A GENERALIZATION OF n-TORSIONFREE MODULES

RYO TAKAHASHI

Abstract. We consider in this paper two approximation theorems for finitely generated modules over a commutative noetherian ring; one is due to Auslander and Bridger, and the other is due to Auslander and Buchweitz. We shall give a result which implies both of these two theorems.

Key Words: Torsionfree, Semidualizing, Cohen-Macaulay approximation, Contravariantly finite.

2000 Mathematics Subject Classification: 13C05, 13C13, 13D02, 16D90.

1. Introduction

In the late 1960s, Auslander and Bridger [2] constructed the notion of a certain approximation, which we call in this paper a spherical approximation. This notion says that each of the modules whose nth syzygies are n-torsionfree is described by using an n-spherical module and a module of projective dimension less than n. On the other hand, about two decades later, the notion of a Cohen-Macaulay approximation was introduced and developed by Auslander and Buchweitz [3]. This notion says that over a Cohen-Macaulay local ring with a canonical module, the category of finitely generated modules is obtained by glueing together the subcategory of maximal Cohen-Macaulay modules and the subcategory of modules of finite injective dimension. Cohen-Macaulay approximations have been playing an important role in commutative algebra. In this paper, we set our sight on these two notions. More precisely, we shall consider and generalize the following two theorems.

Theorem 1 (Auslander-Bridger). The following are equivalent for a finitely generated module M over a commutative noetherian ring R:

(1) Ω^{n}M is n-torsionfree;

(2) There exists an exact sequence 0 → Y → X → M → 0 of finitely generated Rmodules such that Ext_{R}^{i}(X, R) = 0 for 1 ≤ i ≤ n and pd Y < n.

Theorem 2 (Auslander-Buchweitz). Let R be a Cohen-Macaulay local ring with a canonical module. Then for every finitely generated R-module M there exists an exact sequence 0 → Y → X → M → 0 of finitely generated R-modules such that X is maximal Cohen-Macaulay and id Y < ∞.

The detailed version of this paper has been submitted for publication elsewhere.

2. The existence of n-C-spherical approximations

Throughout the present paper, R is always a commutative noetherian ring, and all R-modules are finitely generated. Auslander and Bridger [2] introduced the notion of an n-torsionfree module.

Definition 3. Let n be an integer. An R-module M is called n-torsionfree if Ext_{R}^{i}(TrM, R) = 0 for 1 ≤ i ≤ n.

In this paper, unless otherwise specified, we always denote by n a positive integer, by C an R-module, by (−)^{†} the C-dual functor Hom_{R}(−, C) and by λ_{M} the natural homomorphism M → M^{††} for an R-module M. Note that λ_{R} can be identified with the homothety map R → Hom_{R}(C, C). We can generalize the notion of an n-torsionfree module as follows.

Definition 4. Let M be an R-module. We say that M is 1-C-torsionfree if λ_{M} is a monomorphism. We say that M is n-C-torsionfree, where n ≥ 2, if λ_{M} is an isomorphism and Ext_{R}^{i}(M^{†}, C) = 0 for all 1 ≤ i ≤ n − 2.

We denote by mod R the category of finitely generated R-modules. Let X be a full subcategory of mod R. An R-homomorphism f : X → M is called a right X -approximation of M if X belongs to X and the sequence Hom_{R}(−, X) ⟶^{(−,f)} Hom_{R}(−, M) → 0, where (−, f) = Hom_{R}(−, f), is exact on X. We say that X is contravariantly finite if any X ∈ X has a right X -approximation. For an R-module X, we denote by add X the full subcategory of mod R consisting of all direct summands of finite direct sums of copies of X.

To develop the notion of an n-C-torsionfree module to the utmost extent, we establish the following definition.

Definition 5. We say that C is 1-semidualizing if λ_{R} is a monomorphism and Ext_{R}^{1}(C, C) = 0. We say that C is n-semidualizing, where n ≥ 2, if λ_{R} is an isomorphism and Ext^{i}(C, C) = 0 for all 1 ≤ i ≤ n.

The following proposition, which is essentially proved in [6, Proposition 2.5.1], says that there are a lot of n-semidualizing modules.

Proposition 6. Let R be a Cohen-Macaulay local ring of dimension d ≥ 2 with an isolated singularity. Let I be an ideal of R which is a maximal Cohen-Macaulay R-module. Then λ_{R} is an isomorphism and Ext_{R}^{i}(I, I) = 0 for every 1 ≤ i ≤ d − 2. Hence R is d-I-torsionfree, and I is (d − 2)-semidualizing.

