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The radius of a subcategory of modules

We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to contain only maximal Cohen-Macaulay modules. We also show that the category of maximal Cohen-Macaulay modules has finite radius when R is a Cohen-Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen-Macaulay modules, finite/countable Cohen-Macaulay representation type and the uniform Auslander condition.


Introduction
Let R be a commutative Noetherian local ring and mod R the category of finitely generated modules over R. In this paper we introduce and study a new invariant for subcategories X of mod R which we call the radius of X . Roughly speaking, it is defined as the least number of extensions necessary to build the whole objects in X out of a single object in mod R. (For the precise definition, see Definition 2.3 in this paper.) Our definition is inspired by the notion of dimension of triangulated categories that was introduced by Rouquier in [19].
We obtain strong evidences that the concept of radius is intimately linked to both the representation theory and the singularity of R. For example, over a Gorenstein complete local ring R, the category of maximal Cohen-Macaulay modules has radius zero if and only if R has finite Cohen-Macaulay representation type, in other words, R is a simple hypersurface singularity (when R has an algebraically closed coefficient field of characteristic zero). In addition, the category of maximal Cohen-Macaulay modules over a complete local hypersurface (over an algebraically closed field of characteristic not two) of countable Cohen-Macaulay representation type has radius one. We also observe a tantalizing connection to the uniform Auslander condition, which has attracted researchers over the years.
Perhaps most surprisingly, one of the corollaries of our first main result (Theorem 3.3) states: Theorem I. Let R be a local complete intersection, and let X be a resolving subcategory of mod R. If the radius of X is finite, then X contains only maximal Cohen-Macaulay modules.
We conjecture that the above result holds for all Cohen-Macaulay local rings. Our second main result below supports this conjecture, which follows from a more general results (Theorems 5.7 and 5.11).
Theorem II. Let R be a Cohen-Macaulay complete local ring with perfect coefficient field. Then the category of maximal Cohen-Macaulay modules over R has finite radius.
The structure of the paper is as follows. In Section 1 we set the basic notations and definitions. Section 2 contains our key definition (Definition 2.1) of the radius of a subcategory of mod R, as well as some detailed comparisons to similar notions. We also give several results connecting the radius to the singularities of finite and countable Cohen-Macaulay representation type. Sections 3 and 4 consist of the statement and proof of our main Theorem I, respectively. We also discuss here thickness of resolving subcategories of maximal Cohen-Macaulay modules over a complete intersection. Section 5 contains the proof of (generalizations of) our main Theorem II. Section 6 connects the main results to the uniform Auslander condition and discuss some open questions.

Preliminaries
In this section, we recall the definitions of a resolving subcategory, totally reflexive modules and a thick subcategory. We begin with our convention. Convention 1.1. Throughout this paper, we assume all rings are commutative Noetherian rings with identity. All modules are finitely generated. All subcategories are full and strict. (Recall that a subcategory X of a category C is called strict provided that for objects M, N ∈ C with M ∼ = N, if M is in X , then so is N.) Hence, the subcategory of a category C consisting of objects {M λ } λ∈Λ always mean the smallest strict full subcategory of C to which M λ belongs for all λ ∈ Λ. Note that this coincides with the full subcategory of C consisting of all objects X ∈ C such that X ∼ = M λ for some λ ∈ Λ. Let R be a (commutative Noetherian) ring. Denote by mod R the category of (finitely generated) Rmodules and R-homomorphisms. For a Cohen-Macaulay local ring R, we call a maximal Cohen-Macaulay R-module just a Cohen-Macaulay R-module. We denote by CM(R) the subcategory of mod R consisting of Cohen-Macaulay R-modules.
The following notation is used throughout this paper. Notation 1.2. For a subcategory X of mod R, we denote by add X (or add R X ) the additive closure of X , namely, the subcategory of mod R consisting of direct summands of finite direct sums of modules in X . When X consists of a single module M, we simply denote it by add M (or add R M). For an R-module M, we denote by M * the R-dual module Hom R (M, R). For a homomorphism f : M → N of R-modules, f * denotes the R-dual homomophism N * → M * sending σ ∈ N * to the composition σ · f ∈ M * .
The notion of a resolving subcategory has been introduced by Auslander and Bridger [3]. It can actually be defined for an arbitrary abelian category with enough projective object. The only resolving subcategories we deal with in this paper are ones of mod R. Definition 1.3. A subcategory X of mod R is called resolving if the following hold. (R1) X contains the projective R-modules. (R2) X is closed under direct summands: if M is an R-module in X and N is an R-module that is a direct summand of M, then N is also in X . (R3) X is closed under extensions: for an exact sequence 0 → L → M → N → 0 of R-modules, if L, N are in X , then so is M. (R4) X is closed under kernels of epimorphisms: for an exact sequence 0 A resolving subcategory is a subcategory such that any two minimal resolutions of a module by modules in it have the same length; see [3,Lemma (3.12)]. Note that one can replace the condition (R1) with: Next we recall the notion of a totally reflexive module.
We denote by G(R) the subcategory of mod R consisting of totally reflexive modules.
A totally reflexive module was defined by Auslander [2], and deeply studied by Auslander and Bridger [3]. The R-dual of a totally reflexive R-module is also totally reflexive. Every projective module is totally reflexive, i.e., add R ⊆ G(R). If R is a Cohen-Macaulay local ring, then every totally reflexive R-module is Cohen-Macaulay, i.e., G(R) ⊆ CM(R). When R is a Gorenstein local ring, an R-module is totally reflexive if and only if it is Cohen-Macaulay, i.e., G(R) = CM(R). For more details, see [3] and [10].
