CLASSIFYING DENSE RESOLVING AND CORESOLVINGSUBCATEGORIES OF EXACT CATEGORIES VIA
GROTHENDIECK GROUPSHIROKI MATSUI
Abstract. Classification problems of subcategories have been deeply considered
so far. In this paper, we discuss classifying dense resolving and dense coresolving
subcategories of exact categories via their Grothendieck groups. This study ismotivated by the classification of dense triangulated subcategories of triangulated
categories due to Thomason.

1. Introduction
Let C be a category. Classifying subcategories means for a property P, finding a
one-to-one correspondence{subcategories of C satisgying P} "

f
g

!

S,

where the set S is easier to understand. Classifying subcategories is an important
approach to understand the category C and has been studied in various areas ofmathematics, for example, stable homotopy theory, commutative/noncommutative
ring theory, algebraic geometry, and modular representation theory.
Let A be an additive category and X a full additive subcategory of A. We saythat X is additively closed if it is closed under taking direct summands, and that
X is dense if any object in A is a direct summand of some object of X . We caneasily show that X is additively closed if and only if X = add X and that X isdense if and only if A = add X . Here, add X denotes the smallest full additive
subcategory of A which is closed under taking direct summands and contains X .
Therefore, for any full additive subcategory X of A, X is a dense subcategory of
add X and add X is an additively closed subcategory of A. For this reason, to classify
additive subcategories, it suﬃces to classify additively closed ones and dense ones.
Classification of additively closed or dense subcategories has been deeply studied sofar. For example, the following three kinds of subcategories have been classified by
Gabriel [10], Hopkins and Neeman [13, 18], and Thomason [23], respectively.
(1) The Serre subcategories of finitely generated modules over a commutative
noetherian ring.
2010 Mathematics Subject Classification. 18E10, 18F30, 16G50.
Key words and phrases. exact category, dense subcategory, resolving subcategory, coresolving
subcategory, Grothendieck group.
The author is supported by Grant-in-Aid for JSPS Fellows 16J01067.
1

2

HIROKI MATSUI(2) The thick subcategories of perfect complexes over a commutative noetherianring.
(3) The dense triangulated subcategories of an essentially small triangulated
category.
(1) and (2) are classifications of additively closed subcategories, while (3) is a classification of dense subcategories.
Let us state the precise statement of Thomason’s classification theorem.
Theorem 1.1 (Thomason). Let T be an essentially small triangulated category.
Then there is a one-to-one correspondence{dense triangulated subcategories of T } "fg!

{subgroups of K0 (T )},where f and g are given by f (X ) := 〈[X] | X ∈ X 〉 and g(H) := {X ∈ T | [X] ∈ H},
respectively, and K0 (T ) stands for the Grothendieck group of T .
Motivated by this theorem, we discuss classifying dense resolving and dense coresolving subcategories of exact categories. The notion of a resolving subcategory has
been introduced by Auslander and Bridger [2] and that of a coresolving subcategory
is its dual notion. Resolving and coresolving subcategories have been widely studied
so far, for example, see [1, 3, 8, 14, 21, 22]. The main theorem of this paper is thefollowing.
Theorem 1.2 (Proposition 2.5, Theorem 2.7). Let E be an essentially small exact
category with either a generator or a cogenerator G.
(1) The following subcategories of E are the same:(i) dense G-resolving subcategories(ii) dense G-coresolving subcategories(iii) dense G-2-out-of-3 subcategories(2) There is a one-to-one correspondence
#
$
!
" f !
subgroups of K0 (E)dense G-(co)resolving subcategories of E ",gcontaining the image of G
where f and g are given by f (X ) := 〈[X] | X ∈ X 〉 and g(H) := {X ∈ E | [X] ∈H}, respectively, and K0 (E) stands for the Grothendieck group of E.

Here, the notion of a G-resolving (resp. G-coresolving) subcategory is a slight
generalization of that of a resolving (resp. coresolving) subcategory. Indeed, they
coincide when G consists of the projective (resp. injective) objects. In addition, G-2out-of-3 subcategory is a subcategory which is both G-resolving and G-coresolving.
The precise definitions of these subcategories will be given in Definition 2.3.
The organization of this paper is as follows. In Section 2, we give a proof ofour main theorem and several corollaries which include a correspondence between
dense resolving and dense coresolving subcategories of an exact category and dense
triangulated subcategories of its derived category. In Section 3, we discuss relation
between dense subcategories of exact categories and that of triangulated categories.