For an R-module M, we define Cdim_{R} M, the add C-resolution dimension of M, to be the infimum of nonnegative integers n such that there exists an exact sequence 0 → C_{n} → C_{n−1} → ⋯ → C_{0} → M → 0 with each C_{i} being in add C. Note that add R-resolution dimension is the same as projective dimension. We make the following definition.

Definition 7. Let M be an R-module.

(1) We say that M is n-spherical if Ext^{i}(M, R) = 0 for all 1 ≤ i ≤ n. We call an exact sequence 0 → Y → X → M → 0 of R-modules an n-spherical approximation if X is n-spherical and pd Y < n.

(2) We say that M is n-C-spherical if Ext^{i}(M, C) = 0 for all 1 ≤ i ≤ n. We call an exact sequence 0 → Y → X → M → 0 of R-modules an n-C-spherical approximation if X is n-C-spherical and Cdim Y < n.

We notice that an n-C-spherical approximation gives a right approximation:

Proposition 8. Define two full subcategories of mod R as follows:

X = { X ∈ mod R | X is n-C-spherical },
Y = { Y ∈ mod R | Cdim Y < n }.

Let 0 → Y → X →^{f} M → 0 be an exact sequence of R-modules with X ∈ X and Y ∈ Y. Then the homomorphism f is a right X -approximation of M.

We give here a lemma.

Lemma 9. Suppose that Ext^{1}(C, C) = 0. An R-module M is 1-C-torsionfree if and only if there is an exact sequence 0 → M → C_{0} → N → 0 such that C_{0} ∈ add C and Ext^{1}(N, C) = 0.

Now, we can state and prove the main result of this section.

Theorem 10. Let C be an n-semidualizing R-module. The following are equivalent for an R-module M:

(1) Ω^{n}M is n-C-torsionfree;

(2) M admits an n-C-spherical approximation.

Proof. Let P_{•} be a projective resolution of M.

(1) ⇒ (2): We have an exact sequence 0 → Ω^{i+1}M → P_{i} → Ω^{i}M → 0 for each i. Set X_{0} = Ω^{n}M. Note that X_{0} is n-C-torsionfree. Lemma 9 implies that there exists an exact sequence 0 → X_{0} → C_{0} → Z_{1} → 0 such that C_{0} ∈ add C and Ext^{1}(Z_{1}, C) = 0. We make the pushout diagram:

0  0
↓  ↓
0 ⟶ X_{0} ⟶ C_{0} ⟶ Z_{1} ⟶ 0
↓  ↓  ∥
0 ⟶ P_{n−1} ⟶ X_{1} ⟶ Z_{1} ⟶ 0
↓  ↓
Ω^{n−1}M  Ω^{n−1}M
↓  ↓
0  0

Since Ext^{1}(Z_{1}, C) = 0 = Ext^{1}(P_{n−1}, C), we have Ext^{1}(X_{1}, C) = 0. If n = 1, then the middle column is a desired exact sequence.

Let n ≥ 2. We can easily check that Z_{1} is (n − 1)-C-torsionfree, and that so is X_{1}. According to Lemma 9, there is an exact sequence 0 → X_{1} → C_{1} → Z_{2} → 0 with C_{1} ∈ add C and Ext^{1}(Z_{2}, C) = 0. We make the pushout diagram:

0  0
↓  ↓
C_{0}  C_{0}
↓  ↓
0 ⟶ X_{1} ⟶ C_{1} ⟶ Z_{2} ⟶ 0
↓  ↓  ∥
0 ⟶ Ω^{n−1}M ⟶ Y_{2} ⟶ Z_{2} ⟶ 0
↓  ↓
0  0

Using the bottom row of the above diagram, we make the pushout diagram:

0  0
↓  ↓
0 ⟶ Ω^{n−1}M ⟶ Y_{2} ⟶ Z_{2} ⟶ 0
↓  ↓  ∥
0 ⟶ P_{n−2} ⟶ X_{2} ⟶ Z_{2} ⟶ 0
↓  ↓
Ω^{n−2}M  Ω^{n−2}M
↓  ↓
0  0

From the first diagram, we immediately get Cdim Y_{2} < 2, and Ext^{2}(Z_{2}, C) = 0 because Ext^{1}(X_{1}, C) = 0 = Ext^{2}(C_{1}, C). Hence Ext^{i}(Z_{2}, C) = 0 for i = 1, 2, and we see from the middle row of the second diagram that Ext^{i}(X_{2}, C) = 0 for i = 1, 2. Thus, if n = 2, then the middle column of the second diagram is a desired exact sequence.

Let n ≥ 3. Then similar arguments to the above claims show that both Z_{2} and X_{2} are (n − 2)-C-torsionfree, and Lemma 9 yields an exact sequence 0 → X_{2} → C_{2} → Z_{3} → 0 such that Ext^{1}(Z_{3}, C) = 0. Similarly to the above, we make two pushout diagrams:

0  0
↓  ↓
Y_{2}  Y_{2}
↓  ↓
0 ⟶ X_{2} ⟶ C_{2} ⟶ Z_{3} ⟶ 0
↓  ↓  ∥
0 ⟶ Ω^{n−2}M ⟶ Y_{3} ⟶ Z_{3} ⟶ 0
↓  ↓
0  0

and

0  0
↓  ↓
0 ⟶ Ω^{n−2}M ⟶ Y_{3} ⟶ Z_{3} ⟶ 0
↓  ↓  ∥
0 ⟶ P_{n−3} ⟶ X_{3} ⟶ Z_{3} ⟶ 0
↓  ↓
Ω^{n−3}M  Ω^{n−3}M
↓  ↓
0  0

If n = 3, then the middle column of the second diagram is a desired exact sequence. If n ≥ 4, then iterating this procedure, we eventually obtain an exact sequence 0 → Y_{n} → X_{n} → M → 0 such that Ext^{i}(X_{n}, C) = 0 for 1 ≤ i ≤ n and Cdim Y_{n} < n.

(2) ⇒ (1): Let 0 → Y → X → M → 0 be an n-C-spherical approximation of M. Since

Cdim Y < n, there exists an exact sequence 0 → C_{n−1} →^{d_{n−1}} C_{n−1} →^{d_{n−2}} ⋯ →^{d_{1}} C_{0} →^{d_{0}} Y → 0.

Put Y_{i} = Im d_{i} for each i. We have exact sequences 0 → Y_{i+1} → C_{i} → Y_{i} → 0 and 0 → Ω^{i+1}M → P_{i} → Ω^{i}M → 0 for each i. The following pullback diagram is obtained:

0  0
↓  ↓
ΩM  ΩM
↓  ↓
0 ⟶ Y ⟶ L ⟶ P_{0} ⟶ 0
∥  ↓  ↓
0 ⟶ Y ⟶ X ⟶ M ⟶ 0
↓  ↓
0  0

The projectivity of P_{0} shows that the middle row splits; we have an isomorphism L ≅ Y ⊕ P_{0}. Adding P_{0} to the exact sequence 0 → Y_{1} → C_{0} → Y → 0, we get an exact sequence 0 → Y_{1} → C_{0} ⊕ P_{0} → Y ⊕ P_{0} → 0. Thus the following pullback diagram is obtained:

0  0
↓  ↓
Y_{1}  Y_{1}
↓  ↓
0 ⟶ X_{1} ⟶ C_{0} ⊕ P_{0} ⟶ X ⟶ 0
↓  ↓  ∥
0 ⟶ ΩM ⟶ Y ⊕ P_{0} ⟶ X ⟶ 0
↓  ↓
0  0

Applying a similar argument to the left column of the above diagram, we get exact sequences 0 → X_{i+1} → C_{i} ⊕ P_{i} → X_{i} → 0 for 0 ≤ i ≤ n − 1, where X_{0} = X and X_{n} = Ω^{n}M. The assumption yields Ext^{i}(X_{0}, C) = 0 = Ext^{i}(C_{0} ⊕ P_{0}, C) for 1 ≤ i ≤ n, hence we have an exact sequence 0 → X_{0}^{†} → (C_{0} ⊕ P_{0})^{†} → X_{1}^{†} → 0 and Ext^{1}(X_{1}, C) = 0 for 1 ≤ i ≤ n − 1. Inductively, for each 0 ≤ i ≤ n − 1 an exact sequence 0 → X_{i}^{†} → (C_{i} ⊕ P_{i})^{†} → X_{i+1}^{†} → 0 is obtained and Ext^{j}(X_{i}, C) = 0 for 1 ≤ j ≤ n − i. We have a commutative diagram

0 ⟶ X_{1} ⟶ C_{0} ⊕ P_{0}
λ_{X_{1}} ↓  ^{λ_{C_{0}⊕P_{0}}} ↓
0 ⟶ X_{1}^{††} ⟶ (C_{0} ⊕ P_{0})^{††} –58–

with exact rows. The assumption says that λ_{R} is injective, and we see that λ_{C} = λ_{R^{†}} is injective. Hence the map λ_{C_{0}⊕P_{0}} is injective, and so is λ_{X_{1}}. Therefore X_{1} is 1-C-torsionfree. If n ≥ 2, then λ_{R} is an isomorphism, and so is λ_{C}. There is a commutative diagram

0  0
↓  ↓
0 ⟶ X_{2} ⟶ C_{1} ⊕ P_{1} ⟶ X_{1} ⟶ 0
λ_{X_{2}} ↓  ^{λ_{C_{1}⊕P_{1}}} ↓  ^{λ_{X_{1}}} ↓
0 ⟶ X_{2}^{††} ⟶ (C_{1} ⊕ P_{1})^{††} ⟶ X_{1}^{††}

↓

0

with exact rows and columns, and λ_{X_{2}} is an isomorphism by the snake lemma. Hence X_{2} is 2-C-torsionfree. If n ≥ 3, then we have a commutative diagram

0 ⟶ X_{3} ⟶ C_{2} ⊕ P_{2} ⟶ X_{2} ⟶  0
λ_{X_{3}} ↓  ^{λ_{C_{2}⊕P_{2}}} ↓≅  ^{λ_{X_{2}}} ↓≅
0 ⟶ X_{3}^{††} ⟶ (C_{2} ⊕ P_{2})^{††} ⟶ X_{2}^{††} ⟶ Ext^{1}(X_{3}^{†}, C) ⟶ 0

with exact rows. From this diagram it follows that λ_{X_{3}} is an isomorphism and Ext^{1}(X_{3}^{†}, C) = 0, which means that X_{3} is 3-C-torsionfree. Repeating a similar argument, we see that X_{i} is i-C-torsionfree for every 1 ≤ i ≤ n. Therefore Ω^{n}M = X_{n} is n-C-torsionfree, and the proof of the theorem is completed.

Theorem 1 is a direct corollary of Theorem 10. Theorem 2 is also a corollary of Theorem 10:

Proof of Theorem 2. If d = 0, then 0 → 0 → M →^{=} M → 0 is a desired exact sequence. Let d ≥ 1. Then ω is d-semidualizing, and Ω^{d}M is d-ω-torsionfree. Hence Theorem 10 guarantees the existence of an exact sequence 0 → Y → X → M → 0 such that X is d-ω-spherical and ωdim Y < d. Therefore X is maximal Cohen-Macaulay. On the other hand, noting that ω is an indecomposable R-module, we have an exact sequence 0 → ω^{l_{d−1}} → ω^{l_{d−2}} → ⋯ → ω^{l_{0}} → Y → 0. Decomposing this into short exact sequences and noting that ω has finite injective dimension, one sees that Y also has finite injective dimension.

3. Modules whose nth syzygies are n-C-torsionfree

We begin with stating the following lemma.

Lemma 11. Let M be an R-module.

(1) If R is 1-C-torsionfree, then so is ΩM.

(2) If R is 2-C-torsionfree, then for each n ≥ 2 the map λ_{Ω^{n}M} is a split monomorphism and the cokernel is isomorphic to Ext^{n}(M, C)^{†}.

For R-modules M, N, we define grade(M, N) by the infimum of integers i such that Ext^{i}(M, N) ≠ 0. One has grade(M, N) = inf{ depth N_{𝔭} | 𝔭 ∈ Supp M }. We state a criterion for Ω^{i}M to be n-C-torsionfree for 1 ≤ i ≤ n in terms of grade, which can be shown by Lemma 11 and induction on n.

Proposition 12. Let C be an R-module such that R is (n − 1)-C-torsionfree.

(1) If Ω^{i}M is i-C-torsionfree for every 1 ≤ i ≤ n, then grade(Ext^{i}(M, C), C) ≥ i − 1 for every 1 ≤ i ≤ n.

(2) The converse also holds if R is n-C-torsionfree.

Now we want to consider the difference between this condition and the condition that Ω^{n}M is n-C-torsionfree.

Lemma 13. Let C be an R-module such that λ_{R} is an isomorphism and Ext^{i}(C, C) = 0 for 1 ≤ i ≤ n. If M is an R-module with Cdim M < ∞, then grade(Ext^{i}(M, C), C) ≥ i for any 1 ≤ i ≤ n.

