SINGULARITY CATEGORIES AND SINGULAR EQUIVALENCESFOR RESOLVING SUBCATEGORIESHIROKI MATSUI AND RYO TAKAHASHI
Dedicated to Professor Yuji Yoshino on the occasion of his sixtieth birthday
Abstract. Let X be a resolving subcategory of an abelian category. In this paper we investigate thesingularity category Dsg (X ) = Db (mod X )/Kb (proj(mod X )) of the stable category X of X . We consider
when the singularity category is triangle equivalent to the stable category of Gorenstein projective
objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity
categories. Applying this to the module category of a Gorenstein ring, we prove that the complete
intersections over which the stable categories of resolving subcategories have trivial singularity categories
are the simple hypersurface singularities of type (A1 ). We also generalize several results of Yoshino on
totally reflexive modules.

1. Introduction
One of the most classical subjects in representation theory of algebras is the study of a stable equivalence of selﬁnjective algebras, i.e., a triangle equivalence between the stable categories of ﬁnitely generated
modules over those algebras. This is extended to (inﬁnite-dimensional) Iwanaga-Gorenstein rings [30] under restriction to Cohen-Macaulay modules; a stable equivalence of Iwanaga-Gorenstein rings is a triangle
equivalence between the stable categories of Cohen-Macaulay modules over those rings.
Let R be a noetherian ring. The singularity category of R is by deﬁnition the Verdier quotientDsg (R) = Db (mod R)/Kb (proj(mod R)),
where mod R denotes the category of ﬁnitely generated R-modules, Db (−) the bounded derived category
and Kb (−) the bounded homotopy category. The singularity category Dsg (R) is a triangulated category,
which is also called the stable (stabilized) derived category, triangulated category of singularities andsingular derived category. This has been introduced by Buchweitz [23] in the 1980s, and in recent years
it has been related to the mirror symmetry by Orlov [44]. A lot of studies on singularity categories
have been done in various approaches; see [20, 24, 26, 32, 37, 39, 42, 45, 48, 53, 54, 57] for instance. A
celebrated theorem of Buchweitz [23] shows that if R is an Iwanaga-Gorenstein ring, then the stable
category of Cohen-Macaulay R-modules is triangle equivalent to the singularity category of R. Thus,a stable equivalence of Iwanaga-Gorenstein rings is extended to arbitrary noetherian rings as a triangle
equivalence between their singularity categories, which is called a singular equivalence.
Let C be an additive category with pseudokernels [4]. We denote by mod C the category of ﬁnitely
presented right C-modules, which is a full subcategory of the functor category of C (see Deﬁnition 2.3) andturns out to be an abelian category with enough projective objects. This category has been introduced
by Auslander [4], and in a series of papers [9–14] Auslander and Reiten have established a lot of deepstudies on this category. The notion of singular equivalences of noetherian rings is further extended toadditive categories C by using mod C as follows: We take the Verdier quotientDsg (C) = Db (mod C)/Kb (proj(mod C)),and call this the singularity category of C. We say that two additive categories C, C ′ are singularly
equivalent if there exists a triangle equivalence Dsg (C) ∼= Dsg (C ′ ). Thus, one of the most natural andfundamental questions is the following.
2010 Mathematics Subject Classification. 13C60, 13D09, 16G60, 16G70, 18A25, 18E30.
Key words and phrases. complete intersection, finitely presented functor, functor category, Gorenstein ring, resolving
subcategory, simple hypersurface singularity, singular equivalence, singularity category, stable category.
RT was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
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HIROKI MATSUI AND RYO TAKAHASHI

Question 1.1. When are given two additive categories singularly equivalent?
The main purpose of this paper is to explore this question for certain additive categories in relation
to the Gorenstein projective property in an abelian category (see Deﬁnition 3.3). In fact, we shall ﬁnd a
lot of singular equivalences, which should be interesting and remarkable once one takes into account thefact that triangle equivalences seldom arise over commutative rings. To state our results, let us recall
and introduce several notions.
We say that C is regular (respectively, Gorenstein) if every object of mod C has ﬁnite projective dimension (respectively, Gorenstein projective dimension). A regular (respectively, Gorenstein) category C iscalled of dimension at most n if every object of mod C has projective dimension (respectively, Gorenstein
projective dimension) at most n, or equivalently, the nth syzygies in mod C are projective (respectively,
Gorenstein projective). Any abelian (respectively, triangulated) category is regular (respectively, Gorenstein) of dimension at most 2 (respectively, 0); see Proposition 3.6. Note that C is regular if and only if
C is singularly equivalent to the zero category 0, that is, Dsg (C) ∼= 0. Beligiannis [21] shows that if C is
Gorenstein of dimension at most n for some n ≥ 0, then Dsg (C) is triangle equivalent to Gproj(mod C),
which is a generalization of the theorem of Buchweitz stated above. Here, for an abelian category A withenough projective objects, Gproj A stands for the full subcategory of Gorenstein projective objects of A.
This is a Frobenius category, and its stable category Gproj A is a triangulated category [21].
A full subcategory of an abelian category with enough projective objects is called resolving if it contains
the projective objects and is closed under direct summands, extensions and kernels of epimorphisms. This
notion has been introduced by Auslander and Bridger [7] to prove that the category of totally reﬂexive
modules is a resolving subcategory of the category mod R of ﬁnitely generated modules over a noetherianring R. The category CM(R) of maximal Cohen-Macaulay modules over a Cohen-Macaulay local ring R
is also a resolving subcategory of mod R, and there are many other important subcategories known to beresolving. The studies of resolving subcategories have been done widely so far; see [2, 15, 16, 28, 29, 40,
47, 49–52, 56] for example. For a resolving subcategory X of an abelian category A, let X = X / proj A
be the stable category of X . Then X has pseudokernels, and hence mod X is an abelian category withenough projective objects [16]. In fact, X still has pseudokernels even if one removes the assumptionthat X is closed under direct summands and extensions. So we deﬁne a quasi-resolving subcategory to
be a full subcategory containing the projective objects and closed under ﬁnite direct sums and kernelsof epimorphisms. A resolving subcategory is none other than a quasi-resolving subcategory closed under
direct summands and extensions.
Now, we recall the following two facts. The ﬁrst one is implicitly given by Auslander and Reiten [9],while the second one is essentially obtained by Yoshino [56], based on [16] (see also [22, 36, 38]).
Theorem 1.2 (Auslander-Reiten). Let A be an abelian category with enough projective objects. Let X
be a quasi-resolving subcategory of A. Suppose that every object in X has projective dimension at most n
in A. Then X is regular of dimension at most 3n − 1.
Theorem 1.3 (Yoshino). Let A be an abelian category with enough projective objects. Let X be a quasiresolving subcategory of A. Suppose that X is contained in Gproj A and closed under cosyzygies. Then X
is Gorenstein of dimension at most 0, that is, mod X is a Frobenius category.
Motivated by these two theorems, in this paper we ﬁrst study stable categories of quasi-resolving
subcategories. To be more precise, we shall establish a Gorenstein analogue of Theorem 1.2 which extends
Theorem 1.3. Let X be a quasi-resolving subcategory of an abelian category A and n ≥ 0 an integer.
Denote by Ωn X the full subcategory of A consisting of nth syzygies of objects in X . We introduce thefollowing condition.
(Gn ) Ωn X is contained in Gproj A and closed under cosyzygies.
A typical example of a quasi-resolving subcategory satisfying (Gn ) is the full subcategory of objects ofGorenstein projective dimension at most n. Every resolving subcategory over a local complete intersection
R satisﬁes (Gn ) for A = mod R and n = dim R. We shall obtain the following theorem.
Theorem 1.4. Let A be an abelian category with enough projective objects. Let X be a quasi-resolving
subcategory of A satisfying the condition (Gn ). Then Ωn X = X ∩ Gproj A holds. Denote this by Y.
(1) X is Gorenstein of dimension at most 3n, and there is a triangle equivalence Dsg (X ) ∼= Gproj(mod X ).

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

3(2) Y is a Frobenius subcategory of Gproj A, and Y is a triangulated subcategory of Gproj A.
(3) Y is Gorenstein of dimension at most 0, and there is a triangle equivalence Dsg (Y) ∼= mod Y.
(4) X and Y are singularly equivalent.
The ﬁrst assertion of Theorem 1.4(1) is a Gorenstein analogue of Theorem 1.2, and letting n = 0
recovers Theorem 1.3. In relation to this result, we also consider when the condition Ext>n
R (X , R) = 0 is(modX,
proj(mod
X
))
=
0
for
a
resolving
subcategory
X of modules
equivalent to the condition Ext>3n
mod X
over a commutative ring R. For n = 0 the latter condition is nothing but the condition that mod X
is quasi-Frobenius (in the sense of [56]), and we can recover [56, Theorem 4.2]. One also obtains acharacterization of Gorenstein local rings R in terms of modules over the stable category mod R, whose
artinian case is none other than [56, Corollary 4.3].
The second assertion of Theorem 1.4(1) clariﬁes the structure of the singularity category Dsg (X ); forexample, each object of Dsg (X ) turns out to be isomorphic to a shift of an object in mod X . This is a
key to deduce the remaining assertions in Theorem 1.4.
Using Theorem 1.4(4) one can obtain various singular equivalences. For example, let X be a quasiresolving subcategory of A with Ωn X ⊆ Gproj A ⊆ X for some n ≥ 0, e.g., the category of objects ofGorenstein projective dimension at most n. Then X is singularly equivalent to Gproj A. In particular,
if R is a commutative Gorenstein local ring and X is a quasi-resolving subcategory of mod R containing
CM(R), then X and CM(R) are singularly equivalent. If R is moreover a complete intersection, then
X is singularly equivalent to X ∩ CM(R) for all resolving subcategories X of mod R. Combining thiswith a classiﬁcation of resolving subcategories given in [50] yields that if R is an isolated hypersurface
singularity, then X is singularly equivalent to either CM(R) or the zero category 0, so there are only at
most two singular equivalence classes.
We are thus interested in asking when R admits exactly one singular equivalence class. More precisely,
for a resolving subcategory X of mod R we consider when X is singularly equivalent to 0, or equivalently,
when X is regular. We shall prove the following theorem, which characterizes the regularity of stable
categories of resolving subcategories.
Theorem 1.5. Let R be a d-dimensional nonregular complete local ring with algebraically closed residue
ﬁeld k of characteristic zero. Then the following are equivalent.
(1) R is Gorenstein, and CM(R) is regular.
(2) R is a complete intersection, and X is regular for every resolving subcategory X of mod R.
(3) R is a complete intersection, and X is regular for some resolving subcategory X of mod R that contains
a module of maximal complexity.
(4) R is a simple hypersurface singularity of type (A1 ), namely, R ∼= k[[x0 , . . . , xd ]]/(x20 + · · · + x2d ).
When one of these conditions is satisﬁed, CM(R) is regular of dimension at most 0, namely, mod CM(R)has global dimension 0.
If an additive category C is regular of dimension at most n for some n ≥ 0, then C is regular bydeﬁnition. In view of Theorem 1.5, as of independent interest, we also consider when the converse of thisstatement holds for C = X .
The paper is organized as follows. In Section 2, for later use we state several fundamental results on
the structure of modules over the stable category of a subcategory of an abelian category. In Sections
3 and 4, we compare many kinds of conditions on quasi-resolving subcategories. The ﬁrst assertion ofTheorem 1.4 is shown in this section. In Section 5, we study singular equivalence of the stable categories
of resolving subcategories, and give proofs of the remaining assertions of Theorem 1.4. In Section 6, weconsider when the stable category of a resolving subcategory is regular, and prove Theorem 1.5.
Convention. Throughout this paper, unless otherwise speciﬁed, we use the following convention: All
subcategories are assumed to be strictly full (i.e., full and closed under isomorphism). Let A be an
abelian category with enough projective objects, and denote by proj A the full subcategory of projective
objects of A. Let Ab and 0 stand for the category of abelian groups and the zero category, respectively.
Let R be a commutative noetherian ring, and denote by mod R the category of ﬁnitely generated Rmodules. All R-modules in this paper are assumed to be ﬁnitely generated. A maximal Cohen-Macaulay
module is simply called an MCM module. When R is a Cohen-Macaulay local ring, CM(R) stands forthe full subcategory of mod R consisting of MCM modules over R.