CLASSIFYING DENSE RESOLVING AND CORESOLVING SUBCATEGORIES

3

In Section 4, as applications of our results, we discuss when there are only finitely
many dense resolving subcategories of finitely generated modules over a left noetherian ring.
2. Classification of dense resolving subcategories
In this section, we show our main result. Throughout this paper, let A be an
abelian category, E an exact category, and T a triangulated category. We always
assume that all categories are essentially small, and that all subcategories are fulland additive. For a left noetherian ring A, mod A denotes the category of finitely
generated left A-modules.
We begin with recalling several notions, which are key notions of this paper.
Definition 2.1. Let G be a family of objects of E. We call G a generator (resp. acogenerator) of E if for any object A ∈ E, there is a short exact sequence
A′ ↣ G ↠ A (resp. A ↣ G ↠ A′ )in E with G ∈ G.

Example 2.2. (1) Clearly, E is both a generator and a cogenerator of E
(2) If E has enough projective (resp. injective) objects, then the subcategory proj E(resp. inj E) consisting of projective (resp. injective) objects is a generator (resp.
a cogenerator) of E.
Next we give the definitions of G-resolving and G-coresolving subcategories.

Definition 2.3. Let X be a subcategory of E and G a family of objects of E.
(1) We say that X is a G-resolving subcategory of E if the following three conditions
are satisfied:(i) X is closed under extensions: for a short exact sequence X ↣ Y ↠ Z in
E, if X and Z are in X , then so is Y .
(ii) X is closed under kernels of admissible epimorphisms: for a short exact
sequence X ↣ Y ↠ Z in E, if Y and Z are in X , then so is X.
(iii) X contains G.
If E has enough projective objects, we shall call X simply resolving if it is proj Eresolving.
(2) We say that X is a G-coresolving subcategory of E if the following three conditions
are satisfied:(i) X is closed under extensions: for a short exact sequence X ↣ Y ↠ Z in
E, if X and Z are in X , then so is Y .
(ii) X is closed under cokernels of admissible monomorphisms: for a short exact
sequence X ↣ Y ↠ Z in E, if X and Y are in X , then so is Z.
(iii) X contains G.
(3) We say that X is a G-2-out-of-3 subcategory of E if the following conditions are
satisfied:(i) X satisfies 2-out-of-3 property: for a short exact sequence X ↣ Y ↠ Z in
E, if 2 out of {X, Y, Z} belong to X , then so is the third.
(ii) X contains G.

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HIROKI MATSUI

Remark 2.4. Unlike the definition due to Auslander and Bridger [2], we do notassume that resolving subcategories are closed under direct summands. Therefore,
our definition is rather close to the definitions in [3, 14].
The following proposition shows that dense G-resolving, dense G-coresolving, anddense G-2-out-of-3 subcategories are the same thing.
Proposition 2.5. Let X be a dense subcategory of E.Then X is closed under cokernels of admissible monomorphisms if and only if it is closed under kernels ofadmissible epimorphisms.
Proof. We have only to show the ‘if’ part. The ‘only if’ part is proved by the dual
argument.
fg

Let X ↣ Y ↠ Z be a short exact sequence in E with X, Y ∈ X . Since X isdense, we can take Z ′ ∈ E with Z ⊕ Z ′ ∈ X . Consider a short exact sequence
X ⊕Z

!

f 0
0 idZ
0 0

↣

"

Y ⊕ Z ⊕ Z′

#

g 0 0
0 0 idZ ′↠$Z ⊕ Z ′.