Using this lemma, we can show that under the assumption that C is n-semidualizing, Ω^{i}M is i-C-torsionfree for 1 ≤ i ≤ n if and only if Ω^{n}M is n-C-torsionfree.

Proposition 14. Let C be an n-semidualizing R-module. The following are equivalent for an R-module M:

(1) Ω^{n}M is n-C-torsionfree;

(2) Ω^{i}M is i-C-torsionfree for every 1 ≤ i ≤ n.

Our next aim is to prove the main result of this section. For this, we introduce the following lemma, which will often be used later.

Lemma 15. Let R be a local ring and r a positive integer. Suppose that λ_{R} is an isomorphism and Ext^{i}(C, C) = 0 for all 1 ≤ i < r. Then the following hold.

(1) depth R ≥ r if and only if depth C ≥ r.

(2) Let R be a Cohen-Macaulay local ring with dim R < r. Then C is a maximal Cohen-Macaulay R-module.

Now we can prove the main result of this section.

Theorem 16. Suppose that R is n-C-torsionfree. Then the following are equivalent:

(1) id_{R_{𝔭}} C_{𝔭} < ∞ for any 𝔭 ∈ Spec R with depth R_{𝔭} ≤ n − 2;

(2) Ω^{n}M is n-C-torsionfree for any R-module M.

Proof. When n = 1, the assertion (1) holds because there is no prime ideal 𝔭 of R satisfying depth R_{𝔭} ≤ n−2, and the assertion (2) holds by Lemma 11(1). In the following, we consider the case where n ≥ 2.

(1) ⇒ (2): Fix an R-module M. Induction hypothesis shows that Ω^{i}M is i-C-torsionfree for 1 ≤ i ≤ n − 1. By Proposition 12, we have grade(Ext^{i}(M, C), C) ≥ i − 1 for 1 ≤ i ≤ n − 1, and it suffices to prove that the inequality grade(Ext^{n}(M, C), C) ≥ n − 1 holds. Let 𝔭 ∈ Spec R. If depth C_{𝔭} ≤ n − 2, then depth R_{𝔭} ≤ n − 2 by Lemma 15(1). The assumption says that id_{R_{𝔭}} C_{𝔭} < ∞, and id_{R_{𝔭}} C_{𝔭} = depth R_{𝔭} ≤ n − 2. Therefore Ext_{R}^{n}(M, C)_{𝔭} ≅ Ext_{R_{𝔭}}^{n}(M_{𝔭}, C_{𝔭}) = 0. Thus we see that grade(Ext^{n}(M, C), C) ≥ n − 1, as desired.

(2) ⇒ (1): When n = 2, Lemma 11(2) implies that Ext_{R_{𝔭}}^{2}(M_{𝔭}, C_{𝔭}) = 0 for all Rmodules M and 𝔭 ∈ Ass C, because Ass(Ext_{R}^{2}(M, C)^{†}) = Supp Ext_{R}^{2}(M, C) ∩ Ass C. The isomorphism λ_{R} : R → Hom(C, C) shows that Ass C coincides with Ass R. Hence, setting M = Ω_{R}^{i}(R/𝔭), one has Ext_{R_{𝔭}}^{i+2}(κ(𝔭), C_{𝔭}) ≅ Ext_{R_{𝔭}}^{2}((Ω_{R}^{i}(R/𝔭))_{𝔭}, C_{𝔭}) = 0 for any 𝔭 ∈ Ass R and any i > 0. Therefore id_{R_{𝔭}} C_{𝔭} < ∞ for 𝔭 ∈ Spec R with depth R_{𝔭} = 0.

Let n ≥ 3. Fix an R-module M. We have an exact sequence 0 → Ω^{n+1}M → P → Ω^{n}M → 0 such that P is a projective R-module. From this we get another exact sequence 0 → (Ω^{n}M)^{†} → P^{†} → (Ω^{n+1}M)^{†} → Ext^{n+1}(M, C) → 0. Decompose this into short exact sequences:

\begin{cases}0 → (Ω^{n}M)^{†} → P^{†} → N → 0, \\ 0 → N → (Ω^{n+1}M)^{†} → Ext^{n+1}(M, C) → 0. \end{cases} (1)

Note from the assumption that both Ω^{n}M and Ω^{n+1}M = Ω^{n}(ΩM) are n-C-torsionfree. Since R is n-C-torsionfree, we see from the first sequence in (1) that there is a commutative diagram