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HIROKI MATSUI AND RYO TAKAHASHI

2. Modules over stable categories
In this section we give several fundamental results on the structure of modules over the stable categoryof a subcategory of A. Most of the results are given in [9–14, 16] and [56] at least essentially, but we give
proofs for the convenience of the reader. Let us begin with recalling the deﬁnition of a stable category.
Definition 2.1. (1) For two objects M, N of A we deﬁne the stable hom-set as the quotient group
HomA (M, N ) := HomA (M, N )/PA (M, N ), where PA (M, N ) consists of all morphisms from M to N
that factor through objects in proj A. For A = mod R, we simply write HomR (M, N ) for HomA (M, N ).
(2) Let X be a subcategory of A containing proj A. Then the quotient category X := X / proj A is called
the stable category of X ; the objects of X are the same as those of X , and the hom-set HomX (M, N )of M, N ∈ X is deﬁned as HomA (M, N ). Hence X is a full subcategory of A.
The following lemma is given in the case where A = mod R by [56, Lemma 2.7], whose proof uses
Auslander transposes of modules and does not work for general abelian categories A.
fg

Lemma 2.2. Let 0 → A −→B−→ C → 0 be an exact sequence in A. Let X be an object of A.
(1) The induced sequence HomA (X, A) → HomA (X, B) → HomA (X, C) is exact.
(2) The induced sequence HomA (C, X) → HomA (B, X) → HomA (A, X) is exact if Ext1A (C, proj A) = 0.
Proof. (1) The left-exactness of the functor Hom shows that the induced sequence
HomA (X,f )HomA (X,g)

0 → HomA (X, A) −−−−−−−→ HomA (X, B) −−−−−−−→ HomA (X, C)
is exact. Let u : X → B be a morphism in A such that u is in the kernel of HomA (X, g) : HomA (X, B) →HomA (X, C). Then gu is the composition of some morphisms a : X → P and b : P → C in A, where P
is a projective object in A. There is a morphism c : P → B with gc = b. Hence g(u − ca) = gu − gca =gu − ba = 0, and we ﬁnd a morphism d : X → A such that u − ca = f d. We have u = f d, which is in theimage of HomA (X, f ) : HomA (X, A) → HomA (X, B).
(2) By the left-exactness of Hom the induced sequence
HomA (g,X)

HomA (f,X)

0 → HomA (C, X) −−−−−−−→ HomA (B, X) −−−−−−−→ HomA (A, X)
is exact. Let u : B → X be a morphism with u ∈ Ker HomA (f, X). Then uf is the composition of somemorphisms a : A → P and b : P → X with P ∈ proj A. Since Ext1A (C, proj A) = 0, the map HomA (f, P )is surjective, and there exists a morphism c : B → P such that cf = a. We have (u − bc)f = 0, and ﬁnd
a morphism d : C → X with u − bc = dg. It follows that u = dg ∈ Im Hom(g, X).
■
Next, we recall the deﬁnition of the category of ﬁnitely presented modules over an additive category.
Definition 2.3. Let C be an additive category. Denote by Mod C the functor category of C, that is, theobjects are additive contravariant functors from C to Ab, and the morphisms are natural transformations.
An object and a morphism of Mod C are called a (right) C-module and a C-homomorphism, respectively.
A C-module F is said to be ﬁnitely presented if there is an exact sequence
HomC (−, X) → HomC (−, Y ) → F → 0
in the abelian category Mod C with X, Y ∈ C. The full subcategory of Mod C consisting of ﬁnitely
presented C-modules is denoted by mod C. This is sometimes called the Auslander category of C.
Remark 2.4. (1) For each X ∈ C the functor HomC (−, X) is a projective object of mod C, and conversely,
any projective object of mod C is isomorphic to a direct summand of HomC (−, X) for some X ∈ C.
Yoneda’s lemma asserts that the assignment X 7→ HomC (−, X) makes a fully faithful functor
C ,→ mod C.
This is called the Yoneda embedding of C. Thanks to this, any C-homomorphism HomC (−, X) →HomC (−, Y ) can be described as HomC (−, f ) for some morphism f ∈ HomC (X, Y ).
(2) It is said that C has pseudokernels if for each morphism f : X → Y in C there exists a morphismg : Z → X in C such that the induced sequence
HomC (−,g)

HomC (−,f )HomC (−, Z) −−−−−−−→ HomC (−, X) −−−−−−−→ HomC (−, Y )is exact. This condition is equivalent to saying that mod C is abelian; see [5, Chapter III, §2].

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

5

Here we recall the deﬁnition of the syzygies of an object in an abelian category.
Definition 2.5 (cf. Deﬁnition 3.1). Let M be an object of A, and let n be a positive integer. Let
f : P → M be a right (proj A)-approximation, which is by deﬁnition a morphism such that every
morphism from an object in proj A to M factors through f . Then the kernel of f is called the ﬁrst syzygy
of M and denoted by ΩM . The nth syzygy Ωn M of M is deﬁned inductively as Ω(Ωn−1 M ). We putΩ0 M := M . For a subcategory X of A we denote by Ωn X the subcategory of A consisting of all nth
syzygies. Note that Ωn X contains proj A for n > 0.
Remark 2.6. Let M be an object of A, and let n be a positive integer.
(1) Since A is assumed to have enough projective objects, the nth syzygy of M exists, and one has anexact sequence
0 → Ωn M → Pn−1 → · · · → P0 → M → 0
in A with Pi ∈ proj A for 0 ≤ i ≤ n − 1.
(2) By Schanuel’s lemma, the nth syzygy of M is uniquely determined up to projective summands,
whence in the stable category A it is uniquely determined up to isomorphism.
Now we show the proposition below, which induces from each short exact sequence in A a long exact
sequence in a functor category.
fg

Proposition 2.7. Let σ : 0 → A −→B−→ C → 0 be a short exact sequence in A.
(1) There is a sequence
g2

f1

d

g1

d

f

g

1
[· · · −→ Ω2 C −→
ΩA −→ ΩB −→ ΩC −
→A−
→B−
→ C]

of morphisms in A such that one can form a short exact sequence from any consective two morphisms(consisting three objects) by adding some projective object to the middle object.
(2) Let X be a subcategory of A containing proj A. One has an induced long exact sequence in Mod X :· · · → HomA (−, Ω2 B)|X → HomA (−, Ω2 C)|X → HomA (−, ΩA)|X → HomA (−, ΩB)|X→ HomA (−, ΩC)|X → HomA (−, A)|X → HomA (−, B)|X → HomA (−, C)|X
→ Ext1A (−, A)|X → Ext1A (−, B)|X → Ext1A (−, C)|X → Ext2A (−, A)|X → Ext2A (−, B)|X → · · · .
Proof. (1) Taking a surjection π : P → C with P ∈ proj A and making the pullback diagram of g and π,
we get an exact sequence
(d∗)
(f,∗)
0 → ΩC −−→ A ⊕ P −−−→ B → 0.
Iterating this procedure gives rise to exact sequences(g∗1 )(d,∗)
0 → ΩB −−−→ ΩC ⊕ P ′ −−−→ A → 0,(f∗1 )(g1 ,∗)
0 → ΩA −−−→ ΩB ⊕ P ′′ −−−−→ ΩC → 0,(d∗1 )(f1 ,∗)
0 → Ω2 C −−−→ ΩA ⊕ P ′′′ −−−−→ ΩB → 0,
′

′′

′′′

where P , P , P , P′′′′(g∗2 )(d1 ,∗)
0 → Ω2 B −−−→ Ω2 C ⊕ P ′′′′ −−−−→ ΩA → 0,...,

, . . . are projective. Thus we obtain a sequence
g2

d

f1

g1

d

f

g

1
[· · · −→ Ω2 C −→
ΩA −→ ΩB −→ ΩC −
→A−
→B−
→ C]

of morphisms, which is what we want.
(2) Let X ∈ X . Using Lemma 2.2 for the sequence obtained in (1) yields an exact sequence(2.7.1)· · · → HomA (X, Ω2 C) → HomA (X, ΩA) → HomA (X, ΩB)→ HomA (X, ΩC) → HomA (X, A) → HomA (X, B) → HomA (X, C).

On the other hand, there is an exact sequenceδ

0 → HomA (X, A) → HomA (X, B) → HomA (X, C) −→ Ext1A (X, A) → Ext1A (X, B) → · · · .
Let P be a projective object of A, and let α : X → P and β : P → C be morphisms in A. Then we haveδ(βα) = Ext1A (βα, A)(σ) = Ext1A (α, A)(Ext1A (β, A)(σ)) = 0 as Ext1A (β, A)(σ) = 0. Hence δ(PA (X, C)) =6

HIROKI MATSUI AND RYO TAKAHASHI

0. Therefore, δ induces a homomorphism δ ′ : HomA (X, C) → Ext1A (X, A) such that the following diagram
commutes, where the vertical maps are canonical surjections.
0/ HomA (X, A)/ HomA (X, B)
HomA (X, A)/ HomA (X, B)/ HomA (X, C)/ Ext1A (X, A)
o7
ooo
o
o
ooo ′

ooo δ
/ HomA (X, C)δ/ Ext1A (X, B) → · · ·Using Lemma 2.2 and diagram chasing, we obtain an exact sequence(2.7.2)

δ′

HomA (X, A) → HomA (X, B) → HomA (X, C) −→ Ext1A (X, A) → Ext1A (X, B) → · · · .

Splicing (2.7.1) and (2.7.2) yields an exact sequence as in the assertion.

■

Let us give the deﬁnitions of quasi-resolving and resolving subcategories, which are the main targets
studied in this paper.
Definition 2.8. (1) A subcategory X of A is called quasi-resolving if X satisﬁes the following conditions.
(a) X contains proj A.
(b) X is closed under ﬁnite direct sums, that is, for a ﬁnite number of objects X1 , . . . , Xn in X the
direct sum X1 ⊕ · · · ⊕ Xn is in X .
(c) X is closed under kernels of epimorphisms, that is, for each short exact sequence 0 → L → M →N → 0 in A, if M and N are in X , then so is L.
(2) A quasi-resolving subcategory X of A is called resolving if X satisﬁes the following conditions.
(a) X is closed under direct summands, namely, if X is an object in X and Y is a direct summandof X in A, then Y is in X .
(b) X is closed under extensions, namely, for each short exact sequence 0 → L → M → N → 0 in
A, if L and N are in X , then so is M .
For a subcategory C of A we denote by res C the resolving closure of X , that is, the smallest resolving
subcategory of A containing all the objects in C.
Remark 2.9. The notion of a resolving subcategory has been introduced by Auslander and Bridger [7],but we should remark that in [7] a resolving subcategory is deﬁned to be a quasi-resolving subcategory
closed under extensions. In our sense, a resolving subcategory is also assumed to be closed under direct
summands.
As a trivial example, the subcategory of A consisting of objects with projective dimension less thann for each 1 ≤ n ≤ ∞ is a resolving subcategory. The catgeory CM(R) of MCM modules over a CohenMacaulay local ring R is a resolving subcategory of mod R. There are a lot of other examples of resolving
subcategories; one can ﬁnd some of them in [49, Example 2.4].
Remark 2.10. Let X be a quasi-resolving subcategory of A.
(1) Let M ∈ A and P ∈ proj A. Then the equivalence
M ∈ X ⇐⇒ M ⊕ P ∈ X
holds true. Indeed, it is trivial that if M is in X , then so is M ⊕ P . The opposite implication follows
(10)
(0,1)
from the exact sequence 0 → M −−→ M ⊕ P −−−→ P → 0.
(2) Let f : M → N be a morphism in X . Then one can choose a morphism g : P → N with P ∈ proj A
such that the morphism (f, g) : M ⊕ P → N is an epimorphism. Taking the kernel, one gets a short
exact sequence(f,g)

0 → L → M ⊕ P −−−→ N → 0
in A. Note that all of the three objects in this exact sequence are in X .
Finally, we give a structure result on ﬁnitely presented modules over the stable category of a quasiresolving subcategory. The following proposition extends the result [56, Proposition 3.3] on module
categories to abelian categories. The ﬁrst assertion is also a generalization of the result [16, Proposition
1.1] on resolving subcategories to quasi-resolving ones.
Proposition 2.11. Let X be a quasi-resolving subcategory of A.

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

7(1) The category mod X is an abelian category with enough projective objects.
(2) For each object F ∈ mod X there exists an exact sequence 0 → A → B → C → 0 in A with
A, B, C ∈ X which induces a projective resolution of F in mod X :· · · → HomA (−, Ω2 C)|X → HomA (−, ΩA)|X → HomA (−, ΩB)|X → HomA (−, ΩC)|X→ HomA (−, A)|X → HomA (−, B)|X → HomA (−, C)|X → F → 0.
(f,g)

Proof. (1) Let f : M → N be a morphism in X . There is an exact sequence 0 → L → M ⊕P −−−→ N → 0
in A with L ∈ X and P ∈ proj A (see Remark 2.10). By Lemma 2.2(1) this induces an exact sequence
Hom (−,f )|X

HomA (−, L)|X → HomA (−, M )|X −−−−A−−−−−→ HomA (−, N )|X
in mod X . Thus X has pseudokernels, and the assertion follows from Remark 2.4(2).
ϕ(2) There is an exact sequence HomA (−, B)|X −→ HomA (−, C)|X → F → 0 with B, C ∈ X . By
Remark 2.4(1) we ﬁnd a morphism f : B → C such that ϕ = HomA (−, f )|X . We have an exact sequence(f,g)

0 → A → B ⊕ P −−−→ C → 0 with A ∈ X and P ∈ proj A by Remark 2.10. Applying Proposition 2.7(2)to this short exact sequence, we obtain such an exact sequence as in the assertion.
■
Remark 2.12. We should remark that mod X is not necessarily abelian; X does not necessarily havepseudokernels, even if it is resolving. (For example, consider the subcategory of mod R consisting of
totally reﬂexive modules in the case where R is not Gorenstein.) This is one of the reasons why we takethe stable category X of X and study X rather than X itself.
We recall here the deﬁnition of the singularity category of an additive category.
Definition 2.13. Let C be an additive category such that mod C is abelian. The singularity category of
C is by deﬁnition the Verdier quotientDsg (C) = Db (mod C)/Kb (proj(mod C)).
Note that Dsg (mod R) is nothing but the (usual) singularity category Dsg (R) of R (see the second
paragraph of Section 1).
Let X be a quasi-resolving subcategory of A. Then mod X is an abelian category by Proposition
2.11(1). So one can deﬁne the singularity category Dsg (X ) of X .
3. Several conditions on quasi-resolving subcategories
In this section we compare several conditions on quasi-resolving subcategories, motivated by the workof Yoshino [56]. The point is to investigate the Frobenius subcategory of Gorenstein projective objects,
which is invisible in [56], so that we extend main results of [56]. We also show that one of the conditions
makes a certain equivalence of triangulated categories, which plays an important role in later sections.
To state those conditions, we need to recall the deﬁnitions of a cosyzygy and a Gorenstein projective
object in an abelian category.
Definition 3.1 (cf. Deﬁnition 2.5). Let M be an object of A, and let n be a positive integer. Let
g : M → P be a left (proj A)-approximation, i.e., a morphism such that any morphism from M to an
object in proj A factors through g. Then the cokernel of g is called the ﬁrst cosyzygy of M and denotedby Ω−1 M . The nth cosyzygy Ω−n M of M is deﬁned inductively as Ω−1 (Ω−(n−1) M ). For a subcategory
X of A we denote by Ω−n X the subcategory of A consisting of all nth cosyzygies. This contains proj A.
Remark 3.2. Let M ∈ A and n > 0. Suppose that Ω−n M exists. Then a complex(0 → M → P−1 → · · · → P−n → Ω−n M → 0)
in A with P−i ∈ proj A for 1 ≤ i ≤ n is induced. This complex is not necessarily exact.
Definition 3.3. Let M be an object of A.
(1) The projective dimension of M in A, denoted by pdA M , is deﬁned as the inﬁmum of the integersn ≥ 0 such that there exists an exact sequence
0 → Pn → Pn−1 → · · · → P1 → P0 → M → 0
in A with Pi ∈ proj A for all 0 ≤ i ≤ n.

8

HIROKI MATSUI AND RYO TAKAHASHI(2) An exact sequence
∂

∂

∂

∂−1

2
1
0
P : · · · −→
P1 −→
P0 −→
P−1 −−→ · · ·of projective objects of A is called a complete resolution of M if HomA (P, Q) is again exact for all
Q ∈ proj A and the image of ∂0 is isomorphic to M .
(3) An object admitting a complete resolution is called Gorenstein projective. The subcategory of A
consisting of Gorenstein projective objects is denoted by Gproj A. Since Gproj A contains proj A, thestable category Gproj A := Gproj A is deﬁned.
(4) The Gorenstein projective dimension of M in A, denoted by GpdA M , is deﬁned as the inﬁmum of
the integers n ≥ 0 such that there exists an exact sequence

0 → Gn → Gn−1 → · · · → G1 → G0 → M → 0
in A with Gi ∈ Gproj A for all 0 ≤ i ≤ n.
Remark 3.4. (1) A Gorenstein projective object of mod R is called a totally reﬂexive module. The
Gorenstein projective dimension of an object M of mod R is called the Gorenstein dimension (Gdimension for short) of M , and denoted by GdimR M .
(2) Let M be a Gorenstein projective object of A. Then M admits the nth cosyzygy for all n ≥ 1, whichis uniquely determined up to projective summands, and the induced complex(0 → M → P−1 → · · · → P−n → Ω−n M → 0)
in A with P−i ∈ proj A for 1 ≤ i ≤ n is exact. Furthermore, taking Ω makes an autoequivalence ofthe category Gproj A and taking Ω−1 makes its quasi-inverse.
(3) The category Gproj A is a Frobenius category [33, Chapter I, §2] and a resolving subcategory of A, andthe stable category Gproj A is a triangulated category. These statements are shown in [21, Proposition
2.13] except the fact that Gproj A is closed under direct summands (see Remark 2.9). This fact isproved by noting the following.
(f,g)

Suppose that there is an exact sequence 0 → M ⊕ N −−−→ P → L → 0 in A with P ∈ proj A
and Ext1A (L, proj A) = 0. Letting X, Y be the cokernels of f, g respectively, we get exact sequences
f

g

0→M −
→ P → X → 0 and 0 → N −→ P → Y → 0. There is a commutative diagram0/ M ⊕N

0O

/P
O

/L
O

/0

/ P ⊕PO
1(−1)/ X ⊕Y
O

/0

PO

PO

0

0

(f,g)

/ M ⊕N
(

0

0O

f 0
0 g

)

(1,1)

with exact rows and columns. The exact sequence 0 → P → X ⊕Y → L → 0 splits as Ext1A (L, P ) = 0,
and hence Ext1A (X, proj A) = 0. Using this argument, one observes that if there is an exact sequence
0 → M ⊕ N → P 0 → P 1 → P 2 → · · · whose dual by each projective is also exact, then there is anexact sequence 0 → M → P 0 → P 0 ⊕ P 1 → P 0 ⊕ P 1 ⊕ P 2 → · · · whose dual by each projective isalso exact.
(4) Let M ∈ A and n ≥ 0. Then GpdA M ≤ n if and only if Ωn M ∈ Gproj A.
Let us introduce regularity and Gorensteinness for a quasi-resolving subcategory.
Definition 3.5. Let C be an additive category such that mod C is abelian. We say that C is regular(respectively, Gorenstein) if every object of mod C has ﬁnite projective dimension (respectively, Gorenstein
projective dimension). A regular (respectively, Gorenstein) category C is called of dimension at most nSINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

9

if every object of mod C has projective dimension (respectively, Gorenstein projective dimension) at mostn, or equivalently, Ωn (mod C) coincides with proj(mod C) (respectively, Gproj(mod C)).
Here are typical examples of regular and Gorenstein categories.
Proposition 3.6. (1) Every abelian category is regular of dimension at most 2.
(2) Every triangulated category is Gorenstein of dimension at most 0.
(3) Let Λ be an Iwanaga-Gorenstein ring of (selﬁnjective) dimension at most n. Then the category proj Λof ﬁnitely generated projective Λ-modules is Gorenstein of dimension at most n.
f

Proof. Let C be an additive category, and pick F ∈ mod C. There is an exact sequence HomC (−, M ) −→HomC (−, N ) → F → 0. By Yoneda’s lemma we ﬁnd a morphism g ∈ HomC (M, N ) with f = HomC (−, g).
g(1) Suppose that C is abelian. Then there is an exact sequence 0 → L → M −→ N in C, which induces
an exact sequencef

0 → HomC (−, L) → HomC (−, M ) −→ HomC (−, N ) → F → 0
in mod C. This shows that F has projective dimension at most 2.
g(2) Suppose that C is triangulated. Then there is an exact triangle L → M −→ N ⇝ in C, which
induces an exact sequencef· · · → HomC (−, Σ−1 N ) → HomC (−, L) → HomC (−, M ) −→ HomC (−, N ) → HomC (−, ΣL) → · · ·in mod C, whose dual by each projective object of mod C is also exact by Remark 2.4(1) and Yoneda’s
lemma. Hence this exact sequence gives a complete resolution of F , and so F is Gorenstein projective.
(3) One has Ωn (mod Λ) = Gproj(mod Λ) by [30, Theorem 10.2.14]. The assertion follows from this; seealso Lemma 6.4 stated later.
■
Auslander and Reiten [10] implicitly prove the following.
Theorem 3.7 (Auslander-Reiten). Let X be a quasi-resolving subcategory of A. Suppose that every
object in X has projective dimension at most n in A. Then X is regular of dimension at most 3n − 1.
We are interested in establishing a Gorenstein analogue of this theorem. For this, we introduce thefollowing condition.
Definition 3.8. (1) We say that a subcategory Y of Gproj A is closed under cosyzygies provided that if
Y is an object in Y, then so is Ωi Y for all i < 0.
(2) Let X be a quasi-resolving subcategory of A, and let n ≥ 0 be an integer. We say that X satisﬁes
the condition (Gn ) if Ωn X is contained in Gproj A and closed under cosyzygies.
From now on we ﬁx a quasi-resolving subcategory X of A and an integer n ≥ 0, and name several
statements for convenience.
(A) X satisﬁes (Gn ).
(B) X is Gorenstein of dimension at most 3n.
(C) Every object in X has Gorenstein projective dimension at most n.
(D) Ext>n
A (X , proj A) = 0.
(E) Ext>3n
mod X (mod X , proj(mod X )) = 0.
It is not hard to observe that the implications (A) ⇒ (C) ⇒ (D) and (B) ⇒ (E) hold. In what follows
we consider other implications. Before that, we investigate what the condition (A) means. In fact, thiscondition can be interpreted in several other ways.
Proposition 3.9. Let X be a quasi-resolving subcategory of A. The following are equivalent for n ≥ 0.
(1) X satisﬁes (Gn ).
(2) Ωn X is contained in Gproj A and Ωi (Ωn X ) is contained in X for all i ∈ Z.
(3) Each object in Ωn X has a complete resolution the images of whose diﬀerential morphisms are in X .
n(4) One has Ext>nA (X , proj A) = 0 and every object M in Ω X admits an exact sequence 0 → M → P →′′n
M → 0 with P ∈ proj A and M ∈ Ω X .
n(5) One has Ext>nA (X , proj A) = 0 and there exists a subcategory Y of A with Ω X ⊆ Y ⊆ X such thateach Y ∈ Y admits an exact sequence 0 → Y → P → Y ′ → 0 with P ∈ proj A and Y ′ ∈ Y.

10

HIROKI MATSUI AND RYO TAKAHASHI

Proof. (1) ⇒ (2): The implication is obvious.
(2) ⇒ (3): Let M be an object in Ωn X . Then M is Gorenstein projective, and admits a complete
∂

∂

∂

∂−1

2
1
0
resolution · · · −→
P1 −→
P0 −→
P−1 −−→ · · · . For each i ∈ Z the image of ∂i is isomorphic to Ωi M (upto projective summands), which is in Ωi (Ωn X ) and so in X .

∂

∂

∂

∂−1

2
1
0
(3) ⇒ (4): Let M ∈ Ωn X . Then M has a complete resolution · · · −→
P1 −→
P0 −→
P−1 −−→ · · ·>0such that Im ∂i is in X for all i ∈ Z. Hence ExtA (M, proj A) = 0, and there is an exact sequence
0 → M → P−1 → N → 0 with N := Im ∂−1 . Note that N = Ωn (Im ∂−n−1 ), which belongs to Ωn X .
(4) ⇒ (5): The implication follows by setting Y := Ωn X .
(5) ⇒ (1): Let M be an object in Ωn X . Take a projective resolution(3.9.1)∂∂2
1· · · −→P1 −→P0 → M → 0nof M . It is seen that Im ∂i ∈ Ωn X for all i > 0. As Ext>0
A (Ω X , proj A) = 0, the Q-dual of (3.9.1) isagain exact for all Q ∈ proj A. Since M is in Y, so there is an exact sequence 0 → M → P−1 → M ′ → 0with P−1 ∈ proj A and M ′ ∈ Y. Hence there is an exact sequence 0 → M ′ → P−2 → M ′′ → 0 with
P−2 ∈ proj A and M ′′ ∈ Y. Iterating this procedure gives rise to an exact sequence(3.9.2)∂−1∂−2

0 → M → P−1 −−→ P−2 −−→ · · ·nwith Pi ∈ proj A and Im ∂i ∈ Y for all i < 0. Since Im ∂i = Ωn (Im ∂i−n ) ∈ Ωn X and Ext>0
A (Ω X , proj A) =0, the Q-dual of (3.9.2) is an exact sequence for all Q ∈ proj A. Splicing (3.9.1) and (3.9.2) yields acomplete resolution of M , which shows that M is Gorenstein projective, and Ωi M = Im ∂i belongs toΩn X for all i ∈ Z.
■

The following result shows that (D) implies (E).
Proposition 3.10. Let n be a nonnegative integer, and let X be a quasi-resolving subcategory of A. If
>3n
Ext>n
A (X , proj A) = 0, then Extmod X (mod X , proj(mod X )) = 0.
Proof. Let F be an object in mod X . By virtue of Proposition 2.11 there is an exact sequence(3.10.1)

0 → X2 → X1 → X0 → 0

in A with X0 , X1 , X2 ∈ X which induces a projective resolution· · · → HomA (−, ΩX0 )|X → HomA (−, X2 )|X → HomA (−, X1 )|X → HomA (−, X0 )|X → F → 0
of F in mod X . In view of Remark 2.4(1), we have only to show that Ext3n+1
mod X (F, HomA (−, Y )|X ) = 0 for
Y ∈ X . The induced complex(
Hommod X (HomA (−, Ωn X0 )|X , HomA (−, Y )|X ) → Hommod X (HomA (−, Ωn X1 )|X , HomA (−, Y )|X ))→ Hommod X (HomA (−, Ωn X2 )|X , HomA (−, Y )|X )is isomorphic to the induced complex()
HomA (Ωn X0 , Y ) → HomA (Ωn X1 , Y ) → HomA (Ωn X2 , Y )by Remark 2.4(1) again. This comes from the short exact sequence 0 → Ωn X2 → Ωn X1 → Ωn X0 → 0
obtained by applying Ωn to (3.10.1). Since Ext1A (Ωn X0 , proj A) = 0, it is exact by Lemma 2.2(2).
■The main result of this section is the following theorem, which is a Gorenstein analogue of Theorem 3.7
due to Auslander and Reiten. It not only shows that the implication (A) ⇒ (B) holds, but also establishes
an equivalence of triangulated categories which analyzes the strucuture of singularity categories.
Theorem 3.11. Let n be a nonnegative integer. Let X be a quasi-resolving subcategory of A satisfying(Gn ). Then X is Gorenstein of dimension at most 3n, and there is a triangle equivalence
Dsg (X ) ∼= Gproj(mod X ).
Proof. The second assertion follows from the ﬁrst assertion and [21, Corollary 4.13]. So let us show theﬁrst assertion. It is obvious that Gproj(mod X ) is contained in Ω3n (mod X ). Let F be an object in mod X .
Proposition 2.11 yields a short exact sequence(3.11.1)