Syzygies, cosyzygies and transposes are key tools in this paper. We recall here their precise definitions. Definition 1.5. Let (R, m) be a local ring, and let M be an R-module.
(1) Take a minimal free resolution · · · δ n+1 Then, for each n ≥ 1, the image of δ n is called the n-th syzygy of M and denoted by Ω n M (or Ω n R M). For convention, we set Ω 0 M = M. (2) The cokernel of the R-dual map δ * 1 : F * 0 → F * 1 is called the (Auslander) transpose of M and denoted by TrM (or Tr R M).
− −−− → · · · be a minimal free coresolution of M, that is, an exact sequence with F −n free and Im δ −n ⊆ mF −(n+1) for all n ≥ 1. Then we call the image of δ −n the n-th cosyzygy of M and denote it by Ω −n M (or Ω −n R M). Let R be a local ring. Then by [28,Lemma 3.2] one can replace (R4) with: Totally reflexive modules behave well under taking their syzygies, cosyzygies and transposes. Let R be a local ring. Let M be a totally reflexive R-module. The R-dual of a minimal free resolution (respectively, coresolution) of M is a minimal free coresolution (respectively, resolution) of M * . In particular, a minimal free coresolution of M always exists, and it is uniquely determined up to isomorphism. The n-th syzygy Ω n M and cosyzygy Ω −n M are again totally reflexive for all n. This is an easy consequence of [10, (1.2.9) and (1.4.8)]. The transpose TrM is also totally reflexive; see [3,Proposition (3.8)]. For an R-module M, the n-th syzygy Ω n M for any n ≥ 1 and the transpose TrM are uniquely determined up to isomorphism, since so are a minimal free resolution of M. If M is totally reflexive, then the n-th cosyzygy Ω −n M for any n ≥ 1 is also uniquely determined up to isomorphism, since so is a minimal free coresolution of M.
A lot of subcategories of mod R are known to be resolving. For example, CM(R) is a resolving subcategory of mod R if R is Cohen-Macaulay. The subcategory of mod R consisting of totally reflexive R-modules is resolving by [3, (3.11)]. One can construct a resolving subcategory easily by using vanishing of Tor or Ext. Also, the modules of complexity less than a fixed integer form a resolving subcategory of mod R. For the details, we refer to [23,Example 2.4]. Now we define a thick subcategory of totally reflexive modules.
Definition 1.6. A subcategory X of G(R) is called thick if it is closed under direct summands, and short exact sequences: for an exact sequence 0 → L → M → N → 0 of totally reflexive R-modules, if two of L, M, N are in X , then so is the third.
A typical example of a thick subcategory is obtained by restricting a resolving subcategory to G(R).
The following proposition is shown by an argument dual to [28,Lemma 3.2].
Proposition 1.7. Let R be a local ring. Let X be a subcategory of G(R) containing R. Then X is a thick subcategory of G(R) if and only if X is a resolving subcategory of mod R and is closed under cosyzygies: if M is in X , then so is Ω −1 M.
Let (R, m) be a local ring. We call R a hypersurface if the m-adic completion R of R is a residue ring of a complete regular local ring by a principal ideal. We say that R is a complete intersection if R is a residue ring of a complete regular local ring by an ideal generated by a regular sequence.
We recall the definitions of Gorenstein dimension and complete intersection dimension, which are abbreviated to G-dimension and CI-dimension. These notions have been introduced by Auslander and Bridger [3] and Avramov, Gasharov and Peeva [7], respectively. Definition 1.8. Let R be a local ring, and let M be an R-module. The G-dimension of M, denoted Gdim R M, is defined as the infimum of the lengths of totally reflexive resolutions of M, namely, exact sequences of the form 0 → X n → X n−1 → · · · → X 1 → X 0 → M → 0 with each X i being totally reflexive. The CI-dimension of M is defined as the infimum of ← 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.
Recall that M is said to have complexity c, denoted by cx R M = c, if c is the least nonnegative integer n such that there exists a real number r satisfying the inequality β R i (M) ≤ ri n−1 for all i ≫ 0.
Remark 1.9. For a local ring (R, m, k) and a module M over R, the following are known to hold. For the proofs, we refer to [10] and [7].
(1) Gdim R M = ∞ if and only if M does not admit a totally reflexive resolution of finite length.
is finite, then R is a Gorenstein ring (respectively, a complete intersection).

Definition of the radius of a subcategory
This section contains the key definition and establishes several results. More precisely, we will give the definition of the radius of a subcategory of mod R for a local ring R, and compare it with other notions, such as the dimension of a triangulated category defined by Rouquier. We will also explore its relationships with representation types of a Cohen-Macaulay local ring. (1) For a subcategory X of mod R we denote by [X ] the additive closure of the subcategory of mod R consisting of R and all modules of the form Ω i X, where i ≥ 0 and X ∈ X . When X consists of a single module X, we simply denote it by [X]. (2) For subcategories X , Y of mod R we denote by X • Y the subcategory of mod R consisting of the R-modules M which fits into an exact sequence 0 → (3) Let C be a subcategory of mod R. We define the ball of radius r centered at C as If C consists of a single module C, then we simply denote [C] r by [C] r , and call it the ball of radius r centered at C. We write [C] R r when we should specify that mod R is the ground category where the ball is defined. Some similar notions have already been introduced. In [23, Definition 3.1] the second author defines the subcategory res n X of the resolving closure res X of a given subcategory X of mod R. This is different from ours in that res n X is not closed under syzygies. In [6] the thickening thick n X of a given subcategory X of a triangulated category is defined. This cannot be applied directly to a module category.