0 ⟶ Ω^{n+1}M ⟶ P ⟶ Ω^{n}M ⟶  0
α↓  ^{λ_{P}} ↓≅  ^{λ_{Ω^{n}M}} ↓≅
0 ⟶ N^{†} ⟶ P^{††} ⟶ (Ω^{n}M)^{††} ⟶ Ext^{1}(N, C) ⟶ 0

with exact rows, and Ext^{i}(N, C) = 0 for 2 ≤ i ≤ n − 2. This diagram shows that α is an isomorphism and Ext^{1}(N, C) = 0. The second sequence in (1) gives an exact sequence 0 → Ext^{n+1}(M, C)^{†} → (Ω^{n+1}M)^{††} →^{β} N^{†} → Ext^{1}(Ext^{n+1}(M, C), C) → 0 and Ext^{i}(Ext^{n+1}(M, C), C) = 0 for 2 ≤ i ≤ n − 2. Since the diagram

Ω^{n}M  Ω^{n}M
λ_{Ω^{n}M} ↓≅  ^{α}↓≅
(Ω^{n}M)^{††} ⟶^{β} N^{†}

commutes, the map β is an isomorphism, and Ext^{n+1}(M, C)^{†} = 0 = Ext^{1}(Ext^{n+1}(M, C), C). Thus we have Ext^{i}(Ext^{n+1}(M, C), C) = 0 for every i ≤ n − 2, which means that the inequality grade(Ext^{n+1}(M, C), C) ≥ n − 1 holds. Therefore, if 𝔭 is a prime ideal of R with depth R_{𝔭} ≤ n − 2, then depth C_{𝔭} ≤ n − 2 by Lemma 15(1), and it follows that 𝔭 does not belong to Supp Ext_{R}^{n+1}(M, C), i.e., Ext_{R_{𝔭}}^{n+1}(M_{𝔭}, C_{𝔭}) = 0. Putting M = Ω_{R}^{i}(R/𝔭), we obtain Ext_{R_{𝔭}}^{n+1+i}(κ(𝔭), C_{𝔭}) ≅ Ext_{R_{𝔭}}^{n+1}((Ω_{R}^{i}(R/𝔭))_{𝔭}, C_{𝔭}) = 0 for any i > 0. This implies that id_{R_{𝔭}} C_{𝔭} < ∞, and the proof is completed.

The lemma below says that over a Gorenstein local ring of dimension d ≥ 2, any n-semidualizing module is free for n ≥ d.

Lemma 17. Let (R, 𝔪, k) be a d-dimensional Gorenstein local ring. If λ_{R} is an isomorphism and Ext^{i}(C, C) = 0 for 1 ≤ i ≤ d, then C ≅ R.

Applying the above lemma, we can get a sufficient condition for R and C to satisfy Theorem 16(1).

Proposition 18. Suppose that R is n-C-torsionfree and that R_{𝔭} is Gorenstein for any 𝔭 ∈ Spec R with depth R_{𝔭} ≤ n−2. Then id_{R_{𝔭}} C_{𝔭} < ∞ for any 𝔭 ∈ Spec R with depth R_{𝔭} ≤ n − 2. (Hence Ω^{n}M is n-C-torsionfree for any R-module M.)

We have studied the case where the nth syzygies of all R-modules are n-C-torsionfree. As the last result of this paper, we give a result concerning when the nth syzygy of a given module is n-C-torsionfree.

Proposition 19. Let M be an R-module, and let C be an R-module such that R is n-C-torsionfree. Suppose that pd_{R_{𝔭}} M_{𝔭} < ∞ for any 𝔭 ∈ Spec R with depth R_{𝔭} ≤ n − 2. Then Ω^{n}M is n-C-torsionfree.

This proposition can be proved by induction on n. Apply Proposition 12, Lemmas 11(1) and 15(1).

References

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[3] Auslander, M.; Buchweitz, R.-O. The homological theory of maximal Cohen-Macaulay approximations. Mem. Soc. Math. France (N.S.) No. 38 (1989), 5–37.

[4] Auslander, M.; Reiten, I. Applications of contravariantly finite subcategories. Adv. Math. 86 (1991), no. 1, 111–152.

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[6] Iyama, O. Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories.

   preprint (2004), arXiv:math.RT/0407052.

[7] Takahashi, R. A common generalization of spherical approximations and Cohen-Macaulay approximations. Preprint (2005).

Department of Mathematics

School of Science and Technology Meiji University Kawasaki 214-8571, JAPAN

  E-mail address: takahasi@math.meiji.ac.jp