Then X ⊕ Z is an object of X because X is closed under kernels of admissible
epimorphisms. From the split short exact sequence Z ↣ X ⊕ Z ↠ X, we obtain
Z ∈ X since X is closed under kernels of admissible epimorphisms.
#
Now we recall the definition of the Grothendieck group of an exact category.
Definition 2.6. Let E be an exact category. Let F be the free abelian group
generated by the isomorphism classes of objects of E. Let I be the subgroup of
F generated by the elements of the form [A] − [B] + [C] where A ↣ B ↠ C are
short exact sequences in E. Then we define the Grothendieck group of E, denotedby K0 (E), as the quotient group F/I.
The following theorem is our main result of this paper.
Theorem 2.7. Let E be an essentially small exact category with either a generator
or a cogenerator G. Then there are one-to-one correspondences among the following
sets:(1) {dense G-resolving subcategories of E},(2) {dense G-coresolving subcategories of E},(3) {dense G-2-out-of-3 subcategories of E}, and(4) {subgroups of K0 (E) containing the image of G}.
One-to-one correspondences among (1), (2) and (3) have been already shown in
Proposition 2.5. Thus, it suﬃce to show the bijection between (1) and (4). Moreover,
we will show this bijection only in the case that G is a generator because in thecogenerator case, it can be shown by the dual argument. The following lemma isessential in the proof of our theorem.
Lemma 2.8. Let G be a generator of E and X a dense G-resolving subcategory of
E. Then for an object A in E, A ∈ X if and only if [A] ∈ 〈[X] | X ∈ X 〉.

CLASSIFYING DENSE RESOLVING AND CORESOLVING SUBCATEGORIES

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Proof. Define an equivalence relation ∼ on the isomorphism classes E/∼= of objects′′∼of E, as follows: A ∼ A if there are X, X ∈ X such that A ⊕ X = A′ ⊕ X ′ . Set〈E〉X := (E/∼= )/∼ and denote by 〈A〉 the class of A. Then 〈E〉X is an abelian groupwith 〈A〉 + 〈B〉 := 〈A ⊕ B〉. Indeed, obviously, + is well-defined, commutative,
associative, and 〈0〉 is an identity element. Since X is dense, for any A ∈ E, there
is A′ ∈ E such that A ⊕ A′ ∈ X , and hence 〈A〉 + 〈A′ 〉 = 〈A ⊕ A′ 〉 = 〈0〉. Therefore,〈A′ 〉 is an inverse element of 〈A〉.
fg

Let A ↣ B ↠ C be a short exact sequence in E. Taking A′ , C ′ ∈ E withA ⊕ A′ , C ⊕ C ′ ∈ X and considering a short exact sequence
A ⊕ A′!

f 0
0 idA′0 0↣"

B ⊕ A′ ⊕ C ′

#

g 0 0
0 0 idC ′↠$C ⊕ C ′.

we have B ⊕ A′ ⊕ C ′ ∈ X . This shows 〈B〉 − 〈A〉 − 〈C〉 = 〈B ⊕ A′ ⊕ C ′ 〉 = 〈0〉.
Therefore, there is a group homomorphism
ϕ : K0 (E) → 〈E〉X , [A] (→ 〈A〉.

Note that 〈[X] | X ∈ X 〉 is contained in Ker ϕ.
From the definition of the Grothendieck group, any element of K0 (E) is denotedby [A] − [B]. Moreover, since there is a short exact sequenceB′ ↣ G ↠ B

in E with G ∈ G, [A] − [B] = [A ⊕ B ′ ] − [G]. Thus, any element of K0 (E) is denotedby [A] − [G] with G ∈ G.
Let [A] − [G] with G ∈ G be an element of Ker ϕ. Since X contains G, [A] ∈ Ker ϕ.
This means 〈A〉 = 〈0〉 and there are X, X ′ ∈ X such that A ⊕ X ∼= X ′ . Considering
the split short exact sequence
A ↣ A ⊕ X ↠ X,

we obtain A ∈ X since X is closed under kernels of epimorphisms. Thus, A ∈ X if
and only if [A] ∈ 〈[X] | X ∈ X 〉.
#
Proof of Theorem 2.7. By Lemma 2.5, the set (2) is nothing but the set (1). Therefore, we show that there is a one-to-one correspondence between the sets (1) and(3).
For a dense G-resolving subcategory X , definef (X ) := 〈[X] | X ∈ X 〉,

and for a subgroup H of K0 (E) containing the image of G, defineg(H) := {A ∈ E | [A] ∈ H}.

