Two generalizations of Auslander-Reiten duality and applications
aa r X i v : . [ m a t h . A C ] J u l TWO GENERALIZATIONS OF AUSLANDER–REITEN DUALITY AND APPLICATIONS
ARASH SADEGHI AND RYO TAKAHASHI
Dedicated to Professor Mohammad T. Dibaei on the occasion of his retirement A BSTRACT . This paper extends Auslander–Reiten duality in two directions. As an application, we obtain variouscriteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of knownresults on the Auslander–Reiten conjecture.
1. I
NTRODUCTION
The
Auslander–Reiten conjecture [6] is one of the most celebrated conjectures in the representation theory ofalgebras. This long-standing conjecture is known to hold true over several classes of algebras, including algebrasof finite representation type [6] and symmetric artin algebras with radical cube zero [16]. This conjecture isclosely related to other conjectures such as the
Nakayama conjecture [6, 22] and the
Tachikawa conjecture [7, 24].Although the Auslander–Reiten conjecture was initially proposed over artin algebras, it remains meaningful forarbitrary commutative noetherian rings:
Conjecture 1.1 (Auslander–Reiten) . Let R be a commutative noetherian ring R and let M be a finitely generated R -module. If Ext iR ( M , M ) = = Ext iR ( M , R ) for all i ≥
1, then M is projective.Auslander, Ding and Solberg [5] proved that Conjecture 1.1 holds for any complete intersection local rings. Re-cently there has been various progress towards Conjecture 1.1; see [2, 9–13, 15, 17, 18, 23] for instance. UsingAuslander–Reiten duality, Araya [1] proved that if all Gorenstein local rings of dimension at most one satisfy theAuslander–Reiten conjecture, then so do all Gorenstein local rings.In this paper, we extend Auslander–Reiten duality in two directions; the following two theorems are includedin the main results of this paper. Theorem 1.2.
Let R be a commutative noetherian ring. Let n ≥ be an integer. Let M , N be finitely generatedR-modules. Assume that M is ( n + ) -torsionfree (e.g. totally reflexive) and NF ( M ) ∩ NF ( N ) ⊆ Y n ( R ) . (1) There exists an exact sequence of R-modules → Ext n − R ( Hom R ( M , N ) , R ) → Ext n − R ( N , M ) → Ext nR ( Hom R ( M , N ) , R ) → Ext nR ( Hom R ( M , N ) , R ) . (2) For all integers i ≤ n − one has an isomorphism of R-modules Ext iR ( Hom R ( M , N ) , R ) ∼ = Ext iR ( N , M ) . Theorem 1.3.
Let ( R , m , k ) be a Cohen–Macaulay local ring of dimension d ≥ with canonical module ω . LetM , N be maximal Cohen–Macaulay R-modules such that NF ( M ) ∩ NF ( N ) ⊆ { m } . (1) One has an isomorphism of R-modules
Hom R ( M , N ) ∨ ∼ = Ext d − R ( M ∗ , N † ) . (2) Suppose that M is locally totally reflexive (e.g. locally free) on the punctured spectrum of R. Then for all ≤ i ≤ d − one has an isomorphism of R-modules Ext iR ( M , N ) ∨ ∼ = Ext ( d − ) − iR ( M ∗ , N † ) . Mathematics Subject Classification.
Key words and phrases.
Auslander–Reiten duality, n -torsionfree module, Ext module, Cohen–Macaulay ring, Gorenstein ring, maximalCohen–Macaulay module, Auslander–Reiten conjecture, totally reflexive module, G-dimension.Sadeghi’s research was supported by a grant from IPM. Takahashi was partly supported by JSPS Grant-in-Aid for Scientific Research16K05098 and JSPS Fund for the Promotion of Joint International Research 16KK0099. Let us explain the notation and terminology used above. For an integer n , we denote by Y n ( R ) the set of primeideals p with depth R p ≥ n . An n -torsionfree module is a module M with Ext iR ( Tr M , R ) = ≤ i ≤ n , whereTr M stands for the Auslander transpose of M . We denote by NF ( M ) the non-free locus of an R -module M , thatis, the set of prime ideals p such that M p is not R p -free. For R -modules M , N we denote by Hom R ( M , N ) thestable Hom module, namely, the quotient of Hom R ( M , N ) by the homomorphisms factoring through projectivemodules. Also, ( − ) ∨ = Hom R ( − , E R ( k )) , ( − ) ∗ = Hom R ( − , R ) , and ( − ) † = Hom R ( − , ω ) are the Matlis, algebraicand canonical duals, respectively.Both of the above two theorems recover the following celebrated Auslander–Reiten duality theorem for d ≥ d ≥ Corollary 1.4 (Auslander–Reiten duality) . Let ( R , m ) be a d-dimensional Gorenstein local ring. Let M , N bemaximal Cohen–Macaulay R-modules such that NF ( M ) ∩ NF ( N ) ⊆ { m } . Then for all i ∈ Z there is an isomorphismof Tate cohomology modules c Ext iR ( N , M ) ∨ ∼ = c Ext ( d − ) − iR ( M , N ) . There are also many other applications of the above theorems, including an improved version of a main theoremof Celikbas and Takahashi [10, Theroem 1.1], and the following result. Recall that a commutative noetherian ringis called generically Gorenstein if R p is Gorenstein for all p ∈ Ass R . Corollary 1.5.
Let R be a generically Gorenstein local ring of depth t, and let ≤ n ≤ t be an integer. Let M be afinitely generated R-module such that NF ( M ) ⊆ Y n ( R ) . Then M is free if it satisfies the following three conditions. (1) Ext iR ( M , M ) = for n − ≤ i ≤ t − . (2) Ext iR ( M , R ) = for i > . (3) Ext iR ( Hom R ( M , M ) , R ) = for n ≤ i ≤ t, or Hom R ( M , M ) has finite G-dimension. It is worth mentioning that Corollary 1.5 simultaneously generalizes several known results on the Auslander–Reiten conjecture, including the theorems of Araya [1, Corollary 4], Ono and Yoshino [23, Theorem] and Araya,Celikbas, Sadeghi and Takahashi [2, Corollary 1.6], and substantial parts of the theorems of Huneke and Leuschke[17, Theorem 1.3], Goto and Takahashi [15, Theorem 1.5(2)] and Dao, Eghbali and Lyle [13, Theorem 3.16].Furthermore, Corollary 1.5 yields the following result.
Corollary 1.6.
Let R be a noetherian normal local ring of depth t. A finitely generated R-module M is free, ifeither of the following conditions holds. (1) Ext iR ( M , M ) = Ext jR ( M , R ) = Ext hR ( Hom R ( M , M ) , R ) = for all ≤ i ≤ t − , j ≥ and ≤ h ≤ t. (2) Ext iR ( M , M ) = Ext jR ( M , R ) = for all ≤ i ≤ t − and j ≥ , and that Hom R ( M , M ) has finite G-dimension. In Section 2, we extend Auslander–Reiten duality. We prove our main theorems in this section that yieldTheorems 1.2 and 1.3 as special cases, and recover Corollary 1.4. In Section 3, we apply our theorems to givevarious criteria for freeness of modules in relation to Conjecture 1.1, including Corollaries 1.5 and 1.6. Amongother things, we generalize the main theorem of [2] in this section (Corollary 3.9).C
ONVENTION
Throughout this paper, we assume that all rings are commutative noetherian and all modules are finitely gener-ated. Let R be a ring. We denote by ( − ) ∗ the R -dual Hom R ( − , R ) . If R is local, then m , k , d , t , ( − ) ∨ always standfor the maximal ideal of R , the residue field of R , the (Krull) dimension of R , the depth of R , and the Matlis dualHom R ( − , E R ( k )) , respectively. Whenever R admits a canonical module ω , we denote by ( − ) † the canonical dualHom R ( − , ω ) . For an R -module M , we denote by Ω M and Tr M the (first) syzygy and the (Auslander) transpose of M . We refer the reader to [4] for details on syzygies, transposes, n -torsionfree modules, totally reflexive modulesand G-dimension. For the definition and basic properties of Tate cohomology, we refer the reader to [25, Section7]. We say that an R -module M is locally free (resp. totally reflexive, of finite G-dimension) on a subset A ofSpec R , provided that M p is free (resp. totally reflexive, of finite G-dimension) as an R p -module for all p ∈ A . Foran integer n and an R -module K , let X n ( K ) (resp. Y n ( K ) ) denote the set of prime ideals p of R such that depth R p K p is at most (resp. at least) n . WO GENERALIZATIONS OF AUSLANDER–REITEN DUALITY AND APPLICATIONS 3
2. E
XTENDING A USLANDER –R EITEN DUALITY TO n - TORSIONFREE MODULES
In this section, we extend Auslander–Reiten duality to ( n + ) -torsionfree modules for n ≥
1. First we presentsome examples of such modules.
Example 2.1.
Let n ≥ R -module M is ( n + ) -torsionfree in each of the following cases.(1) M is totally reflexive.(2) R is local, M is locally totally reflexive on the punctured spectrum, depth M ≥ n ≥ n − R ( M ∗ , R ) = M is locally of finite G-dimension on X n − ( R ) and M is a ( n + ) st syzygy.(4) R is Cohen–Macaulay and local, M is locally of finite G-dimension on the punctured spectrum, and M is asyzygy of a maximal Cohen–Macaulay module. Proof. (1) By the definition of total reflexivity Ext > R ( Tr M , R ) =
0. Thus M is i -torsionfree for all i ≥ R p M p ≥ min { n , depth R p } for p ∈ Spec R . The assumption depth R M ≥ n especially says n ≤ d ,and M p is totally reflexive for p ∈ X n − ( R ) . Hence M is n -torsionfree by [14, Proposition 2.4]. As n ≥
2, we haveExt n + R ( Tr M , R ) ∼ = Ext n − R ( M ∗ , R ) =
0. Therefore M is ( n + ) -torsionfree.(3) This is a consequence of [21, Theorem 43].(4) Write M = Ω N with N maximal Cohen–Macaulay. Note that N is also locally of finite G-dimension on thepunctured spectrum. By [14, Proposition 2.4] the module N is a d th syzygy, and hence M is a ( d + ) st syzygy.The assertion now follows from (3). (cid:4) To prove our first main result and some other results given later, we establish a lemma.
Lemma 2.2.
Let n ≥ be an integer. Let M , N , K be R-modules such that NF ( M ) ∩ NF ( N ) ⊆ Y n ( K ) . (1) There exists an exact sequence of R-modules → Ext n − R ( Hom R ( M , N ) , K ) → Ext n − R ( M ∗ ⊗ R N , K ) → Ext nR ( Hom R ( M , N ) , K ) → Ext nR ( Hom R ( M , N ) , K ) . (2) For all integers i ≤ n − one has an isomorphism of R-modules Ext iR ( Hom R ( M , N ) , K ) ∼ = Ext iR ( M ∗ ⊗ R N , K ) .Proof. According to [4, Theorem (2.8)], there is an exact sequence ♦ Tor R ( Tr M , N ) → M ∗ ⊗ R N f −→ Hom R ( M , N ) → Tor R ( Tr M , N ) → . Set H = Ker f , C = Tor R ( Tr M , N ) and L = Im f . Note that the support of H is contained in that of Tor R ( Tr M , N ) .Since NF ( M ) ∩ NF ( N ) ⊆ Y n ( K ) , we see that Ext iR ( C , K ) = = Ext iR ( H , K ) for all i < n by [4, Lemma (4.5)]. Du-alizing the above exact sequence by K yields isomorphisms Ext iR ( Hom R ( M , N ) , K ) ∼ = Ext iR ( L , K ) and Ext jR ( L , K ) ∼ = Ext jR ( M ∗ ⊗ R N , K ) for all i ≤ n − j ≤ n −
1, and an exact sequence0 → Ext n − R ( Hom R ( M , N ) , K ) → Ext n − R ( L , K ) → Ext nR ( C , K ) → Ext nR ( Hom R ( M , N ) , K ) . It remains to note from [27, Lemma (3.9)] that Hom R ( M , N ) ∼ = Tor R ( Tr M , N ) = C . (cid:4) Let K be an R -module. An R -module M is called n-torsionfree with respect to K provided that Ext iR ( Tr M , K ) = ≤ i ≤ n . Note that M is n -torsionfree if and only if it is n -torsionfree with respect to R . For R -modules M , N we denote by grade ( M , N ) the smallest non-negative integer n such that Ext iR ( M , N ) =
0. The following theoremis the first main result of this paper, which includes Theorem 1.2 from the Introduction.
Theorem 2.3.
Let n ≥ be an integer. Let M , N , K be R-modules. Assume that M is ( n + ) -torsionfree withrespect to K, and that NF ( M ) ∩ NF ( N ) ⊆ Y n ( K ) . Then the following hold. (1) There exists an exact sequence of R-modules → Ext n − R ( Hom R ( M , N ) , K ) → Ext n − R ( N , M ⊗ R K ) → Ext nR ( Hom R ( M , N ) , K ) → Ext nR ( Hom R ( M , N ) , K ) . (2) For all integers i ≤ n − one has an isomorphism of R-modules Ext iR ( Hom R ( M , N ) , K ) ∼ = Ext iR ( N , M ⊗ R K ) . ♦ In fact, one can add “0 → ” at the beginning of the exact sequence, that is, Ker f ∼ = Tor R ( Tr M , N ) . ARASH SADEGHI AND RYO TAKAHASHI
Proof.
Thanks to Lemma 2.2, it suffices to show that Ext iR ( M ∗ ⊗ R N , K ) ∼ = Ext iR ( N , M ⊗ R K ) for all integers i ≤ n −
1. There are two spectral sequences converging to the same point: E pq = Ext pR ( Tor Rq ( M ∗ , N ) , K ) = ⇒ H p + q , F pq = Ext pR ( N , Ext qR ( M ∗ , K )) = ⇒ H p + q . As grade ( Tor Ri ( M ∗ , N ) , K ) ≥ n for all i > E pq = q > p < n ,and obtain an isomorphism E i ∼ = H i for all integers i ≤ n . Since Ext iR ( Tr M , K ) = ≤ i ≤ n +
1, we haveExt iR ( M ∗ , K ) = ≤ i ≤ n −
1. Hence F pq = ