of projective R-modules with Hj P = 0 for all
k + 1 " j " k + n − 1 there is a direct sum decomposition P ∼= τ #k P ⊕ τ !k+1 P of R-complexes. In fact, since gldim R " n,
k+1
k+1
k+n
the exact sequence 0 → B
P →P→ ··· → Pshows that the R-module Bk+1 P is projective. Hence the exactkk
k+1
sequence 0 → Z P → P → B
P → 0 splits, and so does the exact sequence 0 → τ #k P → P → τ !k+1 P → 0.
Let X be a thick subcategory of D(Mod R). Let X ∈ X be such that Hj X = 0 for all j with i ∕= j ∈ (i − n, i + n).
Replacing X with a semiprojective resolution, we may assume that X j is projective for each j ∈ Z. As Hj X (resp.
Hj (τ #i X)) vanishes for all i + 1 " j " i + n − 1 (resp. (i − n) + 1 " j " (i − n) + n − 1), the claim shows that there are
direct sum decompositions X ∼= τ #i X ⊕ τ !i+1 X and τ #i X ∼= τ #i−n τ #i X ⊕ τ !i−n+1 τ #i X ∼= τ #i−n X ⊕ Hi X[−i]. Here, to
#i
i−2
i−1
i
apply the claim we replace τ X with a projective resolution (· · · → Q→Q→ Q → 0), so that each component
is a projective R-module. Thus we get X ∼= τ #i−n X ⊕ Hi X[−i] ⊕ τ !i+1 X. As X is closed under direct summands andshifts, we obtain Hi X ∈ X .
#7. A remark on cofinite modules
For an ideal J over a commutative noetherian ring R, Hartshorne [26] introduced the notion of a J-cofinite moduleand posed a question of whether the category of J-cofinite modules forms an abelian category. A counterexample tothis question is given by himself. However, the question is still considered by several authors in the direction of finding
conditions that lead the question to be aﬃrmative, e.g., [7, 16, 38, 39]. In this section, we consider a weakened version ofcofinite modules and n-coherence of the category of such modules. Although our result does not recover known results, it
may provide another approach to the question.
By [16, Corollary 3.5], cofiniteness of modules are related to artinianness of local cohomology modules. Our strategy is
to relax artinianness as follows.
Definition 7.1. We say that an R-module M is ind-artinian if all finitely generated submodules of M have finite length.
Note that any artinian R-module is ind-artinian and an R-module M is ind-artinian if and only if it is an inductive limit
of artinian modules.
The following proposition shows that the family of ind-artinian modules forms a localizing subcategory. More strongly,
it is the smallest localizing subcategory of Mod R containing all artinian modules.
Proposition 7.2. The subcategory of Mod R consisting of ind-artinian modules coincides with Ass−1
Mod R (Max R). This
also coincides with supp−1
Mod R (Max R), and is localizing.
Proof. Let M be an R-module. If p ∈ Ass M is a nonmaximal prime ideal, then M has a submodule isomorphic to R/p,which does not have finite length. This shows that every ind-artinian R-module belongs to Ass−1
Mod R (Max R). Conversely,
assume Ass M ⊆ Max R. If N is a finitely generated submodule of M , then Ass N ⊆ Ass M ⊆ Max R. Since the minimal
primes of M are associated primes, we see that Supp N ⊆ Max R. As N is finitely generated, it has finite length. Thus
the proof of the first assertion of the proposition is completed. The second assertion follows from [36, Theorem 2.6]. #FILTRATIONS IN MODULE CATEGORIES, DERIVED CATEGORIES AND PRIME SPECTRA

15

The following lemma generalizes [36, Proposition 4.5] in a relative setting.
Lemma 7.3. Let Φ be a specialization-closed subset of Spec R. Let M be an R-module.
(1) There is an equality AssR ΓΦ (M ) = AssR M ∩ Φ.
(2) Let n ! 0 be an integer.
−1i(a) Let Ψ be a subset of Spec R. Then HiΦ (M ) ∈ supp−1
Mod R (Ψ) for all i " n if and only if ΓΦ (E (M )) ∈ suppMod R (Ψ)
for all i " n.
(b) Let X be an (s E, ⊕)-closed subcategory of Mod R. Then HiΦ (M ) ∈ X for all i " n if and only if ΓΦ (Ei (M )) ∈ X
for all i " n.
Proof. (1) We use [36, Propositions 3.2(1a) and 2.3(6)] for both inclusions. Let p ∈ Ass M ∩ Φ. Then there is amonomorphism R/p ↩→ M , which induces a monomorphism ΓΦ (R/p) ↩→ ΓΦ (M ). As V(p) ⊆ Φ, we have ΓΦ (R/p) = R/p.
(2a) The equality HiΦ (M ) = Hi (E(M )) and Proposition 7.2 deduce the “if” part. We prove the “only if” part byinduction on n. We havesupp ΓΦ (E0 (M )) = Ass ΓΦ (E0 (M )) = Ass E0 (M ) ∩ Φ = Ass M ∩ Φ = Ass ΓΦ (M ) ⊆ supp ΓΦ (M ) ⊆ Ψ,

where the first equality holds since ΓΦ (E0 (M )) is injective, and the second and fourth ones follow from Lemma 7.3(1).
−1
Now let n > 0, and take an exact sequence 0 → M → E(M ) → ✵M → 0. Then HiΦ (✵M ) = Hi+1
Φ (M ) ∈ suppMod R (Ψ) for0
1
all 1 " i " n − 1. An exact sequence 0 → ΓΦ (M ) → ΓΦ (E (M )) → ΓΦ (✵M ) → HΦ (M ) → 0 is induced. The modules0ΓΦ (M ), H1Φ (M ) belong to supp−1
Mod R (Ψ) by assumption, while so does ΓΦ (E (M )) by the induction basis. It is observed−1that ΓΦ (✵M ) ∈ suppMod R (Ψ). The induction hypothesis implies that ΓΦ (Ei (M )) = ΓΦ (Ei−1 (✵M )) ∈ supp−1
Mod R (Ψ) for
all 1 " i " n. Thus the proof of the assertion is completed.
(2b) The assertion is a direct consequence of (2a) and Corollary 4.11.
#∁
Proposition 7.4. Let Φ, Ψ be subsets of Spec R. Suppose that Φ is specialization-closed. Then supp−1
Mod R (Φ ∪Ψ) coincides
with the subcategory of Mod R consisting of modules M with HiΦ (M ) ∈ supp−1
Mod R (Ψ) for all i ! 0.

Proof. Let M be an R-module. Using Lemma 7.3(2a), we have that HiΦ (M ) ∈ supp−1
Mod R (Ψ) for all i ! 0 if and only if Ψ
contains supp ΓΦ (Ei (M )) = Ass ΓΦ (Ei (M )) = Ass Ei (M ) ∩ Φ for all i ! 0, where the first equality holds since ΓΦ (Ei (M ))is injective, and the second one follows from Lemma 7.3(1). The latter condition is equivalent to saying that Φ∁ ∪ Ψcontains Ass Ei (M ) for all i ! 0, which is equivalent to the inclusion Φ∁ ∪ Ψ ⊇ supp M .
#Remark 7.5. Putting X = 0 in Lemma 7.3(2) and Ψ = ∅ in Proposition 7.4, we recover [36, Proposition 4.5] and [36,Remark 3.8(1)], respectively.
For a specialization-closed subset Φ of Spec R, we denote by HΦ the subcategory of Mod R consisting of modules M
such that HiΦ (M ) is ind-artinian for all i ! 0.
Corollary 7.6. Let Φ be a specialization-closed subset of Spec R.
∁(1) There is an equality HΦ = supp−1
Mod R (Φ ∪ Max R).
(2) Let R have finite positive Krull dimension d. Then HΦ is (d − 1)-wide. In particular, HΦ is abelian if d " 2.
Proof. (1) Apply Proposition 7.4 for Ψ = Max R and use Proposition 7.2.
(2) We have ht p < d for all p ∈ Φ \ Max R. Proposition 4.6(1) implies that (Φ \ Max R)∁ = Φ∁ ∪ Max R is (d − 1)coherent. It follows from Corollary 7.6(1) and Theorem 4.10 that HΦ is (d − 1)-wide. Thus the proof of the first assertionis completed. The second assertion follows from the first one and [36, Remark 4.2(5)].
#Definition 7.7. Let Φ be a specialization-closed subset of Spec R. An R-module M is said to be Φ-cofinite if SuppR M ⊆ Φand the R-module ExtiR (R/I, M ) is finitely generated for all integers i and ideals I of R with V(I) ⊆ Φ. We denote by
Cof Φ the subcategory of Mod R consisting of Φ-cofinite R-modules.
i∨
If R is semilocal, then Cof Φ = {M ∈ Supp−1
Mod R (Φ) | HΦ (M ) is artinian for all i ! 0} by [16, Corollary 3.5], where(−)∨ = HomR (−, ER (R/ rad R)). As an analogue, (for a not-necessarily-semilocal ring R) we definei∨
Cof ′ Φ = {M ∈ Supp−1
Mod R (Φ) | HΦ (M ) is ind-artinian for all i ! 0}.

It is proved in [16, Theorem 3.13(iii)] that if dim R " 2, then Cof Φ is abelian for a specialization-closed subset Φ ofSpec R. We can prove an analogue of this result for Cof ′ Φ.
−1∨∁Corollary 7.8. One has Cof ′ Φ = {M ∈ supp−1
Mod R (Φ) | M ∈ suppMod R (Φ ∪ Max R)} for a specialization-closed subset′
Φ of Spec R. If dim R " 2, then Cof Φ is abelian.
−1∁
Proof. We use [36, Theorem 2.6] and Corollary 7.6. Both supp−1
Mod R (Φ) and suppMod R (Φ ∪ Max R) are abelian, and it isroutine to deduce the last assertion from this.
#16

HIROKI MATSUI AND RYO TAKAHASHI

Acknowledgments. The authors thank Tsutomu Nakamura for reminding them of Lemma 5.7 and the correctness of [32,Theorem 5.2]; without this, the authors would not be able to get the proof of Proposition 5.8. The authors also thankLidia Angeleri Hügel for giving them useful observation on bireflective Giraud subcategories. Part of this work was done
during Takahashi’s visit to the University of Kansas from March 2018 to September 2019, and he thanks them for their
hospitality. Also, the authors very much thank the anonymous referees for their careful reading, valuable comments andhelpful suggestions.
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(H. Matsui) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914,
Japan
Email address: mhiroki@ms.u-tokyo.ac.jp(R. Takahashi) Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
Email address: takahashi@math.nagoya-u.ac.jp
URL: http://www.math.nagoya-u.ac.jp/~takahashi/