0→A→B→C→0

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

11

in A with A, B, C ∈ X which induces a projective resolution· · · → HomA (−, Ω2 C)|X → HomA (−, ΩA)|X → HomA (−, ΩB)|X → HomA (−, ΩC)|X→ HomA (−, A)|X → HomA (−, B)|X → HomA (−, C)|X → F → 0
of F in mod X . Then G := Ker(HomA (−, Ωn A)|X → HomA (−, Ωn B)|X ) is in Ω3n (mod X ). We want toshow that G is Gorenstein projective. Applying Ωn to (3.11.1), we have an exact sequence
f

g

0→L−
→M −
→ N → 0,where L = Ωn A, M = Ωn B and N = Ωn C. Applying Proposition 2.7(1) to the induced short exact
sequence 0 → Ω−i L → Ω−i M → Ω−i N → 0 of objects in Ωn X for each i > 0 (or making a dual argument
to the proof of Proposition 2.7(1)), we obtain a sequence[· · · → Ω2 N → ΩL → ΩM → ΩN → L → M → N → Ω−1 L → Ω−1 M → Ω−1 N → Ω−2 L → · · · ]of morphisms in A with Ωi C ∈ X for all i ∈ Z and C ∈ {L, M, N } such that any consective two morphisms
become a short exact sequence by adding some projective object. Hence we get an exact sequence(3.11.2)· · · → HomA (−, Ω2 N )|X → HomA (−, ΩL)|X → HomA (−, ΩM )|X → HomA (−, ΩN )|X→ HomA (−, L)|X → HomA (−, M )|X → HomA (−, N )|X → HomA (−, Ω−1 L)|X
→ HomA (−, Ω−1 M )|X → HomA (−, Ω−1 N )|X → HomA (−, Ω−2 L)|X → · · ·of projective objects in mod X . Applying HomA (−, Y )|X with Y ∈ X , we have an exact sequence· · · → HomA (Ω−1 L, Y ) → HomA (N, Y ) → HomA (M, Y ) → HomA (L, Y ) → HomA (ΩN, Y ) → · · · ;see Remark 2.4(1) and Lemma 2.2(2). Therefore (3.11.2) gives a complete resolution of G, and thus G isa Gorenstein projective object of mod X .
■In the rest of this section, we give several applications of our results. First, applying Proposition 3.10
and Theorem 3.11 for n = 0 yields the following corollary, which recovers [56, Theorems 3.5 and 3.7].
Corollary 3.12 (Yoshino). Let X be a quasi-resolving subcategory of A.
(1) If Ext>0A (X , proj A) = 0, then mod X is a quasi-Frobenius category.
(2) Suppose that X satisﬁes (G0 ). Then one has mod X = Gproj(mod X ). In particular, mod X is a
Frobenius category, whose stable category is triangle equivalent to Dsg (X ).
Recall that a thick subcategory of a triangulated category is by deﬁnition a triangulated subcategory
closed under direct summands.
Remark 3.13. The above corollary especially says that the following statements hold for a Gorenstein
local ring R, where CM(R) := CM(R) is the stable category of CM(R).
(1) Let X be a resolving subcategory of mod R contained in CM(R). Then mod X is a quasi-Frobenius
category.
(2) Let C be a thick subcategory of CM(R). Then mod C is a Frobenius category.
For each n ≥ 0, denote by Gpdn (A) the subcategory of A consisting of objects having Gorenstein
projective dimension at most n. Using Remark 3.4 and Theorem 3.11, we have the following result.
Corollary 3.14. (1) The subcategory Gpdn (A) of A is resolving and satisﬁes Ωn (Gpdn (A)) = Gproj A.
(2) The subcategory Gpdn (A) of A satisﬁes (Gn ).
(3) The stable category Gpdn (A) is Gorenstein of dimension at most 3n.
(4) There is a triangle equivalence Dsg (Gpdn (A)) ∼= Gproj(mod Gpdn (A)).
Next, we apply our theorem to complete intersections. Let R be a local ring with maximal ideal m.
We say that R is a complete intersection if its m-adic completion is the residue ring of a regular local
ring by an ideal generated by a regular sequence. The complete intersection dimension (CI-dimension
for short) of an R-module M , denoted by CIdimR M , is deﬁned as the inﬁmum of the quantities
pdS (M ⊗R R′ ) − pdS R′ ,

12

HIROKI MATSUI AND RYO TAKAHASHI
fg

where R → R′ ← S runs over the quasi-deformations of R. Here, a diagram R → R′ ← S of homomorphisms of local rings is called a quasi-deformation of R if f is faithfully ﬂat and g is a surjection whose
kernel is generated by an S-sequence. The notion of CI-dimension has been introduced by Avramov,
Gasharov and Peeva [19] to which we refer the reader for details of CI-dimension.
Corollary 3.15. Let R be a local ring.
(1) Let n be a nonnegative integer. Let X be a resolving subcategory of mod R all of whose objects haveCI-dimension at most n. Then X satisﬁes (Gn ). Hence X is Gorenstein of dimension at most 3n,
and there is a triangle equivalence Dsg (X ) ∼= Gproj(mod X ).
(2) Let R be a complete intersection of dimension d. Let X be a resolving subcategory of mod R. Then X
satisﬁes (Gd ). Hence X is Gorenstein of dimension at most 3d, and there is a triangle equivalence
Dsg (X ) ∼= Gproj(mod X ).
Proof. (1) We have GdimR M ≤ CIdimR M ≤ n for all R-modules M by [19, Theorem (1.4)]. According
to Remark 3.4(1)(2), the subcategory Ωn X of mod R is contained in Gproj(mod R). Since every modulein Ωn X has CI-dimension at most 0 by [19, Lemma (1.9)], it follows from [28, Theorem 4.15] thatΩ−1 (Ωn X ) ⊆ res Ωn X ⊆ X
holds. In particular, every module M ∈ Ω−1 Ωn X has CI-dimension at most n, and CIdimR M =GdimR M ≤ 0 by [19, Theorem (1.4)] and the fact that Gproj(mod R) is closed under cosyzygies. Using [28, Theorem 4.15] again, we obtainΩ−2 (Ωn X ) = Ω−1 (Ω−1 Ωn X ) ⊆ res Ω−1 Ωn X ⊆ X .
Iterating this procedure shows that the second condition in Proposition 3.9 is satisﬁed for A = mod R,
and hence X satisﬁes (Gn ). Theorem 3.11 complete the proof of the corollary.
(2) Taking advantage of [19, Theorems (1.3) and (1.4)], one observes that every R-module has CIdimension at most d. The assertion thus follows from (1).
■
4. Vanishing of Ext of modules over stable resolving subcategories
In this section we consider when the converse of Proposition 3.10 holds, in other words, when (E)implies (D). We focus on the category of ﬁnitely generated modules over a commutative noetherian local
ring which is Henselian (e.g. complete). Throughout this section, let R be a Henselian local ring withmaximal ideal m and residue ﬁeld k.
We say that two R-modules M and N are stably isomorphic if M ∼= N in the stable category mod R :=
N
⊕
Q
in
mod
R
for
some
free R-modules P, Q. Also, for
mod R of mod R, or equivalently, if M ⊕ P ∼=an R-module X we denote the (Auslander) transpose of X by TrX. This is deﬁned to be the cokernel
ϕof the map HomR (ϕ, R), where ϕ appears in an R-free presentation F1 −→ F0 → X → 0 of X. Thisis uniquely determined up to free summands, so uniquely determined up to isomorphism in the stable
category mod R.
Let Sn (respectively, Tn ) be the subcategory of mod R consisting of modules M with ExtiR (M, R) = 0(respectively, ExtiR (TrM, R) = 0) for all 1 ≤ i ≤ n. A module in Tn is called an n-torsionfree module.
There is an intimate relationship between these categories; see [35, Proposition 1.1.1 and Corollary 1.1.2].
Lemma 4.1. For all n, m ≥ 0 there is a diagram of functors
Sn+1 ∩ Tm o
O
Tr

Ω
Ω−1

/ S ∩T
n
m+1
O

Tr


Sm ∩ Tn+1 o

Tr
Ω−1
Ω

/S

Tr



m+1 ∩ Tn

which commutes up to isomorphism, whose horizontal arrows are equivalences and whose vertical arrows
are dualities.
The following is the main result of this section, which asserts that (E) implies (D) in a certain setting.

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

13

Theorem 4.2. Let n ≥ 0 be an integer, and let X be a resolving subcategory of mod R satisfying thefollowing three conditions.
(1) The R-modules of projective dimension less than n are in X .
(2) For each R-sequence x = x1 , . . . , xn the module R/xR belongs to X .
(3) For all X ∈ X the module Ωn+1 X is (n + 1)-torsionfree.
>n
If Ext3n+1
mod X (mod X , proj(mod X )) = 0, then ExtR (X , R) = 0.

This theorem is proved in ﬁve steps. The ﬁrst four steps are done by the following four lemmas,
respectively.
Lemma 4.3. Let n be a nonnegative integer. Let X be a quasi-resolving subcategory of mod R with
Ext3n+1
mod X (mod X , proj(mod X )) = 0. Let 0 → Y → E → X → 0 be an exact sequence of modules in X
with pdR Y ≤ n and pdR E > n such that X is an indecomposable R-module belonging to Sn . Then Ωn E
is stably isomorphic to Ωn X.
(f,r)

Proof. Taking nth syzygies, we have an exact sequence 0 → Ωn Y → Ωn E ⊕P −−−→ Ωn X → 0, where P isa free R-module. (This follows from the horseshoe and snake lemmas.) Since Y has projective dimension
at most n, the R-module Ωn Y is free. Proposition 2.7(2) gives rise to an exact sequenceα

0 = HomR (−, Ωn Y )|X → HomR (−, Ωn E)|X −→ HomR (−, Ωn X)|X → HomR (−, Ωn−1 Y )|X→ · · · → HomR (−, E)|X → HomR (−, X)|X → F → 0.
nThen F belongs to mod X . As Ωn E is in X , the assumption implies Ext3n+1
mod X (F, HomR (−, Ω E)|X ) = 0.
Therefore the map

Hommod X (HomR (−, Ωn X)|X , HomR (−, Ωn E)|X ) → Hommod X (HomR (−, Ωn E)|X , HomR (−, Ωn E)|X )induced by α is surjective, which shows that the map HomR (f, Ωn E) : HomR (Ωn X, Ωn E) →HomR (Ωn E, Ωn E) is also surjective (see Remark 2.4(1)). Hence there exists a homomorphism g : Ωn X →Ωn E of R-modules such that 1 = gf in HomR (Ωn E, Ωn E). Therefore Ωn E is stably isomorphic to a direct
summand of Ωn X, and there is an isomorphism Ωn X ∼= Ωn E ⊕ C in Tn . Sending this isomorphism by−n
the nth cosyzygy functor Ω , we obtain an isomorphismX∼= Ω−n Ωn E ⊕ Ω−n C
in Sn by Lemma 4.1. In view of the assumption that E has projective dimension more than n, we
observe that X is a nonzero indecomposable object and Ω−n Ωn E is a nonzero object in mod R. As
mod R is a Krull-Schmidt category, X is stably isomorphic to Ω−n Ωn E, which yields isomorphismsΩn X ∼■= Ωn (Ω−n Ωn E) ∼= Ωn E in mod R by Lemma 4.1.
Lemma 4.4. Let n ≥ 0 be an integer. Let X be a quasi-resolving subcategory of mod R closed under
direct summands such that Ext3n+1
mod X (mod X , proj(mod X )) = 0. Let X be an R-module in X ∩ Sn with
pdR X = n + 1. Then the R-module TrΩn X is stably isomorphic to a k-vector space.
Proof. Let X = X1 ⊕ · · · ⊕ Xa ⊕ Xa+1 ⊕ · · · ⊕ Xb be a direct sum decomposition of X into indecomposable
modules with pdR Xi = n + 1 for 1 ≤ i ≤ a and pdR Xi ≤ n for a + 1 ≤ i ≤ b. Since X is closed under
direct summands, Xi is in X ∩ Sn for all i, and TrΩn X ∼= TrΩn X1 ⊕ · · · ⊕ TrΩn Xa in mod R. Thus we
may assume that X is an indecomposable R-module.
π
Let ε : 0 → ΩX → F −→ X → 0 be an exact sequence with F free. Fix an element a ∈ m. There is apullback diagramπ
ε : 0 −−−−→ ΩX −−−−→ F −−−−→ X −−−−→ 0
x
x

a


aε : 0 −−−−→ ΩX −−−−→ Y −−−−→ X −−−−→ 0,
and it is seen that Y is in X . Using the exact sequence(4.4.1)(π,a)

0 → Y → F ⊕ X −−−→ X → 0

14

HIROKI MATSUI AND RYO TAKAHASHI

and the fact that a is an element in m, we easily observe that Y has projective dimension n + 1. Hence
we can apply Lemma 4.3 to the short exact sequence aε to get an isomorphismΩn Y ∼= Ωn X in mod R.

(4.4.2)

Applying Ext∗R (−, R) to (4.4.1) induces an exact sequence
a

β

n+2
Extn+1→ Extn+1→ Extn+1
R (X, R) −R (X, R) −R (Y, R) → ExtR (X, R).

Since X has projective dimension n + 1, the module Extn+2
R (X, R) vanishes, which implies that the map
β is surjective. Using (4.4.2), we have isomorphisms
1
1
n+1
n
n
∼
∼
∼
Extn+1
R (X, R) = ExtR (Ω X, R) = ExtR (Ω Y, R) = ExtR (Y, R).

According to [41, Theorem 2.4], the map β is an isomorphism, and hence aExtn+1
R (X, R) = 0. It follows
n+1
n+1
1
n
that mExtR (X, R) = 0, which means that ExtR (Ω X, R) = ExtR (X, R) is a k-vector space. As Ωn X
has projective dimension 1, there is an exact sequence
0 → P1 → P0 → Ωn X → 0
with P1 , P0 free. It is seen from this that Ext1R (Ωn X, R) is stably isomorphic to TrΩn X. Consequently,
the R-module TrΩn X is stably isomorphic to a k-vector space.
■
Lemma 4.5. Let n be a nonnegative integer. Let X be a resolving subcategory of mod R satisfying
Ext3n+1
mod X (mod X , proj(mod X )) = 0. Then X ∩ Sn contains no module of projective dimension n + 1.
Proof. Suppose that the subcategory X ∩ Sn contains a module X of projective dimension n + 1. Thenit follows from Lemma 4.4 that TrΩn X is stably isomorphic to k ⊕t for some t ≥ 0. As the projective
dimension of X exceeds n, the integer t has to be positive. There are isomorphisms
X∼= TrΩn TrΩn X ∼= TrΩn k ⊕t
in mod R, where the ﬁrst isomorphism follows from Lemma 4.1. Since t is positive, X is in X ∩ Sn and
X is closed under direct summands, the module TrΩn k is in X ∩ Sn .
Let e be the embedding dimension of R; note e ≥ 1. Write m = (x1 , x2 , . . . , xe ), and take the subideal
I = (x21 , x2 , . . . , xe ). Then m/I is isomorphic to k, which means that there is an exact sequence
0 → k → R/I → k → 0.
As depth R ≥ pdR X = n + 1, we have ExtnR (k, R) = 0. By [29, Lemma 2.3], applying TrΩn induces anexact sequence
0 → TrΩn k → TrΩn (R/I) → TrΩn k → 0.
Since the subcategory X ∩ Sn of mod R is closed under extensions, TrΩn (R/I) belongs to X ∩ Sn . It
follows from [29, Lemma 2.5] that the R-module TrΩn (R/I) has projective dimension n + 1. Thus one
can apply Lemma 4.4 to this module to see that TrΩn (TrΩn (R/I)) is stably isomorphic to k ⊕r for somer ≥ 0. As R/I has ﬁnite length and depth R ≥ n + 1, we see that R/I is in Sn . Lemma 4.1 implies
that TrΩn (TrΩn (R/I)) is stably isomorphic to R/I. Hence R/I is stably isomorphic to k ⊕r . Using the
Krull-Schmidt theorem, we observe that R/I is isomorphic to k as an R-module. Taking the annihilators
yields I = m, which is a contradiction. We conclude that X ∩ Sn does not contain a module of projective
dimension n + 1.
■
Lemma 4.6. Let n be a nonnegative integer. Let X be a resolving subcategory of mod R such that
Ext3n+1
mod X (mod X , proj(mod X )) = 0. If R/xR belongs to X for all R-sequences x = x1 , . . . , xn andn+1Ω(X ∩ Sn ) ⊆ Tn+1 , then one has Extn+1
R (X ∩ Sn , R) = 0.
Proof. Let X be a module in X ∩ Sn . We want to show that Extn+1
R (X, R) = 0. Since X ∩ Sn isclosed under direct summands, we may assume that X is indecomposable. According to Lemma 4.5,
the R-module X does not have projective dimension n + 1. If pdR X ≤ n, then it obviously holds that
n+1
Extn+1
X belongs to Tn+1 by
R (X, R) = 0. So we may assume that pdR X ≥ n + 2. The module Ω
n+1
assumption, and we have grade ExtR (X, R) ≥ n by [7, Proposition (2.26) and Corollary (4.18)].

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

15

n+1
Suppose that Extn+1
R (X, R) ̸= 0. Then the annihilator AnnR ExtR (X, R) is a proper ideal of R, andone can take an R-sequence x = x1 , . . . , xn in it. Letting K = K(x, R) be the Koszul complex of R withrespect to x, one gets a free resolution
δ

0 → Kn −→ Kn−1 → · · · → K1 → K0 → R/xR → 0
of the R-module R/xR. For an R-module M we denote by F i (M ) the composition
i+2
2
ExtiR (M, R/xR) → Exti+1
R (M, Ω(R/xR)) → ExtR (M, Ω (R/xR))
i+n−1
i+n
→ · · · → ExtR
(M, Ωn−1 (R/xR)) → Exti+n
R (M, Kn ) = ExtR (M, R)of connecting homomorphisms.
We claim that F 1 (X) : Ext1R (X, R/xR) → Extn+1
R (X, R) is an isomorphism. In fact, since X is in Sn ,the composition f : Ext1R (X, R/xR) → Ext2R (X, Ω(R/xR)) → · · · → ExtnR (X, Ωn−1 (R/xR)) is bijectiven+1
and the map g : ExtnR (X, Ωn−1 (R/xR)) → Extn+1
R (X, R) is injective. Since x annihilates ExtR (X, R),
n+1
n+1
n+1
the map ExtR (X, δ) : ExtR (X, Kn ) → ExtR (X, Kn−1 ) is a zero map, and we see that the injective
map g is surjective. Therefore F 1 (X) = gf is an isomorphism.
1
As we assume that Extn+1
R (X, R) does not vanish, neither does ExtR (X, R/xR), and there exists a
nonsplit short exact sequenceθσ : 0 → R/xR −→E→X→0of R-modules. By assumption R/xR is in X , and X is closed under extensions. Hence E is also in X .
One has pdR (R/xR) = n, and pdR E ≥ n + 2 since pdR X ≥ n + 2. Therefore we can apply Lemma 4.3
to see that Ωn E is stably isomorphic to Ωn X. There is a commutative diagramα

HomR (R/xR, R/xR) −−−−→ Ext1R (X, R/xR) −−−−→ Ext1R (E, R/xR) −−−−→ Ext1R (R/xR, R/xR)








F 0 (R/xR)y
F 1 (X)y∼F 1 (E)yF 1 (R/xR)y=
ExtnR (R/xR, R)

β

−−−−→

Extn+1
R (X, R)γ−−−−→

Extn+1R (E, R)−−−−→

Extn+1R (R/xR, R)

with exact rows. The fact that R/xR has projective dimension n implies Extn+1
R (R/xR, R) = 0, whichshows that the map γ is surjective. As Ωn X is stably isomorphic to Ωn E, the module Extn+1
R (X, R) is
n+1
isomorphic to ExtR (E, R). It is seen from [41, Theorem 2.4] that γ is an isomorphism, and hence β = 0.
Since F 1 (X) is an isomorphism, diagram chasing implies that α = 0. Hence the map HomR (θ, R/xR) :HomR (E, R/xR) → HomR (R/xR, R/xR) is surjective, which means that the exact sequence σ splits.
This contradiction shows that Extn+1■
R (X, R) = 0.
Now we have reached (the last ﬁfth step of) the proof of the theorem.
Proof of Theorem 4.2. Since X is closed under syzygies, it suﬃces to deduce that Extn+1
R (X, R) = 0 foreach R-module X in X . The assumption (3) implies that Ωn X is n-torsionfree; see [7, Corollary (4.18)].
It follows from [7, Proposition (2.21)] that there is an exact sequence
0→Y →Z→X→0nnof R-modules with Z = TrΩ TrΩ X such that Y has projective dimension less than n. The assumption(1) shows that Y is in X , and hence so is Z. As Ωn X is in Tn , Lemma 4.1 implies that Z is in Sn . Thus
we have Z ∈ X ∩ Sn . Thanks to the assumptions (2) and (3), one can apply Lemma 4.6 to see thatn+1
nExtn+1
R (Z, R) = 0. Since ExtR (Y, R) = 0, the above short exact sequence shows that ExtR (X, R) = 0,
which is what we want.
■
From now on to the end of this section, we give several applications of Theorem 4.2. The ﬁrst one is
the following corollary, where the assumption on R is satisﬁed for instance when R is a Cohen-Macaulay
local ring with an isolated singularity. (Recall that a local ring R is said to have an isolated singularity
if for each nonmaximal prime ideal p of R the local ring Rp is regular.)Corollary 4.7. Let R be a d-dimensional Cohen-Macaulay local ring that is locally Gorenstein on the
punctured spectrum. Let X be a resolving subcategory of mod R containing the modules of ﬁnite projective>3d
dimension. Then Ext>dR (X , R) = 0 if and only if Extmod X (mod X , proj(mod X )) = 0.

16

HIROKI MATSUI AND RYO TAKAHASHI

Proof. The ‘only if’ part follows from Proposition 3.10. As to the ‘if’ part, in view of Theorem 4.2, it isenough to prove that Ωd+1 X is contained in Td+1 . We haveΩd+1 X ⊆ Ωd+1 (mod R) = ΩCM(R) ⊆ Td+1 .
Indeed, the ﬁrst inclusion is obvious. It follows from [31, Theorem 3.8] that CM(R) = Td = Ωd (mod R),
which implies the equality. As for the second inclusion, let M be an MCM R-module. Then there is anexact sequence
0→F →N →M →0of R-modules such that Ext1R (N, R) vanishes and F is free; see [7, Proposition (2.21)]. This short exact
sequence shows that N is an MCM R-module, and we have N ∈ S1 ∩ Td . Lemma 4.1 implies thatΩM = ΩN is in Td+1 , and thus the inclusion considered follows.
■
The next application is a characterization of Gorenstein rings. Note that the second condition in the
result below corresponds to (B) for n = d.
Corollary 4.8. Let R be a d-dimensional Cohen-Macaulay local ring which is locally Gorenstein on the
punctured spectrum. The following are equivalent.
(1) R is Gorenstein.
(2) mod R is Gorenstein of dimension at most 3d.
(3) Ext3d+1
mod(mod R) (mod(mod R), proj(mod(mod R))) = 0.
Proof. (1) ⇒ (2): Since R is Gorenstein, one has Ωd (mod R) = CM(R) = Gproj(mod R), and this is closed
under cosyzygies. Theorem 3.11 shows the implication.
(2) ⇒ (3): The implication is straightforward.
(3) ⇒ (1): By virtue of Corollary 4.7, one obtains Ext>dR (mod R, R) = 0, which especially says that
Ext>d■
R (k, R) = 0. Therefore R is a Gorenstein ring.
Let n = 0 in Theorem 4.2. Then the three conditions (1)–(3) in the theorem are trivially satisﬁed;see [31, Lemma 3.4]. Hence the following result holds, whose two assertions are nothing but [56, Theorem
4.2] and [56, Corollary 4.3], respectively. (The second assertion is shown along the same lines as in the
proof of Corollary 4.8.)
Corollary 4.9. (1) Let X be a resolving subcategory of mod R. If mod X is a quasi-Frobenius category,then Ext>0
R (X , R) = 0.
(2) The following statements are equivalent.
(a) R is an artinian Gorenstein ring.
(b) mod(mod R) is a Frobenius category.
(c) mod(mod R) is a quasi-Frobenius category.
5. A sufficient condition for singular equivalence
In this section we study singular equivalences among stable quasi-resolving subcategories. First of all,let us make the precise deﬁnition of singular equivalence.
Definition 5.1. Let C and C ′ be additive categories such that mod C and mod C ′ are abelian. We saythat C and C ′ are singularly equivalent if there exists a triangle equivalence Dsg (C) ∼= Dsg (C ′ ).
Our main interest is to ask when this is the case for the stable categories of quasi-resolving subcategories. The main result of this section gives an answer to this question. We begin with a proposition.
Proposition 5.2. Let X and Y be quasi-resolving subcategories of A. Let n be a nonnegative integer.
The following are equivalent.
(1) One has Ωn X ⊆ Y ⊆ X ∩ Gproj A and Y is closed under cosyzygies.
(2) One has Ωn X ⊆ Y ⊆ X ∩ Gproj A and Ωn X is closed under cosyzygies.
(3) One has Ωn X = Y = X ∩ Gproj A and this is closed under cosyzygies.
When this is the case, X ∩ Gproj A is closed under extensions.