Proposition 2.2. Let R be a local ring.
(1) Let X , Y be subcategories of mod R. The following are equivalent for an R-module M: (a) M belongs to X • Y; (3) Let C be a subcategory of mod R, and let a, b be positive integers. Then one has Taking syzygies and direct sums, we have an exact sequence The left and right terms are in [X ] and [Y], respectively. The middle term contains an R-module isomorphic to M. Thus the statement (b) follows.
(2) First, let M be an R-module in (X • Y) • Z. By the assertion (1) there is an exact such that M is a direct summand of V . By (1) again, we have an exact sequence 0 → X → U → Y → 0 with X ∈ [X ] and Y ∈ [Y] such that W is a direct summand of U. Writing U = W ⊕ W ′ , we make the following pushout diagram.
The bottom row implies that T is in Y • Z, and it follows from the middle column that Then it follows from (1) that there is an exact We prove this by induction on b. It holds by definition when b = 1. Let b ≥ 2. Then we have where the second equality follows from (2), and the induction hypothesis implies the third equality.
Let C be a subcategory of mod R and r ≥ 0 an integer. By the second and third assertions of Proposition 2.2, without danger of confusion we can write: Now we can make the definition of the radius of a subcategory. Definition 2.3. Let R be a local ring, and let X be a subcategory of mod R. We define the radius of X , denoted by radius X , as the infimum of the integers n ≥ 0 such that there exists a ball of radius n + 1 centered at a module containing X . By definition, radius X ∈ N ∪ {∞}.
The definition of the radius of a resolving subcategory looks similar to that of the dimension of a triangulated category which has been introduced by Rouquier (cf. [19,Definition 3.2]). The stable category CM(R) of Cohen-Macaulay modules over a Gorenstein local ring R is triangulated by [9,14], and the dimension of CM(R) in the sense of Rouquier is defined. It might look the same as the radius of CM(R) in our sense.
However there are (at least) two differences in the definitions: (1) A defining object for dim CM(R) is required to be inside the category CM(R), but a defining object for radius CM(R) is not, i.e., it is enough to be an object of mod R. More precisely, dim CM(R) (respectively, radius CM(R)) is defined as the infimum of the integers n ≥ 0 such that CM(R) = G n+1 for some object G (respectively, CM(R) ⊆ [C] n+1 for some object C). Then G must be an object of CM(R), while C may not be an object of CM(R), just being an object of mod R. (2) Let X and Y be subcategories of CM(R) and CM(R), respectively. Then the subcategory Y of CM(R) is closed under taking cosyzygies of Cohen-Macaulay modules in it, but the subcategory [X ] of CM(R) is not in general. (In fact, this difference is a reason why we can prove Proposition 2.5 below but do not know whether the analogue for dimension holds or not; see Question 2.6 below.) Thus these two notions are different, but they are still related to each other. In fact, we can show that the following relationship exists between them.
In the following, we show this claim. Since Ω −d Ω d C is a Cohen-Macaulay R-module, and CM(R) is a resolving subcategory of mod R, the inclusion CM(R) ⊇ [Ω −d Ω d C] n+1 hold. To get the opposite inclusion, it is enough to prove that for every m ≥ 1 and Let us prove this by induction on m. The case m = 1 is obvious, so let m ≥ 2. According to Proposition 2.2(1), there is an (2) Let n := dim CM(R) < ∞. We find a Cohen-Macaulay R-module G such that CM(R) = G n+1 . We want to prove that CM(R) = [G] n+1 holds, and it suffices to show that for every m ≥ 1 and M ∈ G m we have M ∈ [G] m . Let us use induction on m. The case m = 1 follows from the fact that ΩN ∼ = Ω −1 N up to free summand for each Cohen-Macaulay R-module N, since R is a hypersurface. When m ≥ 2, there exists an exact triangle X → Y → Z → ΣX in CM(R) with X ∈ G m−1 and Z ∈ G such that M is a direct summand of Y . Then we have an exact sequence 0 → X → Y ⊕ R ⊕h → Z → 0 of R-modules, and we are done by applying the induction hypothesis.
In the rest of this section, we will study the relationships between the representation types of a Cohen-Macaulay local ring and the radius of the category of Cohen-Macaulay modules. Recall that a Cohen-Macaulay local ring R is said to be of finite (respectively, countable) Cohen-Macaulay representation type if CM(R) has only finitely (respectively, countably but not finitely) many indecomposable modules up to isomorphism.
We can describe the property of finite Cohen-Macaulay representation type in terms of a radius.
Proposition 2.5. Let R be a Gorenstein Henselian local ring. The following are equivalent: (1) One has radius CM(R) = 0; (2) The ring R has finite Cohen-Macaulay representation type.
Note that since R is Henselian, the Krull-Schmidt theorem holds, i.e., each R-module uniquely decomposes into indecomposable R-modules up to isomorphism. Let C 1 , . . . , C n be the indecomposable direct summands of C. We may assume that C = C 1 ⊕ · · · ⊕ C n . Since R is Gorenstein, taking syzygies preserves indecomposability of nonfree Cohen-Macaulay R-modules. We see that the set of nonisomorphic indecomposable Cohen-Macaulay R-modules is We may assume that for all i ≥ 0 and 1 ≤ j = j ′ ≤ n we have Ω i C j ∼ = C j ′ , because if Ω i C j ∼ = C j ′ for some such i, j, j ′ , then we can exclude C j ′ from C. Now fix an integer j with 1 ≤ j ≤ n. As taking cosyzygies preserves indecomposability of nonfree Cohen-Macaulay R-modules, Ω −1 C j is isomorphic to Ω a C b for some a ≥ 0 and 1 ≤ b ≤ n. Taking the a-th cosyzygies, we have Ω −1−a C j ∼ = C b , hence C j ∼ = Ω 1+a C b . This forces us to have b = j, which says that C j is periodic. Hence there are only finitely many indecomposable Cohen-Macaulay R-modules. Question 2.6. Does the equality in Proposition 2.4(1) hold true? If it does, then Proposition 2.5 will say that a Gorenstein Henselian local ring R has finite Cohen-Macaulay representation type if and only if dim CM(R) ≤ 0. This statement is a partial generalization of Minamoto's theorem [18,Theorem 0.2], which asserts that the same statement holds for a finite-dimensional selfinjective algebra over a perfect field, extending Yoshiwaki's recent theorem [27,Corollary 3.10].