We show that f and g give mutually inverse bijections between (1) and (3).
First note that g(H) := {A ∈ E | [A] ∈ H} is a dense G-resolving subcategoryof E for a subgroup H of K0 (E) containing the image of G. Indeed, for any object
A ∈ E, take a short exact sequence A′ ↣ G ↠ A in E with G ∈ G. Then
[A ⊕ A′ ] = [A] + [A′ ] = [G] ∈ H, and hence A ⊕ A′ ∈ g(H). Thus g(H) isdense. Obviously, g(H) contains G. Furthermore, for any short exact sequence

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HIROKI MATSUIA ↣ B ↠ C, the relation [A] − [B] + [C] = 0 implies that g(H) is G-resolving.
Besides, f (X ) is clearly a subgroup of K0 (E) containing the image of G. As a result,f and g are well-defined maps between the sets (1) and (3)Let H be a subgroup of K0 (E) containing the image of G. Then the inclusion
f g(H) ⊂ H is trivial. For any [A] − [G] ∈ H with G ∈ G, [A] = ([A] − [G]) + [G] ∈ H
implies A ∈ g(H), and thus [A] − [G] ∈ f g(H). Therefore, f g(H) = H.
Let X be a dense resolving subcategory of E containing G. Then the inclusion
X ⊂ gf (X ) is trivial. Conversely, for any A ∈ gf (X ), since [A] ∈ f (X ) = 〈[X] |X ∈ X 〉, we have A ∈ X by Lemma 2.8. Therefore, gf (X ) = X . Consequently, f
and g are mutually inverse bijections between (1) and (3).
#
3. Relations with dense triangulated subcategories
In this section, we consider some combinations of Theorem 1.1 and Theorem
2.7. Let us start with the definition of the Grothendieck group for a triangulated
category.
Definition 3.1. Let T be a triangulated category. Let F be the free abelian group
generated by the isomorphism classes of objects of T . Let I be the subgroup generated by the elements of the form [A]−[B]+[C] where A → B → C → A[1] are exact
triangles in T . Then we define the Grothendieck group of T , denoted by K0 (T ), as
the quotient group F/I.
First, we discuss dense subcategories of exact categories and their derived categories. Please refer to [7, 17] for the definition of the derived category of an exact
category.
Lemma 3.2. [24, Lemma 9.2.4] Let E be an essentially small exact category. Thenthe canonical functor E → Db (E) induces an isomorphism ϕ : K0 (E) → K0 (Db (E)).
Combining Theorem 1.1, Theorem 2.7 and this lemma, we have the following
corollary.
Corollary 3.3. Let E be an essentially small exact category with either a generator
or a cogenerator G. Then there are one-to-one correspondences among the following
sets:(1) {dense G-resolving subcategories of E},(2) {dense triangulated subcategories of Db (E) containing G}, and(3) {subgroups of K0 (E) containing the image of G}.
Taking G = proj E in this corollary gives the dense version of the following theorem
due to Krause and Stevenson:
Theorem 3.4. [15, Theorem 1] Let E be an exact category with enough projective
objects. Then there is one-to-one correspondence between(1) {thick subcategories of E containing proj E} and(2) {thick triangulated subcategories of Db (E) containing proj E}.

CLASSIFYING DENSE RESOLVING AND CORESOLVING SUBCATEGORIES

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Next, we give a more concrete corollary.
Let S be an Iwanaga-Gorenstein ring (i.e. S is noetherian on both sides and S isof finite injective dimension as a left S-module and a right S-module). Let us give
several remarks about Iwanaga-Gorenstein rings (cf. [6, 25]).
Remark 3.5. (1) [6, Lemma 4.4.2] We say that a finitely generated left S-module
X is maximal Cohen-Macaulay if ExtiS (X, S) = 0 for all integers i > 0. CM(S)denotes the subcategory of mod S consisting of maximal Cohen-Macaulay Smodules. Then it is a Frobenius category, and hence, its stable category CM(S)is triangulated.
(2) Natural inclusions CM(S) ↩→ mod S ↩→ Db (mod S) induce isomorphisms
K0 (CM(S)) ∼= K0 (mod S) ∼= K0 (Db (mod S)).
Here, the first isomorphism is shown in [25, Lemma 13.2] and the second isomorphism is by Lemma 3.2(3) [6, Theorem 4.4.1] Composition of the natural inclusion CM(S) ↩→ Db (mod S)
and the quotient functor Db (mod S) → Dsg (S) := Db (mod S)/Kb (proj(mod S))
induces a triangle equivalence
CM(S) ∼= Dsg (S).
Corollary 3.6. Let S be an Iwanaga-Gorenstein ring. Then there are one-to-one
correspondences among the following sets:(1) {dense resolving subcategories of CM(S)},(2) {dense resolving subcategories of mod S},(3) {dense triangulated subcategories of Db (mod S) containing proj(mod S)},(4) {dense triangulated subcategories of CM(S) ∼= Dsg (S)},(5) {subgroups of K0 (mod S) containing the image of proj(mod S)}, and(6) {subgroups of K0 (CM(S))}.
Proof. One-to-one correspondences among (1), (2), (3), and (5) follow from theabove remarks and Corollary 3.3. The bijection between (4) and (6) follows from
Thomason’s result, Theorem 1.1. Thus, we show the one-to-one correspondence
between (5) and (6).
The localization sequence
Db (proj(mod S)) → Db (mod S) → Dsg (S)yields the exact sequence

K0 (Db (proj(mod S))) → K0 (Db (mod S)) → K0 (Dsg (S)) → 0.
The equivalence Dsg (S) ∼= CM(S) and Lemma 3.2 turns this sequence into the exact
sequence
K0 (proj(mod S)) → K0 (mod S) → K0 (CM(S)) → 0.
Then, the equivalence is clear.
#In the last two corollaries, we constructed a triangulated category Db (E) from angiven exact category E and discussed their dense subcategories. Next, we consider

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HIROKI MATSUI

the opposite direction. More precisely, we construct an abelian category from agiven triangulated category, and then we discuss their dense subcategories.
Let us recall the definition and some basic properties of t-structures; for details,see [11].
Definition 3.7. (1) A t-structure on T is a pair (T ≤0 , T ≥0 ) of subcategories in T
satisfying the following conditions:(i) HomT (T ≤−1 , T ≥0 ) = 0.
(ii) For any object X ∈ T , there exists an exact triangle X ′ → X → X ′′ →X ′ [1] in T with X ′ ∈ T ≤−1 and X ′′ ∈ T ≥0 .
(iii) T ≤−1 ⊂ T ≤0 and T ≥0 ⊂ T ≥−1 .
Here, T ≤−n := T ≤0 [n] and T ≥−n := T ≥0 [n]. Moreover, the intersection T ≤0 ∩T ≥0 has the structure of an abelian category and we call it the heart of thet-structure.
%(2) A t-structure (T ≤0 , T ≥0 ) on T is called bounded if T = i,j∈Z T ≤i ∩ T ≥j .
Example 3.8. Let A be an abelian category and put

Db (A)≤0 := {X ∈ Db (A) | Hi (X) = 0 (∀i > 0)},Db (A)≥0 := {X ∈ Db (A) | Hi (X) = 0 (∀i < 0)}.
Then (Db (A)≤0 , Db (A)≥0 ) defines a bounded t-structure on Db (A) and its heart isA.
The next proposition is a variant of Lemma 3.2.

Proposition 3.9. [19] Let (T ≤0 , T ≥0 ) be a bounded t-structure on T with heart A.
Then the inclusion functor induces an isomorphism K0 (A) ∼= K0 (T ).
From this proposition and Theorem 2.7, we have the following corollary.
Corollary 3.10. Let T be an essentially small triangulated category, (T ≤0 , T ≥0 ) abounded t-structure on T with heart A, and either a generator or a cogenerator G
of A. Then there are one-to-one correspondences among the following sets:(1) {dense G-resolving subcategories of A},(2) {dense triangulated subcategories of T containing G}, and(3) {subgroups of K0 (T ) containing the image of G}.
4. Examples
In this section, we give some examples of module categories which have only
finitely many dense resolving subcategories.
Let us start with the following remark.
Remark 4.1. Let L be an abelian group. Then there are only finitely many subgroups of L if and only if L is a finite group. Indeed, ‘if part’ is clear. Suppose that
there are only finitely many subgroups of L. Then L is an noetherian Z-module andin particular, finitely generated. Therefore, there is an isomorphism
n
&
⊕r
∼
L=Z ⊕
(Z/nZ)⊕mi ,i=1