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

17

Proof. It is trivial that (3) implies (1), and we see from Proposition 3.9 that (1) implies (2). Pick an
object M ∈ X ∩ Gproj A. Then M is Gorenstein projective, so we have M ∼= Ω−n (Ωn M ). As M is in X ,the syzygy Ωn M is in Ωn X . If Ωn X is closed under cosyzygies, then Ω−n (Ωn M ) belongs to Ωn X and sodoes M . This shows that (2) implies (3), and consequently, the conditions (1)–(3) are equivalent.
fSuppose that one of the three equivalent conditions is satisﬁed. Let 0 → L −→ M → N → 0 be angexact sequence in A with L, N ∈ X ∩ Gproj A. There is an exact sequence 0 → L −→ P → Ω−1 L → 0
1with P ∈ proj A. As ExtA (N, P ) = 0, the pushout diagram of f and g gives rise to an exact sequence
0 → M → P ⊕ N → Ω−1 L → 0.
The object P ⊕ N is in X , and so is Ω−1 L since X ∩ Gproj A is closed under cosyzygies. By deﬁnition X is
closed under kernels of epimorphisms, and hence M belongs to X . As Gproj A is closed under extensions,■
M is in X ∩ Gproj A. Thus X ∩ Gproj A is closed under extensions.
As an immediate consequence of Proposition 5.2, we have:
Corollary 5.3. Let X be a quasi-resolving subcategory of A. If X satisﬁes (Gn ). then Ωn X = X ∩Gproj A.
Now we state the main result of this section.
Theorem 5.4. Let X be a quasi-resolving subcategory of A satisfying the condition (Gn ) for some n ≥ 0.
(1) The equality Ωn X = X ∩ Gproj A holds. Denote this subcategory by Y.
(2) One has the following.
(a) Y is a quasi-resolving subcategory of A satisfying (G0 ).
(b) Y is a Frobenius subcategory with proj Y = proj A.
(c) Y is a triangulated category.
(d) Y is a strictly full subcategory of Gproj A.
(3) There are triangle equivalences
Dsg (X ) ∼= Gproj(mod X ) = Ω3n (mod X ) ∼= mod Y = Gproj(mod Y) ∼= Dsg (Y),where the middle equivalence is induced by the restriction functor F 7→ F |Y . In particular, X and Y
are singularly equivalent.
Proof. (1) The assertion follows from Propositions 5.2 and 3.9.
(2) Proposition 5.2 implies that Y is closed under cosyzygies and extensions. Since X and Gproj A are
quasi-resolving, so is Y. Thus (a) and (b) hold, while (c) is a consequence of (b). The induced functor
Y → Gproj A is fully faithful, and via this functor we can regard Y as a full subcategory of Gproj A. In
view of Remark 2.10(1), we easily observe that Y is closed under isomorphism as a full subcategory ofGproj A. This shows (d).
(3) Note that Y satisﬁes (G0 ). By virtue of Theorem 3.11, we have equalities Gproj(mod X ) =Ω3n mod X and Gproj(mod Y) = mod Y, and triangle equivalences Dsg (X ) ∼= Gproj(mod X ) and Dsg (Y) ∼=Gproj(mod Y). The restriction F 7→ F |Y makes a covariant exact functor
Φ : Mod X → Mod Y
of abelian categories. We establish several claims.
Claim 1. The functor Φ sends HomA (−, X)|X with X ∈ X to HomA (−, Y )|Y with Y = Ω−n Ωn X ∈ Y.
Hence one has Φ(proj(mod X )) ⊆ proj(mod Y).
Proof of Claim. Using Proposition 2.7(2) and the fact that Ext>0
A (Y, proj A) = 0, we have isomorphisms
1
2
2
∼
∼
Hom (−, X)|Y = Ext (−, ΩX)|Y = Ext (−, Ω X)|Y ∼= ··· ∼
= Extn (−, Ωn X)|Y ,
A

A

A

A

HomA (−, Ω−n Ωn X)|Y ∼= Ext1A (−, Ω1−n Ωn X)|Y ∼= Ext2A (−, Ω2−n Ωn X)|Y ∼= ··· ∼
= ExtnA (−, Ωn X)|Y .
Therefore there is an isomorphism HomA (−, X)|Y ∼= HomA (−, Ω−n Ωn X)|Y . Since Ωn X ∈ Y and Ω−1 Y ⊆−n n□Y, we have Ω Ω X ∈ Y.
It is easy to see from Claim 1 that Φ induces an exact functor mod X → mod Y and a triangle functor
ϕ : Ω3n mod X −→ mod Y.

18

HIROKI MATSUI AND RYO TAKAHASHI

Claim 2. The functor ϕ is dense.
p

Proof of Claim. Let G be an object in mod Y. Take a projective presentation HomA (−, Y1 )|Y −→HomA (−, Y0 )|Y → G → 0 with Y0 , Y1 ∈ Y. One can write p = HomA (−, f )|Y for some morphism
f : Y1 → Y0 (see Remark 2.4(1)). Let F be the cokernel of the morphism
HomA (−, Ω−n f )|X : HomA (−, Ω−n Y1 )|X → HomA (−, Ω−n Y0 )|X .
As Ω−n Y0 and Ω−n Y1 are in Y and hence in X , it is seen that HomA (−, Ω−n f )|X is an X -homomorphism(f,g)

of projective X -modules, and F belongs to mod X . There is an exact sequence 0 → Y2 → Y1 ⊕ P −−−→
Y0 → 0 with P ∈ proj A and Y2 ∈ Y (see Remark 2.10), which induces an exact sequence 0 → Ω−n Y2 →Ω−n Y1 → Ω−n Y0 → 0. Proposition 2.7(2) shows that there is an exact sequence
Hom (−,f )|X

HomA (−, Y1 )|X −−−−A−−−−−→ HomA (−, Y0 )|X → HomA (−, Ω−1 Y2 )|X → · · ·Hom (−,Ω−n f )|X→ HomA (−, Ω−n Y1 )|X −−−−A−−−−−−−−→ HomA (−, Ω−n Y0 )|X → F → 0.
Letting H be the cokernel of HomA (−, f )|X , we observe that H is the (3n)th syzygy of F . Thus H is inΩ3n mod X , and it is obvious that ϕ(H) = G. Consequently, ϕ is a dense functor.
□
Claim 3. The functor ϕ is full.
Proof of Claim. Take objects F, G ∈ Ω3n mod X and a morphism ρ ∈ Hommod Y (F |Y , G|Y ). There exists
an object F ′ ∈ mod X whose (3n)th syzygy is F . Using Proposition 2.11(2), we obtain an exact sequence
0 → X2 → X1 → X0 → 0 of objects in X which induces a projective resolution
Hom (−,f )|X

HomA (−, Ωn X1 )|X −−−−A−−−−−→ HomA (−, Ωn X0 )|X → HomA (−, Ωn−1 X2 )|X→ · · · → HomA (−, X2 )|X → HomA (−, X1 )|X → HomA (−, X0 )|X → F ′ → 0of F ′ in mod X , where f : Ωn X1 → Ωn X0 is a morphism in A. By Schanuel’s lemma, we may assume
that F is the cokernel of HomA (−, f )|X . Applying the same argument to G, we get two exact sequences
Hom (−,f )|X

HomA (−, Ωn X1 )|X −−−−A−−−−−→ HomA (−, Ωn X0 )|X → F → 0,
Hom (−,g)|X

HomA (−, Ωn X1′ )|X −−−−A−−−−−→ HomA (−, Ωn X0′ )|X → G → 0
with X0′ , X1′ ∈ X and g ∈ HomA (Ωn X1′ , Ωn X0′ ). As Ωn Xi and Ωn Xi′ are in Y for i = 0, 1, the objects
HomA (−, Ωn Xi )|Y and HomA (−, Ωn Xi′ )|Y are projective in mod Y. Hence there is a commutative diagram
HomA (−,f )|Y

HomA (−, Ωn X1 )|Y −−−−−−−−−→ HomA (−, Ωn X0 )|Y −−−−→ F |Y −−−−→ 0






ρy
HomA (−,h1 )|Y y
HomA (−,h0 )|Y y
HomA (−,g)|Y

HomA (−, Ωn X1′ )|Y −−−−−−−−−→ HomA (−, Ωn X0′ )|Y −−−−→ G|Y −−−−→ 0,
where hi : Ωn Xi → Ωn Xi′ is a morphism in A for i = 0, 1. Substituting Ωn X1 for “−”, we observe thath0 f = gh1 . Hence the square in the diagram below commutes.
Hom (−,f )|X

HomA (−, Ωn X1 )|X −−−−A−−−−−→ HomA (−, Ωn X0 )|X −−−−→ F −−−−→ 0
HomA (−,h1 )|X y
HomA (−,h0 )|X y
Hom (−,g)|X

HomA (−, Ωn X1′ )|X −−−−A−−−−−→ HomA (−, Ωn X0′ )|X −−−−→ G −−−−→ 0.
Let ξ : F → G be the induced morphism by this diagram. Then ξ is in Hommod X (F, G) and we haveϕ(ξ) = ξ|Y = ρ. This shows that ϕ is a full functor.
□
Claim 4. The functor ϕ is faithful.

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

19

Proof of Claim. Let ξ : F → G be a morphism in mod X with F, G ∈ Ω3n mod X . Suppose that ξ is sent
to 0 by the functor ϕ. Then ξ|Y = 0, which says that the morphism ξ|Y : F |Y → G|Y factors through someP ∈ proj(mod Y). As in the proof of Claim 3, we may assume that F, G have projective presentations
Hom (−,f )|X

HomA (−, Y1 )|X −−−−A−−−−−→ HomA (−, Y0 )|X → F → 0,
Hom (−,g)|X

HomA (−, Y1′ )|X −−−−A−−−−−→ HomA (−, Y0′ )|X → G → 0
with Yi , Yi′ ∈ Y and f ∈ HomA (Y1 , Y0 ), g ∈ HomA (Y1′ , Y0′ ), and there is a commutative diagram
Hom (−,f )|X

HomA (−, Y1 )|X −−−−A−−−−−→ HomA (−, Y0 )|X −−−−→ F −−−−→ 0






HomA (−,h1 )|X y
HomA (−,h0 )|X yξy
Hom (−,g)|X

HomA (−, Y1′ )|X −−−−A−−−−−→ HomA (−, Y0′ )|X −−−−→ G −−−−→ 0,
where hi ∈ HomA (Yi , Yi′ ) for i = 0, 1. We also have an equality h0 f = gh1 . Sending the above diagramby the functor ϕ, we obtain the following diagram in mod Y:
HomA (−,f )|Y/ F |Y
s
s
s
α ss
s
n n
ss
n
s
HomA (−,ℓ)|Y n nsy s
HomA (−,h1 )|Y
HomA (−,h0 )|Y
ξ|Y
n
P KK
m
n
KK β
n
ε m m
K
n
KK
m
KK
n n
m m
KK 
m

vn n

v
%
HomA (−,g)|Y
δ
′
′
/
/ G|Y
HomA (−, Y0 )|Y
HomA (−, Y1 )|Y
HomA (−, Y1 )|Y/ HomA (−, Y0 )|Y
n n

/0

γ

/0

Since P is a projective object of mod Y, the morphism β factors through δ; we ﬁnd a morphism ε :P → HomA (−, Y0′ )|Y with β = δε. Hence there exists a morphism HomA (−, ℓ)|Y : HomA (−, Y0 )|Y →HomA (−, Y1′ )|Y with ℓ ∈ HomA (Y0 , Y1′ ) such that
HomA (−, h0 )|Y = HomA (−, g)|Y · HomA (−, ℓ)|Y + εαγ.
Composition with HomA (−, f )|Y shows that HomA (−, h0 f )|Y = HomA (−, gℓf )|Y , which implies h0 f =gℓf . The left triangle in the diagram
Hom (−,f )|X
A/ HomA (−, Y0 )|X/
HomA (−, Y1 )|X
oF
RRR
o
RRR
o
RRR
oo
RRR
o
RRR
o
0
ξ
RRRHomA (−,h0 −gℓ)|X
o oζ
RRR
o
RRR
o
o
RRR
R(

wo o

HomA (−,g)|X′′/ Hom (−, Y0 )|X/G
Hom (−, Y1 )|X

A

A

/0

/0

commutes, which implies that there exists a morphism ζ : F → HomA (−, Y0′ )|X such that the right upper
triangle commutes, and so does the right lower one. It follows that the morphism ξ factors through the
projective object HomA (−, Y0′ )|X of mod X , and we have ξ = 0. The functor ϕ is thus faithful.
□
Combining Claims 2, 3 and 4 implies that the triangle functor ϕ is an equivalence, and the proof of
the theorem is completed.
■
Remark 5.5. The assumption of Theorem 5.4 does not necessarily imply that there is an equivalence
Db (mod X ) ∼= Db (mod Y).
Indeed, let R be a regular local ring of positive Krull dimension d. Set n = d and A = X = mod R. Then
Gproj A = CM(R) consists of free R-modules, and the assumption of Theorem 5.4 is satisﬁed. There areembeddings X ,→ mod X ,→ Db (mod X ), and since X ̸= 0, we have Db (mod X ) ̸= 0. On the other hand,since Y = 0, we have Db (mod Y) = 0. Hence Db (mod X ) ≇ Db (mod Y).

20

HIROKI MATSUI AND RYO TAKAHASHI

The remainder of this section is devoted to stating immediate applications of our Theorem 5.4. First
we apply the theorem to a quasi-resolving subcategory containing the Gorenstein projective objects.
Corollary 5.6. (1) Let X be a quasi-resolving subcategory of A with Ωn X ⊆ Gproj A ⊆ X for somen ≥ 0. Then X is singularly equivalent to Gproj A.
(2) One has Gpdn (A) is singularly equivalent to Gproj A for each n ≥ 0. In particular, Gpdi (A) andGpdj (A) are singularly equivalent for all i, j ≥ 0.
Proof. (1) Since Gproj A is closed under cosyzygies, Proposition 3.9 guarantees that X satisﬁes (Gn ).
Theorem 5.4(3) implies the assertion.
(2) The assertion follows by applying (1) to X = Gpdn (A), or by directly combining Corollary 3.14(2)with Theorem 5.4(3).
■Applying the above result to the module category of a Gorenstein ring, one observes that all resolving
subcategories containing the MCM modules are singularly equivalent:
Corollary 5.7. Let R be a Gorenstein local ring. Let X be a quasi-resolving subcategory of mod R
containing CM(R). Then X and CM(R) are singularly equivalent.
Proof. Putting d = dim R, we have Ωd X ⊆ CM(R) ⊆ X . Since CM(R) = Gproj(mod R), the assertion
follows from Corollary 5.6(1).
■Here we give an example of a singular equivalence which is analogous to Corollary 5.7.
Example 5.8. Let R be a d-dimensional Gorenstein local ring. Let mod0 (R) stand for the category of
R-modules that are locally free on the punctured spectrum of R, and set CM0 (R) = CM(R) ∩ mod0 (R).
Let X be a quasi-resolving subcategory of mod R contained in mod0 (R) and containing CM0 (R). Then
one has Ωd X ⊆ CM0 (R) ⊆ X and CM0 (R) is closed under cosyzygies. Proposition 3.9 implies that X
satisﬁes (Gn ), and therefore X is singularly equivalent to CM0 (R) by Theorem 5.4.
Remark 5.9. In general, the existence of singular equivalences between X and Y for quasi-resolving
subcategories X , Y does not imply that there is an inclusion relation between X and Y, as is shown by
the example below.
Let R be a Gorenstein local integral domain of Krull dimension at least 2 with unique maximal ideal/ AssR M
m. Let X (respectively, Y) be the subcategory of mod R consisting of modules M satisfying m ∈(respectively, AssR M ⊆ {0, m}), where AssR M stands for the set of associated prime ideals of R. Then
it is easy to observe that X and Y are resolving subcategories of mod R containing CM(R). In fact, inthe bijection constructed in [47, Corollary 8.9] the resolving subcategory X (respectively, Y) corresponds
to the grade-consistent function f (respectively, g) deﬁned as follows.
{{
ht p
if p ∈ Spec R \ {m},ht p
if p ∈ {0, m},
f (p) =
g(p) =
ht p − 1 if p = m,
ht p − 1 if p ∈ Spec R \ {0, m}.
It follows from Corollary 5.7 that X and Y are singularly equivalent; they are also singularly equivalent
to CM(R). On the other hand, the module R/m is in Y but not in X , and the module R/p is in X but
not in Y for any prime ideal p of R diﬀerent from 0 and m (such a prime ideal exists since R has Krull
dimension at least 2). Therefore, X and Y have no inclusion relation.
From now on, we apply our theorem to complete intersection local rings.
Corollary 5.10. Let R be a local ring. Let X be a subcategory of mod R.
(1) Assume that X is resolving and that all modules in X have CI-dimension at most n. Then Ωn X =X ∩ Gproj(mod R) holds, and Ωn X is resolving. One has that X is singularly equivalent to Ωn X .
(2) If R is a complete intersection, then res Ωi X and res Ωj X are singularly equivalent for all i, j ≥ 0.
Proof. (1) The category Gproj(mod R) consists of totally reﬂexive R-modules, which is a resolving subcategory of mod R. Hence X ∩ Gproj(mod R) is also a resolving subcategory of mod R. Corollary 3.15(1)and Theorem 5.4 yield Ωn X = X ∩ Gproj(mod R), and its stable category is singularly equivalent to X .

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

21(2) It suﬃces to prove that res Ωi X is singularly equivalent to res Ωd X for all i ≥ 0, where d := dim R.
By [19, Theorems (1.3) and (1.4)] and (1) we observe that Ωd (res Ωi X ) is resolving, and res Ωi X is
singularly equivalent to Ωd (res Ωi X ). It is enough to verify thatΩd (res Ωi X ) = res Ωd X .

(5.10.1)
For this, we establish a claim.

Claim. For any subcategory C of mod R and any integer n ≥ 0 one has the inclusionΩn (res C) ⊆ res Ωn C.
Proof of Claim. Let M be the subcategory of mod R consisting of modules whose nth syzygies are inres Ωn C. Then it is easily seen that M is a resolving subcategory containing C. Hence M contains res C,
which deduces the claim.
□
Let us prove the equality (5.10.1). Since Ωi X is contained in res X , so is res Ωi X . Hence Ωd (res Ωi X )is contained in Ωd (res X ), which is contained in res Ωd X by the claim. On the other hand, since Ωd X iscontained in CM(R), it is observed by [28, Corollary 4.16] that Ω−i Ωd X is contained in res Ωd X . HenceΩd X = Ωi (Ω−i Ωd X ) is contained in Ωi (res Ωd X ), which is contained in res Ωd+i X by the claim. Thus
res Ωd X is contained in res Ωd+i X = res Ωd (Ωi X ), which is contained in res Ωd (res Ωi X ) = Ωd (res Ωi X ).
■This implies that Ωd (res Ωi X ) contains res Ωd X .
Remark 5.11. It is not known that ﬁniteness of CI-dimension is preserved by taking extensions; see[43, Remark 5.1] for example. If this turns out to be true, then the second assertion of Corollary
5.10 will extend to arbitrary local rings R and subcategories X all of whose objects have ﬁnite CIdimension, because all modules in res Ωi X will have ﬁnite CI-dimension. (The same proof will work bysome appropriate replacement; d = dim R should be replaced with depth R, and so on.)The following result immediately follows from Corollary 5.10(1). This is regarded as a stronger version
of Corollary 5.7 for resolving subcategories. Thanks to this result, to classify singular equivalence classes
over a complete intersection, one has only to consider resolving subcategories consisting of MCM modules.
Corollary 5.12. Let R be a local complete intersection. Let X be a resolving subcategory of mod R.
Then X is singularly equivalent to X ∩ CM(R).
Let R be a local ring with maximal ideal m. Recall that R is called a hypersurface if the m-adic
completion of R is isomorphic to the quotient ring of a regular local ring by a principal ideal. Needless tosay, any hypersurface is a complete intersection. Over a hypersurface with an isolated singularity, there
are at most two singular equivalence classes.
Corollary 5.13. Let R be a local hypersurface with an isolated singularity. Let X be a resolving subcategory of mod R. Then X is singularly equivalent to either CM(R) or 0.
Proof. It follows from Corollary 5.12 that X is singularly equivalent to X ∩ CM(R). As X ∩ CM(R)is a resolving subcategory of mod R contained in CM(R), it is equal to either proj(mod R) or CM(R)by [50, Corollary 6.9(1)]. Hence X ∩ CM(R) coincides with either 0 or CM(R).
■
6. Regularity for resolving subcategories and simple singularities
We have learned in Corollary 5.13 that over an isolated hypersurface singularity the stable category X
of a resolving subcategory X is singularly equivalent to either CM(R) or 0. So it is natural to ask when
X is singularly equivalent to 0. The main purpose of this section is to give an answer to this question.
The global dimension (respectively, ﬁnitistic dimension) of A is deﬁned to be the supremum of the
projective dimensions (respectively, ﬁnite projective dimensions) of objects of A. Let us begin withinvestigating the relationships of the condition that X is singularly equivalent to 0 with several otherconditions, including ﬁniteness of the global dimension of mod X .
Proposition 6.1. Let X be a quasi-resolving subcategory of A. Consider the following four conditions.
(1) mod X has ﬁnite global dimension.
(2) X is singularly equivalent to 0.
(3) X is regular.

22

HIROKI MATSUI AND RYO TAKAHASHI(4) Every object of X has ﬁnite projective dimension in A.
Then the implications (1) ⇒ (2) ⇔ (3) ⇐ (4) hold. The implication (1) ⇐ (2) holds if X satisﬁes thecondition (Gn ). The implication (1) ⇐ (4) holds if the ﬁnitistic dimension of A is ﬁnite.
Proof. First of all, note that the condition (2) is equivalent to the equality Db (mod X ) = Kb (proj(mod X )).
(2) ⇒ (3): It is straightforward to verify the implication.
(3) ⇒ (2): We observe that mod X is contained in Kb (proj(mod X )) as subcategories of Db (mod X ). Let
C = (0 → C a → C a+1 → · · · → C b → 0) be any object of Db (mod X ). Then C belongs to the smallest
thick subcategory T of Db (mod X ) containing M := C a ⊕C a+1 ⊕· · ·⊕C b ∈ mod X . Since Kb (proj(mod X ))is a thick subcategory of Db (mod X ) containing M , we see that Kb (proj(mod X )) contains T . Hence C
belongs to Kb (proj(mod X )). Thus Db (mod X ) coincides with Kb (proj(mod X )).
(1) ⇒ (3): The implication is trivial.
(4) ⇒ (3): Let F be an object in mod X . Using Proposition 2.11(2), we have an exact sequence
0 → A → B → C → 0 in A with A, B, C ∈ X which induces a projective resolution· · · → HomA (−, ΩC)|X → HomA (−, A)|X → HomA (−, B)|X → HomA (−, C)|X → F → 0.
Since C has ﬁnite projective dimension, the nth syzygy Ωn C is projective for some n ≥ 0. Hence theabove resolution has length less than 3n.
(2) ⇒ (1): Suppose that X satisﬁes (Gn ). Then Theorem 3.11 yields triangle equivalences0∼= Gproj(mod X ) = Ω3n (mod X )= Dsg (X ) ∼for some n ≥ 0. From this we obtain Ω3n (mod X ) = proj(mod X ), which implies that mod X has global
dimension at most 3n.
(4) ⇒ (1): Assume that A has ﬁnitistic dimension m. With the notation of the proof of the implication(4) ⇒ (3), the syzygy Ωm C is projective, whence the length of the resolution is less than 3m.
■
Thanks to Proposition 6.1, now we know that the stable category of a quasi-resolving subcategory is
singularly equivalent to 0 if and only if it is regular. Thus, in the rest of this section we consistently use
the terminology of regularity instead of singular equivalence to 0.
Applying Proposition 6.1 to categories of modules over rings, we have the following.
Corollary 6.2. (1) Let R have ﬁnite Krull dimension. Let X be a resolving subcategory of mod R whose
objects have ﬁnite projective dimension as R-modules. Then mod X has ﬁnite global dimension.
(2) Let R be a local complete intersection. Let X be a resolving subcategory of mod R. Then X is regular
if and only if mod X has ﬁnite global dimension.
Proof. (1) The ﬁnitistic dimension of mod R is at most the Krull dimension of R (see [46, Seconde partie,Théorème (3.2.6)]), so it is ﬁnite. Combining this with Proposition 6.1 shows the assertion.
(2) The assertion follows from Proposition 6.1 and Corollary 3.15(2).
■The following lemma plays an essential role in the proof of the main result of this section.
Lemma 6.3. Let R be a Gorenstein Henselian local ring. Let X be a quasi-resolving subcategory of mod R
contained in CM(R) and closed under cosyzygies. Assume that there exists a nonsplit exact sequencefg

σ:0→X−→Y −→Z→0
of R-modules with X, Y, Z ∈ X such that X, Z are indecomposable. If X is regular, then Y is free, and
X is isomorphic to ΩZ.
Proof. The short exact sequence σ induces an exact sequence
Hom (−,f )|X

Hom (−,g)|X

π

HomR (−, X)|X −−−−R−−−−−→ HomR (−, Y )|X −−−−R−−−−−→ HomR (−, Z)|X −→F →0in mod X ; see Proposition 2.7(2). The functor F belongs to mod X , and it is not isomorphic to 0. Indeed,
if F ∼= 0 in mod X , then there exists a homomorphism h : Z → Y such that gh = 1. So there arehomomorphisms α : Z → P and β : P → Z with P a free R-module such that 1 − gh = βα. One has anequality 1 = gh + βα in the local ring EndR (Z). Hence either gh or βα is an automorphism, and in eachcase it is deduced that σ splits. This contradiction shows F ≇ 0 in mod X .

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

23

Taking advantage of Theorem 3.11, we havemod X = Gproj(mod X ) ∼= Dsg (X ) ∼= 0,
which means that all the objects of mod X are projective. Hence F is a projective object of mod X , which
implies that π is a split epimorphism. We establish a claim.
Claim. The morphism π : HomR (−, Z)|X → F is an isomorphism.
Proof of Claim. The claim follows from a similar argument to the proof of [3, Lemma IV.6.5]. Let
θ : F → HomR (−, Z)|X be a splitting of π, i.e., a morphism with πθ = 1. Then θπ : HomR (−, Z)|X →HomR (−, Z)|X is represented as HomR (−, ℓ)|X for some ℓ ∈ EndR (Z); see Remark 2.4(1). We have(θπ)2 = θ(πθ)π = θπ, which shows that HomR (−, ℓ2 )|X = HomR (−, ℓ)|X . Hence ℓ2 = ℓ, namely, ℓ is an
idempotent in EndR (Z). Since Z is a nonfree R-module, we have EndR (Z) ̸= 0, and EndR (Z) is a local
ring as Z is indecomposable. Therefore ℓ is either 0 or 1. If ℓ = 0, then θπ = 0, and F = 0 as θ is amonomorphism and π is an epimorphism. So we must have ℓ = 1, which shows that π is an isomorphism.
□
The claim shows HomR (−, g)|X = 0, and hence HomR (−, f )|X : HomR (−, X)|X → HomR (−, Y )|X issurjective, and so Y is isomorphic in CM(R) to a direct summand of X. Since X is indecomposable in
CM(R), the object Y is isomorphic to either 0 or X in CM(R). Hence the R-module Y is either free or
stably isomorphic to X.
Suppose that Y is stably isomorphic to X. Then HomR (−, Y )|X is isomorphic to HomR (−, X)|X ,and it is seen from [41, Theorem 2.4] that HomR (−, f )|X is an isomorphism. Let HomR (−, λ)|X :HomR (−, Y )|X → HomR (−, X)|X be a inverse morphism of HomR (−, f )|X . Then we have λf = 1,
which implies that λf + βα = 1 in EndR (X) for some homomorphisms α : X → L and β : L → X with
L free. Since EndR (X) is a local ring, either λf or βα is an automorphism, and in either case the short
exact sequence σ splits, which is a contradiction. Consequently, the R-module Y has to be free.
■
For an object X in an additive category C we denote by addC X (or simply add X if there is noconfusion) the additive closure of X in C, that is, the full subcategory of C containing X which is closed
under ﬁnite direct sums and direct summands. We say that a Cohen-Macaulay local ring R has ﬁnite
CM-representation type if there exist only a ﬁnite number of isomorphism classes of indecomposable MCM
modules over R.
Lemma 6.4. Let C be an additive category, and let X be an object of C. Then the assignment F 7→ F (X)makes an equivalencemod(addC X) → mod EndC (X).
In particular, if R is a Cohen-Macaulay local ring of ﬁnite CM-representation type, then CM(R) =addCM(R) G for some MCM R-module G, and one has an exact functor mod CM(R) → mod EndR (G) of
abelian categories which is an equivalence.
Proof. Let Λ be any ring. Applying [6, Proposition 2.5] to the inclusion {Λ} ⊆ proj Λ shows that theassignment F 7→ F (Λ) makes an equivalence mod(proj Λ) → mod Λ. Note that the assignment M 7→
HomC (X, M ) makes an equivalence addC X ∼= proj EndC (X). Now letting Λ = EndC (X) completes theproof of the ﬁrst assertion. The second assertion follows from the ﬁrst assertion and the fact that any
equivalence of abelian categories is an exact functor; for instance, see the proof of [1, Proposition 21.4]. ■Let R be a local ring. Recall that M is said to have complexity c, denoted by cxR M = c, if c is the least
nonnegative integer n such that there exists a real number r satisfying the inequality βiR (M ) ≤ rin−1 for
all i ≫ 0. It is known that if R is a complete intersection, then the codimension of R is the maximum of
the complexities of R-modules. For details on the complexity of a module, we refer the reader to [17, §4.2].
Let R be a d-dimensional Gorenstein local ring with algebraically closed residue ﬁeld k of characteristic
zero. Then R contains a ﬁeld isomorphic to k, and it is known that R has ﬁnite CM-representation typeif and only if R is a simple (hypersurface) singularity [55, §8], namely, R is isomorphic to a hypersurface
k[[x0 , . . . , xd ]]/(f ),24

HIROKI MATSUI AND RYO TAKAHASHI

where f is one of the following.
(An )x20 + xn+1
+ x22 + · · · + x2d ,
1

(Dn )

x20 x1 + xn−1
+ x22 + · · · + x2d ,
1

(E6 )

x30 + x41 + x22 + · · · + x2d ,(E7 )x30 + x0 x31 + x22 + · · · + x2d ,(E8 )x30 + x51 + x22 + · · · + x2d .

For each T ∈ {An , Dn , E6 , E7 , E8 }, a simple hypersurface singularity of type (T) is shortly called a (T)singularity.
Now we can state and prove the main result of this section, which characterizes the regularity of stable
categories of resolving subcategories.
Theorem 6.5. Let R be a d-dimensional Gorenstein nonregular complete local ring with algebraicallyclosed residue ﬁeld k of characteristic zero. Then the following are equivalent.
(1) CM(R) is regular.
(2) R is a complete intersection, and X is regular for every resolving subcategory X of mod R.
(3) R is a complete intersection, and X is regular for some resolving subcategory X of mod R that contains
a module of maximal complexity.
(4) R is an (A1 )-singularity.
When one of these conditions holds, mod CM(R) has global dimension zero.
Proof. (4) ⇒ (2): An (A1 )-singularity is an isolated hypersurface singularity. In view of Corollary 5.13, itis enough to show that CM(R) is regular. By Knörrer’s periodicity [55, Theorem (12.10)] we can assume
that R is isomorphic to either k[[x]]/(x2 ) or k[[x, y]]/(xy).
First, let R = k[[x]]/(x2 ). Then all the nonfree indecomposable R-modules are isomorphic to k. Hence
CM(R) = mod R = addmod R (k), and we have mod CM(R) ∼= mod EndR (k) by Lemma 6.4. Note that
EndR (k) is isomorphic to the ﬁeld k, whose singularity category is 0. Therefore CM(R) is regular.
Next, let R = k[[x, y]]/(xy). The nonisomorphic nonfree indecomposable MCM R-modules are R/(x)and R/(y), whence Lemma 6.4 implies mod CM(R) ∼= mod EndR (R/(x) ⊕ R/(y)). It is seen by using [55,Lemma (3.9)] that
) (()
HomR (R/(y), R/(x)) ∼ k 0 ∼EndR (R/(x))∼
EndR (R/(x) ⊕ R/(y)) === k × k,EndR (R/(y))
HomR (R/(x), R/(y))
0 kand we have Dsg (k × k) = 0. Thus CM(R) is regular.
(2) ⇒ (1): The implication follows by letting X = CM(R).
(1) ⇒ (3): Take an MCM approximation
0→Y →X→k→0
of k, i.e., a short exact sequence of R-modules such that X is MCM and Y has ﬁnite projective dimension;see [8, Theorem 1.8]. Let M be a nonfree indecomposable MCM R-module. There is an exact sequence
Ext1R (M, X) → Ext1R (M, k) → Ext2R (M, Y ),1
and Ext>0
R (M, Y ) = 0 since M is MCM and Y has ﬁnite projective dimension. As ExtR (M, k) does not1
1′
vanish, neither does ExtR (M, X). Hence ExtR (M, X ) ̸= 0 for some indecomposable direct summand X ′of X, and we ﬁnd a nonsplit exact sequence 0 → X ′ → E → M → 0. As M and X ′ are MCM, so is E.
Applying Lemma 6.3 to X = CM(R), the module X ′ is isomorphic to ΩM . Hence M is isomorphic to a
direct summand of Ω−1 X, which shows that

CM(R) = add(R ⊕ Ω−1 X).
Therefore R has ﬁnite CM-representation type, and is a simple hypersurface singularity by [55, Corollary(8.16)]. In particular, R is a complete intersection. By assumption CM(R) is regular, and CM(R) containsΩd k which has maximal complexity (see also [17, Remarks 8.1.1(2)])(3) ⇒ (4): Let M be an R-module in X that has maximal complexity. Then Ωd M is in X ∩CM(R) andhas the same complexity as M . By virtue of Corollary 5.12, replacing X with X ∩ CM(R), we may assume
that X is contained in CM(R). It follows from [28, Corollary 4.16] that X is closed under cosyzygies.

SINGULARITY CATEGORIES FOR RESOLVING SUBCATEGORIES

25

There exists an indecomposable direct summand N of M having the same complexity as M . Replacing
M with N , we can assume that M is indecomposable.
Let X be a nonfree indecomposable module in X , and set c = codim R. Suppose that ExtiR (M, X) = 0
R
for all 1 ≤ i ≤ c + 1. Then we have Ext>0
R (M, X) = 0 by [18, Theorem 4.7], and Tor>d (M, X) = 0
by [18, Theorem III]. The fact that M has maximal complexity forces X to have ﬁnite projective
dimension by [25, Proposition 2.7]. This contadicts the fact that X is a nonfree MCM module. Hence
ExtℓR (M, X) ̸= 0 for some 1 ≤ ℓ ≤ c + 1, which implies that there exists a nonsplit exact sequence
0 → X → E → Ωℓ−1 M → 0.
Note that Ωℓ−1 M is a nonfree indecomposable module in X . Since X is in X , so is E. Lemma 6.3 showsthat X is isomorphic to Ω(Ωℓ−1 M ) = Ωℓ M . It follows that the equality
X = add(R ⊕ ΩM ⊕ Ω2 M ⊕ · · · ⊕ Ωc+1 M )holds, which especially says that X is contravariantly ﬁnite in mod R. By virtue of [51, Theorem 1.2], wehave X = CM(R). Hence R has ﬁnite CM-representation type, and is a simple singularity. In particular,
R is an isolated singularity; see [34, Corollary 2].
Let C be a nonfree indecomposable MCM R-module. Take an Auslander-Reiten sequence
0 → τC → L → C → 0
ending in C; see [55, Theorem (3.2)]. This is a nonsplit exact sequence of MCM modules with C and τ Cindecomposable. Hence we can apply Lemma 6.3 to see that L is a free R-module. Thus for each nonfree
indecomposable MCM R-module D there are no irreducible homomorphisms from D to C and no suchones from τ C to D. This means that in the Auslander-Reiten quiver of CM(R) there is no arrow between
two vertices diﬀerent from R. The classiﬁcation of the Auslander-Reiten quivers of the MCM modules
over simple singularities [55, Chapters 8–12] together with [55, Corollary (12.11.3)] implies that the only
simple singularities R where CM(R) possesses such an Auslander-Reiten quiver are (A1 )-singularities.
It now follows that the conditions (1)–(4) in the theorem are equivalent. The last assertion of thetheorem is shown in the proof of the implication (4) ⇒ (2) stated above.
■
Remark 6.6. The condition in Theorem 6.5(3) that X contains a module of maximal complexity cannot
be removed. In fact, let R be any nonregular complete intersection local ring. Let X be a resolving
subcategory of mod R whose objects have ﬁnite projective dimension (e.g., the subcategory consistingof free modules, the subcategory consisting of modules of ﬁnite projective dimension, and so on). Thenthe stable category X is regular by Proposition 6.1. However, of course, R is not necessarily an (A1 )singularity. The reason for this is that all modules in X have complexity zero, and R has positive
codimension, so X does not contain a module of maximal complexity.
Let R be a simple hypersurface singularity. Theorem 5.4 especially says that CM(R) is not regularunless R is an (A1 )-singularity. One can actually conﬁrm this for a 1-dimensional (A2 )-singularity bydirect calculation.
Proposition 6.7. Let k be an algebraically closed ﬁeld of characteristic zero. Let R be an (A2 )-singularityof dimension 1 over k. Then there is a triangle equivalence
Dsg (CM(R)) ∼= Dsg (k[t]/(t2 )).
In particular, CM(R) is not regular.
Proof. One has R ∼= k[[x0 , x1 ]]/(x20 + x31 ), and all the nonisomorphic indecomposable MCM R-modules
are isomorphic to the maximal ideal m of R; see [55, Proposition (5.11)]. Hence Dsg (CM(R)) is triangle
equivalent to Dsg (EndR (m)) by Lemma 6.4.
The ring R is isomorphic to the numerical semigroup ring k[[t2 , t3 ]], which is a subring of the formal
power series ring S = k[[t]]. Note that R and S have the common quotient ﬁeld K. One has
EndR (m) ∼= (m :K m) = (m :S m) = S,RR
∼ ⊕2 ,∼∼EndR (m) ∼= TorR
1 (Tr(m), m) = Tor1 (m, m) = Tor2 (m, k) = k
(a)

(b)

where (a) follows from [55, Lemma (3.9)], and (b) holds since Tr(m) ∼= m. Thus EndR (m) is isomorphic toa quotient ring of S that has dimension 2 as a k-vector space, so EndR (m) ∼= S/(t2 ) = k[t]/(t2 ). Therefore

26

HIROKI MATSUI AND RYO TAKAHASHI

Dsg (CM(R)) is triangle equivalent to Dsg (k[t]/(t2 )), which is nonzero since k[t]/(t2 ) is an artinian ring of
inﬁnite global dimension.
■
Remark 6.8. Let R, S be Gorenstein local rings. Even if CM(R) and CM(S) are singularly equivalent,
the numbers of indecomposable MCM modules over R and S are not necessarily equal. In fact, let
R = k[[x, y]]/(xy) where k is an algebraically closed ﬁeld of characteristic zero. Then CM(R) is singularly
equivalent to 0 = CM(k) by Theorem 6.5. The isomorphism classes of indecomposable MCM R-modules
are those of R, R/xR and R/yR, so there are three. But k is the only indecomposable MCM k-module.
Acknowledgments. The authors are grateful to Kazuhiko Kurano for his comment on the proof of Theorem
6.5, to Henning Krause for his question on Remark 5.9 and to Osamu Iyama for his suggestions on allparts of the paper. The authors also thank the referee for his/her kind and helpful advice.
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Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan
E-mail address: m14037f@math.nagoya-u.ac.jp
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan
E-mail address: takahashi@math.nagoya-u.ac.jp
URL: http://www.math.nagoya-u.ac.jp/~takahashi/