The next result hints at further relationship between finite radius of CM(R) and more well-known classification of singularities.
Proposition 2.7. Let R be a complete local hypersurface over an algebraically closed field of characteristic not two. Assume that R is of countable Cohen-Macaulay representation type. Then radius CM(R) = 1.
Proof. It follows from [1, Theorem 1.1] that there exists an R-module X such that for every indecomposable module M ∈ CM(R) there is an exact sequence 0 → L → M ⊕ R n → N → 0 with L, N ∈ {0, X, ΩX}. This shows that CM(R) = [X] 2 . Now we see from Proposition 2.5 that the radius of CM(R) is equal to one.

Finiteness of the radius of a resolving subcategory
In this section we state our guiding conjecture and first main result.
Conjecture 3.1. Let R be a Cohen-Macaulay local ring. Let X be a resolving subcategory of mod R with finite radius. Then every R-module in X is Cohen-Macaulay.
Remark 3.2. The converse of Conjecture 3.1 also seems to be true. We consider this in Section 5.
Let X be a subcategory of mod R. We denote by res X (or res R X ) the resolving closure of X , namely, the smallest resolving subcategory of mod R containing X . If X consists of a single module M, then we simply denote it by res M (or res R M). For a prime ideal p of R, we denote by X p the subcategory of mod R p consisting of all modules of the form X p , where X ∈ X . The first main result of this paper is the following theorem. Theorem 3.3. Let R be a commutative Noetherian ring. Let X be a resolving subcategory of mod R. Suppose that there exist a prime ideal p of R with ht p > 0 and an R p -module M with 0 = M ∈ add Rp X p which satisfy one of the following conditions.
Then X has infinite radius.
The proof of this theorem will be given in the next section. As a direct consequence of the above theorem, we obtain two cases in which our conjecture holds true. Proof. Conjecture 3.1 trivially holds in the case where R is Artinian, so let (R, m, k) be a Cohen-Macaulay local ring of positive dimension. Then we have ht m > 0. Let X be a resolving subcategory of mod R, and suppose that X contains a non-Cohen-Macaulay R-module M.
(1) We have 0 < dim R − depth R M = depth R − depth R M = CIdim R M < ∞, and Theorem 3.3 (4) implies that X has infinite radius.
(2) We have 0 < n : The module Ω n R M is Cohen-Macaulay, and by assumption res R (Ω n R M) is a thick subcategory of CM(R) = G(R). Theorem 3.3(3) implies that X has infinite radius.

Proof of Theorem I
This section is devoted to give the proof of Theorem 3.3 (hence of Theorem I from Introduction), which we break up into several parts. Most of them also reveal properties of subcategories of mod R which are interesting in their own right.
First of all, we make a remark to reduce our theorem to the local case.
Remark 4.1. Let C be an R-module, n ≥ 0 an integer and p a prime ideal of R. Then for a subcategory X of mod R the implication   Proof. Let us prove the lemma by induction on n.
When n = 1, the module M is isomorphic to a direct summand of (R ⊕ a i=0 Ω i C) ⊕b for some a, b ≥ 0. Hence Γ I (M) is isomorphic to a direct summand of (Γ I (R) ⊕ a i=0 Γ I (Ω i C)) ⊕b . For i ≥ 1, the syzygy Ω i C is a submodule of some free module R ⊕c i , and Γ I (Ω i C) is a submodule of Γ I (R) ⊕c i , which implies that Ann R Γ I (Ω i C) contains Ann R Γ I (R). Hence we obtain , and Proposition 2.2(1) says that there is an exact sequence 0 → X → Y → Z → 0 with X ∈ [C] n−1 and Z ∈ [C] such that M is a direct summand of Y . We have an exact sequence 0 → Γ I (X) → Γ I (Y ) → Γ I (Z), and therefore we obtain which is what we want.
Let (R, m) be a local ring, and let M be an R-module. We denote by ℓℓ(M) the Loewy length of M, which is by definition the infimum of the intergers n ≥ 0 such that m n M = 0. Obviously, ℓℓ(M) is finite if and only if M has finite length. There is a relationship between finite radius and Loewy length: Let (R, m) be a local ring, and let X be a resolving subcategory of mod R. If radius X < ∞, then sup X∈X {ℓℓ(Γ m (X))} < ∞.
Proof. Put r = radius X . By definition, there exists an R-module C such that [C] r+1 contains X . Let X be a module in X . It follows from Lemma 4.2 that the annihilator Ann Γ m (X) contains the ideal (Ann Γ m (R) · Ann Γ m (C)) r+1 . As Γ m (R) and Γ m (C) have finite length, they have finite Loewy length. Set a = ℓℓ(Γ m (R)) and b = ℓℓ(Γ m (C)). Then Ann Γ m (X) contains m (a+b)(r+1) , which means that ℓℓ(Γ m (X)) is at most (a + b)(r + 1). Since the number (a + b)(r + 1) is independent of the choice of X, The following is the essential part of Theorem 3.3(1).