CLASSIFYING DENSE RESOLVING AND CORESOLVING SUBCATEGORIES

9

where r, n and mi are non-negative integers. We obtain r = 0 due to our assumption
as Z has infinitely many subgroups. For this reason, L is isomorphic to a finite direct
sum of finite abelian groups, and thus is a finite group.
From this remark and Theorem 2.7, for a left noetherian ring A, the following two
conditions are equivalent:(1) There are only finitely many dense resolving subcategories of mod A.
(2) K0 (mod A)/〈[P ] | P ∈ proj(mod A)〉 is a finite group.
4.1. The case of finite dimensional algebras. First we consider the case offinite dimensional algebras. Let A be a basic finite dimensional algebra over a fieldk with a complete set {e1 , . . . , en } of primitive orthogonal idempotents. Denote
Si := Aei /radA (Aei ) by the simple A-module corresponds to ei . Then by [4, Theorem
3.5], {[S1 ], . . . , [Sn ]} forms a free basis of the Grothendieck group K0 (mod A), andhence there is an isomorphism of abelian groups:
K0 (mod A) ∼= Z⊕n .
The Cartan matrix of A is an n × n-matrix CA := (dimk ei Aej )i,j=1,...,n . Then theabove isomorphism induces the following isomorphism (see [4, Proposition 3.8]).
CA
K0 (mod A)/〈[P ] | P ∈ proj(mod A)〉 ∼= Coker(Z⊕n −−→ Z⊕n ).

Therefore if CA has elementary divisors (m1 , · · · , mr , 0, · · · , 0), then we obtain adecomposition:
K0 (mod A)/〈[P ] | P ∈ proj(mod A)〉 ∼= Z⊕n−r ⊕ Z/(m1 ) ⊕ · · · ⊕ Z/(mr ),where m1 , . . . , mr are not zero. Furthermore, one has'
0(r < n)det CA =m1 · m2 · · · mn (r = n).
As a result, the abelian group K0 (mod A)/〈[P ] | P ∈ proj(mod A)〉 is a finite group
if and only if the determinant of CA is not zero.
From this argument, we have the following corollary.
Corollary 4.2. Let A be a basic finite dimensional algebra over a field k. Thenmod A has only finitely many dense resolving subcategories if and only if its Cartan
matrix has non-zero determinant. This is the case, the number of dense resolving
subcategories is d(m1 ) · · · d(mn ). Here, (m1 , . . . , mn ) are elementary divisors of CA
and d(l) denotes the number of divisors of l.
Remark 4.3. For the case of gentle algebras, Holm [12] gives a characterization ofalgebras with non-zero Cartan determinant det CA .

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HIROKI MATSUI

4.2. The case of simple singularities. Next we consider the case of simple singularities. Let k be an algebraically closed field of characteristic 0. We say thata commutative noetherian local ring R := k[[x, y, z]]/(f ) has a simple (surface)singularity if f is one of the following form:(An ) x2 + y n+1 + z 2 (n ≥ 1),(Dn ) x2 y + y n−1 + z 2 (n ≥ 4),(E6 ) x3 + y 4 + z 2 ,(E7 ) x3 + xy 3 + z 2 ,(E8 ) x3 + y 5 + z 2 .
In this case, the Grothendieck group of mod R is given as follows (see [25, Proposition
13.10]):(An )(Dn ) (n = even)(Dn ) (n = odd)
(E6 )
(E7 )
(E8 )

K0 (mod R)#{ dense resolv. subcat. of mod R}
Z ⊕ Z/(n + 1)Z
the number of divisors of n + 1⊕2Z ⊕ (Z/2Z)
5
Z ⊕ Z/4Z
3
Z ⊕ Z/3Z2
Z ⊕ Z/2Z2Z
1Here, Z appearing in K0 (mod R) is generated by [R]. Owing to Theorem 2.7, there
are only finitely many dense resolving subcategories of mod R. Hence the following
natural question arises.
Question 4.4. Let R be a Gorenstein local ring of dimension two. Then does
the condition #{dense resolving subcategories of mod R} < ∞ imply that R has asimple singularity?
Remark 4.5. 1-dimensional simple singularities may have infinitely many dense
resolving subcategories (see [25, Proposition 13.10]).
Let R be a noetherian normal local domain with residue field k. Denote by Cl(R)the divisor class group of R. Then there is a surjective homomorphism
( )
rk
u=
: K0 (mod R) → Z ⊕ Cl(R),c1where rk is the rank function and c1 is the first Chern class. Moreover, u([R]) =t(1, 0) and the kernel of u is the subgroup of K0 (mod R) generated by modules ofcodimension at least 2; see [5]. In particular, if R is a 2-dimensional noetherian
normal local domain with residue field k, we obtain a short exact sequence
%

rk
c1

&

0 → 〈[k]〉 → K0 (mod R) −−−→ Z ⊕ Cl(R) → 0

CLASSIFYING DENSE RESOLVING AND CORESOLVING SUBCATEGORIES

11

of abelian groups. This sequence induces the following short exact sequence sincerk(R) = 1 and c1 (R) = 0:c

1
0 → 〈[k], [R]〉/〈[R]〉 → K0 (mod R)/〈[R]〉 →Cl(R) → 0.

Therefore, we have an isomorphism
Cl(R) ∼
= K0 (mod R)/〈[k], [R]〉and the following result is deduced from Theorem 2.7:
Theorem 4.6. Let R be a noetherian normal local domain of dimension two. Then
there is a one-to-one correspondence#$fdense resolving subcategories of mod R!
"{subgroups of Cl(R)}gcontaining kwhere f and g are given by f (X ) := 〈c1 (X) | X ∈ X 〉 and g(H) := {X ∈ mod R |c1 (X) ∈ H} respectively.
The following answers Question 4.4 for domains.
Corollary 4.7. Let R be a 2-dimensional complete non-regular Gorenstein normal
local domain with algebraically closed residue field k of characteristic 0. Then thefollowing are equivalent:(1) R has a simple singularity.
(2) There are only finitely many dense resolving subcategories of mod R.
(3) There are only finitely many dense resolving subcategories of mod R containingk.
Proof. (1) ⇒ (2): If R has a simple singularity, then K0 (mod R)/〈[R]〉 is a finitegroup; see [25, Proposition 13.10]. Thus, Theorem 2.7 shows that mod R has only
finitely many dense resolving subcategories.
(2) ⇒ (3): This implication is trivial.
(3) ⇒ (1): From Theorem 4.6, Cl(R) has finitely many subgroups. Therefore,
Cl(R) is a finite group, and thus by [9, Corollary 3.3] we have ΩCM(R) = add G forsome module G, where ΩCM(R) stands for the category of first syzygies of maximal
Cohen-Macaulay R-modules. Now, since R is Gorenstein, CM(R) = ΩCM(R) has
only finitely many indecomposable objects up to isomorphism. Consequently, R has
a simple singularity from [25, Theorem 8.10].
#
Example 4.8. Let R be a 2-dimensional simple singularity of type (A1 ). Namely,
R = k[[x, y, z]]/(x2 + y 2 + z 2 ). Then the indecomposable
maximal Cohen-Macaulay√
R-modules are R and the ideal I = (x + −1y, z) up to isomorphism. Thus,
every maximal Cohen-Macaulay module is of the form R⊕n ⊕ I ⊕m . Then the dense
resolving subcategories of mod R are:• mod R, and• {M ∈ mod R | Ω2 M ∼= R⊕n ⊕ I ⊕2m for some m, n ∈ Z≥0 }.

12

HIROKI MATSUIProof. Set G := K0 (mod R) and let H be the subgroup generated by [R].
First note that there is a non-split short exact sequence 0 → I → R⊕2 → I → 0,see [25, Chapter 10]. Therefore, [R] and [I] satisfy 2[R] = 2[I] in G. Moreover,the isomorphism G ∼= K0 (CM(R)) shows that G and H are only subgroups of G
containing [R].
Using the notation of Theorem 2.7, we know that g(G) = mod R. It thus suﬃces
to show that g(H) = X . Let M be an object of X . From the exact sequence
0 → Ω2 M → R⊕n1 → R⊕n0 → M → 0, one has[M ] ≡ [Ω2 M ] ≡ 0 mod H.

This shows that M ∈ g(H). Next, take M ∕∈ X . Then Ω2 M ∼= R⊕n ⊕ I ⊕(2m+1) forsome n, m ∈ Z≥0 . Using the similar argument, one has[M ] ≡ [Ω2 M ] ≡ (2m + 1)[I] ≡ [I] mod H.

Hence if [M ] is in H, then so is [I]. This gives a contradiction to G ∕= H. Therefore,[M ] cannot be in H. Thus, we are done.
#
Acknowledgments. The author is grateful to his supervisor Ryo Takahashi for his
many helpful comments. Also, the author very much thank the anonymous referee
for his/her careful reading, valuable comments and helpful suggestions.
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Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku,
Nagoya, Aichi 464-8602, Japan
Email address: m14037f@math.nagoya-u.ac.jp