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Syzygies of Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings

We study syzygies of (maximal) Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings. We compare these modules to Cohen-Macaulay modules over the endomorphism ring of the maximal ideal. After this comparison, we give several characterizations of almost Gorenstein rings in terms of syzygies of Cohen-Macaulay modules.

(1) One has ΩCM(E) ⊆ ΩCM ′ (R) ⊆ CM(E) ⊆ CM ′ (R) as subcategories of mod(R). (2) Assume that the residue field R/m is infinite. Then ΩCM(E) = ΩCM ′ (R) if and only if R has minimal multiplicity. (3) Assume that the completion R of R is generically Gorenstein. Then CM(E) = ΩCM ′ (R) if and only if R is almost Gorenstein in the sense of [7].
The purpose of this paper is to give a proof of this theorem. As an application, we construct some examples of rings R such that ΩCM(R) is of finite type. We also deduce the following results. We remark that there is an example of a ring R which is not almost Gorenstein such that ΩCM(R) is of finite type (see Example 2.16). As another application, we also obtain a result on the dimension of a triangulated category, which has been introduced by Rouquier [16]. It turns out that each integer is realized as the dimension of the stable category of Cohen-Macaulay modules.  We denote by Q(R) the total quotient ring of R and by R the integral closure of R in Q(R). Then we can identify the ring E(= End R (m)) with m : Q(R) m in the natural way. Note that E is a one-dimensional Cohen-Macaulay semilocal ring contained in R because E is a Cohen-Macaulay R-module. Using the propositions in previous section, we have the following identifications.
Proof. Since R is 1-dimensional and generically Gorenstein, the assumption of Lemma 2.1 (2) is satisfied. Thus ΩCM(R) = F 2 (R). The ring E satisfies the same condition, because the total quotient ring of E coincides with Q(R). Therefore ΩCM(E) = F 2 (E). Finally, we have To prove Theorem 1.3, we will need the following lemma which is observed in Bass's "ubiquity" paper [2]. For the proof, see [14,Lemma 4.9].   Proof. Since M is in ΩCM ′ (R), there is an exact sequence 0 → M β − → R ⊕a γ − → M ′ → 0 with a Cohen-Macaulay R-module M ′ . As M has no free summand, γ is a minimal free cover. In particular, µ(M ′ ) = a. Taking the pushout of homomorphisms β and γ, we have the following diagram.
The surjections R ⊕a → P and P → M ′ show that a = µ(M ′ ) ≤ µ(P ) ≤ a. Thus δ is a minimal free cover and L = ΩP . Lemma 2.5 implies that L has no free summand and hence L is in ΩCM ′ (R).
Lemma 2.7. The R-modules m and E belong to ΩCM ′ (R).
Proof. Take a non zerodivisor x of R. Then R/(x) has depth zero. Therefore there is a short exact sequence Applying horseshoe lemma to this sequence, we have an exact sequence 0 → Ωk → R ⊕n → ΩN → 0 for some n ≥ 1. Since ΩN is Cohen-Macaulay and Ωk = m, it follows that m ∈ ΩCM(R). This implies that m is reflexive over R by Lemma 2.2. By [1, Corollary 5.7], m has no free summand. It yields that In particular, E is in ΩCM(R). If E has a nonzero free summand, then, since m is reflexive and E = Hom R (m, R), m must have a nonzero free summand. This is a contradiction. As a consequence, m and E ∈ ΩCM ′ (R). The R-module m plays an important role to prove the theorem. Next lemma says that m can be regarded as a "cogenerator" of ΩCM ′ (R). Lemma 2.8. Let M be an R-module in ΩCM ′ (R). Then there is an exact sequence We have the following commutative diagram with exact rows and columns.
Lemma 2.9. Let A be a 1-dimensional Cohen-Macaulay generically Gorenstein ring and I be an ideal of A containing a non zerodivisor such that Hom A (I, I) is naturally isomorphic to A. If I is reflexive, then it is isomorphic to A.
Proof. Denote by σ the natural homomorphism I → Hom A (Hom A (I, A), A), which maps a to [φ → φ(a)] for a ∈ I and φ ∈ Hom A (I, A). Then σ is isomorphism by the reflexivity of I. Let ev : I ⊗ A Hom A (I, A) → A be the evaluation homomorphism, which maps a ⊗ φ to φ(a). Let K be the kernel of ev, C be the cokernel of ev, and T be the image of ev. we have the following exact sequences. Consider the composition Ψ • Hom A (ev, A) : A → Hom A (I, I). Then it maps 1 A to id I . Therefore Ψ • Hom A (ev, A) is an isomorphism by the assumption. It yields that Hom A (ev, A) and f are isomorphisms and hence Ext 1 A (C, A) = 0. Since A is 1-dimensional, we obtain that C = 0. In this case, ev is a surjection and we can see that there is a surjective map I → A. This yields that I is isomorphic to A. Since Hom E (m, m) = End R (m) = E, we can apply Lemma 2.9 to see that m is isomorphic to E. By [15, Proposition 2.5], it follows that R has minimal multiplicity.
Conversely, assume that R has minimal multiplicity. Then m is isomorphic to E by using [  Proof. The "only if" part is clear. Now we assume E † ∈ ΩCM ′ (R). Let M be in CM(E). Taking a free cover of M † over E, we get an exact sequence 0 → N → E ⊕n → M † → 0 with some E-module N. Since M † , E ∈ CM(R), N is also in CM(R). Applying (−) † to this sequence, we have an exact sequence 0 → M → (E † ) ⊕n → N † → 0. Using Lemma 2.6, M is in ΩCM ′ (R). This shows that CM(E) = ΩCM ′ (R).
If the completion R of R is generically Gorenstein, then R has a canonical module by [7,Proposition 2.7]. In this situation, we see in the next lemma that the condition CM(E) = ΩCM ′ (R) is stable under flat local extension. Proof. Let E ′ = End R ′ (m ′ ). Note that R ′ is also generically Gorenstein by [7,Proposition 2.12]. In addition, ω ⊗ R R ′ is a canonical module of R ′ . Therefore, by Lemma 2.10, The later condition is equivalent to the equations Ext 1 is faithfully flat and the Auslander transpose is preserved under a base change.
Using the above lemma, we can replace R with the completion R. We have one more equivalent condition to being CM(E) = ΩCM ′ (R). Lemma 2.12. Assume that the completion R of R is generically Gorenstein. Then E † ∈ ΩCM ′ (R) if and only if E † ∼ = m.
Proof. Thanks to Corollary 2.11, we can assume that R is complete. If E † ∼ = m, then we have E † ∈ ΩCM ′ (R). Conversely, we assume E † ∈ ΩCM ′ (R). Using Lemma 2.8, we get an exact sequence of modules in CM(E). By the Krull-Schmidt theorem on R, we have a unique decomposition m = m 1 ⊕ · · · ⊕ m n , where m i are indecomposable R-modules. Then we obtain E = End R (m 1 ) × · · · × End R (m n ) as an R-algebra. The components E i = End R (m i ) of E are local rings because of the indecomposablity of m i . Set n i the maximal ideal of E corresponding to the maximal ideal of E i . Note that the localization (E † ) n i = (Hom R (E, ω)) n i is the canonical module of E i and the localization m n i is equal to (m i ) n i . Thus, after localizing at n i , the sequence (2.12.1) becomes split exact and (E † ) n i is a direct summand of (m i ) ⊕m n i . The modules (E † ) n i and (m i ) n i are both indecomposable. Hence we obtain an isomorphism (E † ) n i ∼ = (m i ) n i by the Krull-Schmidt theorem. The homomorphism α is a split injection, since it becomes a split injection after localizing at n i for all i = 1, . . . , n.
The following lemma will be used to prove Theorem 1.3 (3).
Lemma 2.13. Let A be a ring with total quotient ring T , A be the integral closure of A in T , and X be an A-submodule of A containing A. If there is an isomorphism φ : A → X of A-modules, then X = A.
Proof. Let i : A → X be the inclution homomorphism. Then φ −1 • i : A → A is an endmorphism of A. Hence it is a multiplication map by r for some r ∈ A. Since r = φ −1 • i : A → A is injective, 1/r is in T . We have 1 = i(1) = φ(r) = rφ(1) in A and hence 1/r = φ(1) ∈ A. It means that 1/r is integral over A. Therefore we have an equation of integral dependence (1/r) n + a 1 (1/r) n−1 + · · · + a n = 0, where a i ∈ A for all i = 1, . . . , n. Multiplying r n , we get 1 + r(a 1 + · · · + a n r n−1 ) = 0. This equation yields that r is a unit of A. Thus the endmorphism r = φ −1 • i : A → A is an automorphism, i is an isomorphism, and A = X.
Assume that R is complete and has a inifite residue field. Then there is an R-submodule K of Q(R) such that R ⊂ K ⊂ R, and as an R-module, K is a canonical module of R; see [7,Corollary 2.9]. Using this module K, we have the following theorem, which essensially proves Theorem 1.3 (3).
Theorem 2.14. Assume that R is complete and the residue field R/m is infinite. Let K be an R-submodule of Q(R) such that R ⊂ K ⊂ R, and as an R-module, K is a canonical module of R. Then the following conditions are equivalent.
(1) R is almost Gorenstein. Proof. (1) =⇒ (2): If R is Gorenstein, then there is nothing to prove. Thus we assume that R is almost Gorenstein but not Gorenstein. In this case, [7,Theorem 3.16] says that (E =)m : m = K : m. (4) ⇐⇒ (5) and (5)  In this section, as an application of our results obtained in the preceding sections, we consider the dimension of the stable category of Cohen-Macaulay R-modules in the case where R is Gorenstein. We shall construct for each integer n an example of a Gorenstein ring R such that dim CM(R) = n.
Assume that R is a complete domain. Then E is a finite product of local rings because R is henselian and E is module-finite over R. Since E is a subring of R, it is also a domain. Thus E is a complete local domain. Using this argument inductively, we obtain a family {(R n , m n )} n of local rings with R n = E if n = 1, End R n−1 (m n−1 ) otherwise.
If R is Gorenstein, then as is well known, CM(R) is a Frobenius exact category. Thus its stable category CM(R) is a triangulated category; see [8,Chapter 1] for details. In this case, we can consider the dimension of the triangulated category CM(R) in the sense of Rouquier [16].
Proposition 3.1. Assume that R is a complete Gorenstein domain. If CM(R n ) is of finite type, then dim CM(R) ≤ n − 1.
We claim that CM(R) = G n . In fact, let M 1 ∈ CM(R). To show that M 1 ∈ G n , we may assume that M 1 is in CM ′ (R). Since R is Gorenstein, CM ′ (R) = ΩCM ′ (R) and hence CM ′ (R) = CM(R 1 ) by Theorem 1.3 (1). Therefore M 1 ∈ CM(R 1 ). Taking a minimal free resolution of M 1 over R 1 , we have an exact sequence 0 → M 2 → R ⊕ 1 → M → 0. Here M 2 is in ΩCM ′ (R 1 ) by Lemma 2.5 and thus in CM(R 2 ) by Theorem 1.3 (1). Applying this argument inductively, we obtain R-modules M i ∈ CM(R i ) with exact sequences (3.1.1) 0 → M i → R ⊕ i−1 → M i−1 → 0 for each integer i ≥ 2. Since M n ∈ CM(R n ), it is obvious that M n ∈ G 1 . Then induction shows that M i ∈ G n−i+1 by considering the exact triangle in CM(R) induced by the sequence (3.1.1) for each i. In particular, M 1 ∈ G n . This claim yields that dim CM(R) ≤ n − 1.