Theorem 4.4. Let (R, m, k) be a local ring of positive dimension. Let X be a resolving subcategory of mod R. If X contains k, then X has infinite radius.
Proof. We claim that R/m i belongs to X for all integers i > 0. Indeed, there is an exact sequence 0 → m i−1 /m i → R/m i → R/m i−1 → 0, and the left term belongs to X as it is a k-vector space. Induction on i shows the claim.
We have ℓℓ(Γ m (R/m i )) = ℓℓ(R/m i ) = i, where the second equality follows from the assumption that dim R > 0. Therefore it holds that sup X∈X {ℓℓ(Γ m (X))} ≥ sup i>0 {ℓℓ(Γ m (R/m i ))} = sup i>0 {i} = ∞, which implies radius X = ∞ by Proposition 4.3. Now, let us prove Theorem 3.3(1). By Remark 4.1, we may assume that (R, p) is a local ring with dim R > 0 and that M is a nonzero R-module in X . The assumption pM = 0 says that M is a nonzero k-vector space, where k = R/p is the residue field of R. Since X is closed under direct summands, k belongs to X . Theorem 4.4 yields radius X = ∞.

Proof of Theorem 3.3(2)(3).
Establishing several preliminary lemmas and propositions are necessary, which will also be used in the proof of Theorem 3.3(4).
We begin with stating an elementary lemma, whose proof we omit. For an ideal I of R, we denote by V (I) (respectively, D(I)) the closed (respectively, open) subset of Spec R defined by I in the Zariski topology, namely, V (I) is the set of prime ideals containing I and D(I) = Spec R \ V (I). For an R-module M we denote by NF(M) the nonfree locus of M, that is, the set of prime ideals p of R such that the R p -module M p is nonfree. It is well-known that NF(M) is a closed subset of Spec R.
The next result builds, out of each module in a resolving subcategory and each point in its nonfree locus, another module in the same resolving subcategory whose nonfree locus coincides with the closure of the point. Such a construction has already been given in [23,Theorem 4.3], but we need in this paper a more detailed version. Indeed, the following lemma yields a generalization of [23, Theorem 4.3].
Lemma 4.6. Let M be an R-module. For every p ∈ NF(M) there exists X ∈ res M satisfying NF(X) = V (p) and depth X q = inf{depth M q , depth R q } for all q ∈ V (p).
Proof. Note that V (p) is contained in NF(M). If V (p) = NF(M), then we can take X := M ⊕ R. Suppose V (p) is strictly contained in NF(M). Then there is a prime ideal r in NF(M) that is not in V (p). Choose an element x ∈ p \ r. By [23, Proposition 4.2], we have a commutative diagram with exact rows The second row shows that N belongs to res M. Since r is in D((x)), it is not in NF(N), and we have V (p) ⊆ NF(N) NF(M). Now we claim that depth N q = inf{depth M q , depth R q } for all q ∈ V (p). Indeed, localizing the above diagram at q and taking long exact sequences with respect to Ext, we get a commutative diagram with exact rows . As x is an element of q, the maps (1),(2) are zero maps, and so are (3),(4). Thus we have a short exact sequence 0 → Ext i (κ(q), (ΩM) q ) → Ext i (κ(q), N q ) → Ext i (κ(q), M q ) → 0 for each integer i. It is easy to see from this that the first equality in the following holds, while the second equality is obtained by applying Lemma 4.5 to the exact sequence Thus the claim follows.
If V (p) = NF(N), then we can take X := N. If V (p) is strictly contained in NF(N), then the above procedure gives rise to an R-module L ∈ res N (hence L ∈ res M) with V (p) ⊆ NF(L) NF(N) NF(M) such that depth L q = inf{depth N q , depth R q } = inf{depth M q , depth R q } for all q ∈ V (p). Since Spec R is a Noetherian space, iteration of this procedure must stop in finitely many steps, and we eventually obtain such a module X as in the lemma.
The next lemma will play a crucial role in the proofs of our theorems. The main idea of the proof is similar to that of Lemma 4.2, but a much closer examination is necessary to be made.
Proof. For each integer i ≤ 1 and R-module L, set a i L = Ann S Ext i S (L, N). Note that a h L = S for all h > m since Ext h S (L, N) = 0. It suffices to prove that a i M ⊇ ( m j=i a j R a j C ) n . Let us proceed by induction on n.
When n = 0, we have M = 0, and the above two ideals coincide with S. Let n = 1. Then M is isomorphic to a direct summand of a finite direct sum of copies of R ⊕ ( l j=0 Ω j C). Hence Ext i S (M, N) is isomorphic to a direct summand of a finite direct sum of copies of Ext for all j ≥ 0. Thus we obtain: which completes the proof of the lemma.
Here we prepare a lemma, which is an easy consequence of Krull's intersection theorem. Now we can prove the following proposition, which will be the base of the proofs of our theorems. Actually, all of them will be proved by making use of this proposition. Proposition 4.9. Let (R, m, k) be a local ring. Let X be a resolving subcategory of mod R. If X contains a module M such that 0 < pd R M < ∞, then one has radius X = ∞.
Proof. Applying Lemma 4.6 to m ∈ NF(M), we find a module X ∈ res M ⊆ X satisfying NF(X) = {m} and depth X = inf{depth M, depth R}. Since M has finite projective dimension, the depth of M is at most that of R. Hence we have depth X = depth M. Note that the subcategory of mod R consisting of R-modules of finite projective dimension is resolving. Since it contains M, it also contains res M. This implies that X has finite projective dimension, and we have pd R X = depth R − depth X = depth R − depth M = pd R M. Thus, replacing M with X, we may assume that M is locally free on the punctured spectrum of R. Taking the n-th syzygy of M where n = pd R M − 1 ≥ 0, we may also assume that the projective dimension of M is equal to 1. Now Ext 1 R (M, R) is a nonzero R-module of finite length, and we can choose a socle element 0 = σ ∈ Ext 1 R (M, R). It can be represented as a short exact sequence: The module N belongs to X , is locally free on the punctured spectrum of R and has projective dimension at most 1. Hence pd R N = 1 if and only if Ext 1 R (N, R) = 0. Applying the functor Hom R (−, R), we get an exact sequence R Hence we obtain an exact sequence Replacing M by N and repeating this process if length(Ext 1 R (N, R)) > 0, we can assume that Ext 1 R (N, R) = 0. Therefore Ext 1 R (M, R) ∼ = k. Since pd R M = 1, we easily get an isomorphism TrM ∼ = k. Taking the transpose of this isomorphism, we see that Trk is isomorphic to M up to free summand (cf. [3, Proposition (2.6)(d)]). It follows that Trk belongs to X .
We claim that TrL is in X for any R-module L of finite length. This is shown by induction on length L. If length L > 0, then there is an exact sequence 0 → L ′ → L → k → 0, and applying [3,Lemma (3.9)] (see also [25,Proposition 3.3(3)]), we have an exact sequence where the equality follows from the fact that R has positive depth. (As R possesses a module of finite positive projective dimension, the depth of R is positive.) The induction hypothesis implies TrL ′ ∈ X , and the above exact sequence shows TrL ∈ X , as desired. Now, assume that we have radius X = r < ∞. We want to deduce a contradiction. There is a ball [C] R r+1 that contains X . Since X contains Tr R (R/m i ) for all i > 0, the ball [C] R r+1 also contains it. Taking the completions, we have Tr R ( R/m i R) ∈ [ C] R r+1 for all i > 0. By virtue of Cohen's structure theorem, there exists a surjective homomorphism S → R such that S is a Gorenstein local ring with dim S = dim R =: d. Let n denote the maximal ideal of S and note that we have R/m i R = R/n i R for any i > 0. Lemma 4.7 gives an inclusion relation . Fix an integer i > 0 and let a 1 , . . . , a m be a system of generators of the ideal n i of S.
There is an exact sequence R ⊕m (a 1 ,...,am) −−−−−→ R → R/n i R → 0 of R-modules. Dualizing this by R induces an exact sequence where the equality follows since depth R = depth R > 0. This makes an exact sequence which yields an injection Hom S ( R, S)/n i Hom S ( R, S) → Ext 1 S (Tr R ( R/n i R), S). Thus and we obtain where the equality follows from Lemma 4.8.
Let I be the kernel of the surjection S → R. Since dim S = d = dim R, the ideal I of S has height zero. Hence there exists a minimal prime ideal p of S which contains I. Since we have a ring epimorphism from the Artinian Gorenstein local ring S p to R p , the R p -module Hom S ( R, S) p = Hom Sp ( R p , S p ) is isomorphic to the injective hull of the residue field of R p , which is in particular nonzero. This implies that p contains the ideal Ann S Hom S ( R, S). Therefore, for some integer 1 ≤ l ≤ d the ideal p contains either Ann S Ext l S ( R, S) or Ann S Ext l S ( C, S). If p contains Ann S Ext l S ( R, S), then we have Ext l Sp ( R p , S p ) = 0, which contradicts the fact that S p is injective as an S p -module. Similarly, we have a contradiction when p contains Ann S Ext l S ( C, S). This contradiction proves that radius X = ∞.

Now we can show the essential part of Theorem 3.3(2)(3).
Theorem 4.10. Let R be a local ring with dim R > 0. Let X be a resolving subcategory of mod R. One has radius X = ∞ if there exists a module M ∈ X with 0 < Gdim R M < ∞ that satisfies either of the following conditions.
(2) res(Ω n M) is a thick subcategory of G(R) for some n ≥ 0.
Proof. According to Proposition 4.9, it suffices to show that X contains a module of projective dimension one.
(1) We consider a construction whose idea essentially comes from the Auslander-Buchweitz approximation theorem [4, Theorem 1.1]. There are exact sequences 0 → where the latter is possible as Ω g M is a totally reflexive module. We make the following pushout diagram.
As Ω −1 Ω g M is totally reflexive, we have Ext 1 R (Ω −1 Ω g M, R) = 0. Hence the second column in the above diagram splits, and we get an exact sequence 0 As Ω g−1 M and Ω −2 Ω g M are in X , the module L is also in X . The second row shows that L has projective dimension at most 1. Since Ω −2 Ω g M is totally reflexive but Ω g−1 M is not, it follows from the second column that L is nonfree. Therefore the projective dimension of L is equal to 1.
(2) Since by assumption res Ω n M is a subcategory of G(R), the module Ω n M is totally reflexive. Hence n ≥ g := Gdim R M.
We claim that res Ω n M = res Ω g M. In fact, since n − g ≥ 0 and Ω n M = Ω n−g (Ω g M), we observe that res Ω n M is contained in res Ω g M. There is an exact sequence of totally reflexive R-modules with each F i being free. Since res Ω n M is assumed to be thick in G(R), decomposing the above exact sequence into short exact sequences of totally reflexive modules, we see that Ω g M belongs to res Ω n M. Therefore res Ω g M is contained in res Ω n M. Thus the claim follows.
Set X = res Ω n M = res Ω g M. Our assumption implies that X is closed under cosyzygies, whence Ω −2 Ω g M ∈ X . By (1), we conclude that X contains a module of projective dimension 1. Now, using Remark 4.1 and Theorem 4.10, we deduce Theorem 3.3(2)(3).

Proof of Theorem 3.3(4).
We use the notion of a module of reducible complexity, which has been introduced by Bergh [8]. Let us recall the definition.
Definition 4.11. The subcategory C r R of mod R is defined inductively as follows. (1) Every module of finite projective dimension belongs to C r R .
The result below is shown in [8, Proposition 2.2(i)], which is implicitly stated in [7]. Proposition 4.12. Let R be a local ring. Every R-module of finite CI-dimension has reducible complexity.
In a resolving subcategory, for any fixed integer n ≥ 0, existence of modules of CIdimension n is equivalent to existence of modules of projective dimension n. Lemma 4.13. Let R be a local ring. Let X be a resolving subcategory of mod R. Suppose that there is a module M ∈ X such that CIdim R M < ∞. Then X contains a module N with pd R N = CIdim R M.
Proof. Since M has finite CI-dimension, it has finite complexity. It follows from Proposition 4.12 that M has reducible complexity. If cx M = 0, then pd M < ∞, and we can take N := M. Hence we may assume cx M > 0. There exists an exact sequence Replacing M with K and iterating this procedure, we can eventually arrive at a module N ∈ X with CIdim N = CIdim M and cx N = 0. The module N has finite projective dimension, and we have pd N = CIdim N = CIdim M. Lemma 4.13 and Proposition 4.9 immediately yield the following theorem. This is not only the essential part of Theorem 3.3(4) but also a generalization of Proposition 4.9.
Theorem 4.14. Let R be a local ring with dim R > 0. Let X be a resolving subcategory of mod R. Suppose that there exists a module M ∈ X with 0 < CIdim R M < ∞. Then the radius of X is infinite.

4.4.
Another proof of Theorem 4.14. In the next theorem, we study the thickness of resolving subcategories of modules of CI-dimension at most zero. This will give another proof of Theorem 4.14.
Theorem 4.15. Let R be a local ring.
(1) Let M be an R-module of CI-dimension at most zero. Then Ω −1 M belongs to res M.
(2) Let X be a resolving subcategory of mod R. Suppose that every module in X has CI-dimension at most zero. Then X is a thick subcategory of G(R). Proof.
(1) Proposition 4.12 implies that M has reducible complexity. Let K 0 = M and let K i+1 be a reduction in complexity of K i for each i ≥ 0. Then we have a short exact [7,Proposition 7.2]), and eventually we must have cx R K e = 0 for some e ≥ 0. Then K e has finite projective dimension. As CIdim R M = 0, we have depth K e = depth M = depth R. Therefore K e is a free module. Note that the above exact sequence also shows that If e = 0, then M is free and we have Ω −1 M = 0 ∈ res M. So we may assume e ≥ 1. We claim that for each 0 ≤ i ≤ e − 1 the cokernel C i of the composite map f i · · · f 1 f 0 : M → K i+1 belongs to res M. Let us show this claim by induction on i. When i = 0, we have C i = Ω t 0 −1 M ∈ res M. Let i ≥ 1. We have the following commutative diagram with exact rows and columns. 0 The induction hypothesis implies that C i−1 belongs to res M. Since Ω t i −1 K i is in res M by (4.15.1), the right column shows that C i is also in res M. Thus the claim follows. Now we have a short exact sequence 0 → M where K e is free and C e−1 is in res M by the claim. Since M is totally reflexive and G(R) is a resolving subcategory of mod R, all the modules in res M are totally reflexive. Hence all modules appearing in the above exact sequence belong to G(R). It is easy to verify that there exists an isomorphism C e−1 ∼ = Ω −1 M ⊕ F with F being free. Consequently, Ω −1 M belongs to res M.
(2) By assumption, X is a subcategory of G(R). Thanks to Proposition 1.7, it is enough to show that X is closed under cosyzygies. Let M be an R-module in X . Then M is of CI-dimension at most zero, and Ω −1 M belongs to res M by (1). Since X is resolving and contains M, it also contains res M. Thus Ω −1 M is in X .  (1) implies that the radius of X is infinite.

Proof of Theorem II
In this section we prove the main Theorem II from Introduction. In fact, we can prove significantly more general statements (Theorems 5.7 and 5.11). In order to state and prove such results we need to first introduce a couple of definitions related to the concept of radius.
Definition 5.1. Let X , Y be subcategories of mod R. We put |X | = add X , and set X * Y = ||X | • |Y||. (The notation "•" was introduced in Definition 2.1.) For an integer r > 0, set Let X , Y, Z be subcategories of mod R. We observe that an object M ∈ mod R is in X * Y if and only if there is an exact sequence 0 → X → E → Y → 0 with X ∈ |X | and Y ∈ |Y| such that M is a direct summand of E. Also, one has (X * Y) * Z = X * (Y * Z) and |X | a * |X | b = |X | a+b for all a, b > 0.
Definition 5.2. For a subcategory X of mod we define size X (respectively, rank X ) to be the infimum of integers n ≥ 0 such that X ⊆ |G| n+1 (respectively, X = |G| n+1 ) for some G ∈ mod R.
It always holds that size X ≤ rank X . Since |X | n ⊆ [X ] n for all n > 0, one has dim X ≥ radius X ≤ size X . If X is resolving, then dim X ≤ rank X .
For an R-module M, we denote by M ⊕ an object in add R M. (1) There is an exact sequence 0 → I ⊕ → Ω R M → Ω R/I M → 0.
Proof. (1) Take a surjection from a free R-module F to M. Then this factors through a surjection F/IF → M. The assertion follows from this.
For n ≥ 0 we denote by Ω n R (mod R) the subcategory of mod R consisting of n-th syzygies of R-modules. For an ideal of R, let Ω n R (mod R/I) be the subcategory of mod R consisting of n-th syzygies of R-modules annihilated by I.
Proof. There is a filtration R = I 0 I 1 · · · I n = 0 of ideals of R such that for each i one has I i /I i+1 ∼ = R/p i with p i ∈ Spec R. Choose a minimal prime q i contained in p i .
Corollary 5.5. Let I be an ideal of R and n ≥ 0 an integer. Then one has size Ω n R (mod R/I) < (n + 1)(size Ω n R/I (mod R/I) + 1). In particular, if size Ω n R/I (mod R/I) is finite, then so is size Ω n R (mod R/I). Proof. This is a consequence of Proposition 5.3(2). Lemma 5.6. Let M be an R-module.
Let x ∈ R be R-regular.
Theorem 5.7. Let (R, m) be a d-dimensional complete local ring with perfect coefficient field. Then one has size Ω d (mod R) < ∞. Hence radius Ω d (mod R) < ∞.
Proof. We use induction on d. When d = 0, we have mod R = |k| ℓℓ(R) , hence size(mod R) < ℓℓ(R) < ∞. Let d ≥ 1. Take a filtration R = I 0 · · · I n = 0 of ideals such that for each i one has Corollary 5.5 implies size Ω d R (mod R/p i ) < ∞. For each R-module M there is an exact sequence 0 → I i+1 M → I i M → I i M/I i+1 M → 0, which gives an exact sequence Thus, we may assume R is a domain.
If R is regular, then Ω d (mod R) = |R| 1 and size Ω d (mod R) = 0 < ∞, so we may assume that R is singular. By [26, 5.15] there is an ideal J ⊆ m with Sing R = V(J) and J Ext d+1 R (mod R, mod R) = 0. Since R is a domain, we find an element 0 = x ∈ J. The induction hypothesis guarantees Ω d−1 R/xR (mod R/xR) ⊆ |G| R/xR n for some R/xR-module G and an integer n > 0. Let M be an R-module and put N = Ω d R M. Note that x is N-regular as d > 0. Hence N is isomorphic to a direct summand of Ω R (N/xN) Hence N/xN is in |G| R n , which implies Ω R (N/xN) ∈ |Ω R G| R n . Therefore N belongs to |Ω R G| R n , and we obtain Ω d (mod R) ⊆ |Ω R G| R n . It now follows that size Ω d (mod R) < ∞.
Corollary 5.9. Let R be a Cohen-Macaulay complete local ring with perfect coefficient field. Then size CM(R) < ∞.
Proposition 5.10. Let R be a Cohen-Macaulay local ring with a canonical module ω.
Theorem 5.11. Let R be a Cohen-Macaulay local ring admitting perfect coefficient field. Assume either that R is complete or that R excellent with an isolated singularity. Then rank CM(R) < ∞.
Proof. The former case follows from Corollary 5.9 and Proposition 5.10(1). The latter case is handled by the former case and an argument similar to [12, Remark 6.5].

Some discussions and open questions
In this section we relate our results to the uniform Auslander condition and discuss some open questions. For a local ring R, Jorgensen and Şega [17] introduced the uniform Auslander condition: (UAC) : There exists an integer n such that for all R-modules M, N with Ext i R (M, N) = 0 for all i ≫ 0 one has Ext i R (M, N) = 0 for all i ≥ n. It is known that this condition is satisfied if the local ring R is a complete intersection, a Golod ring, a Gorenstein ring with mult R = codim R + 2, or a Gorenstein ring with codim R ≤ 4. Here mult R denotes the multiplicity of R. These are proved in [17, Proposition 1.4], [5,Theorem 4.7], [15,Theorem 3.5] and [20,Theorem 3.4], respectively. More information can be found in [11,Appendix A]. On the other hand, there exists an example of a Gorenstein local ring which does not satisfy (UAC); see [17,Theorem].
The result below says that over a Gorenstein local ring the condition (UAC) is closely related to the thickness of resolving subcategories of Cohen-Macaulay modules. Proposition 6.1. Let R be a Gorenstein local ring. Assume every resolving subcategory of mod R contained in CM(R) is a thick subcategory of CM(R). Then R satisfies (UAC).
Proof. Let t ≥ 0 be an integer, and let M, N be R-modules with Ext i R (M, N) = 0 for all i > t. We define a subcategory X of mod R to consist of all Cohen-Macaulay Rmodules X satisfying Ext i R (X, N) = 0 for all i > t. Then X is a resolving subcategory of mod R contained in CM(R). By assumption, X is a thick subcategory of CM(R). Set d = dim R. Since Ω d M is in X , so is Ω −t Ω d M. There is also a connection between thickness of resolving subcategories of totally reflexive modules and closure under R-duals. Here we say that a subcategory X of mod R is closed under R-duals if for each module M in X its R-dual M * is also in X . Proposition 6.2. (1) Let R be local. Let X be a resolving subcategory of mod R contained in G(R). If X is closed under R-duals, then X is a thick subcategory of G(R). (2) Let R be a local hypersurface. Then every resolving subcategory of mod R contained in CM(R) is closed under R-duals. Proof.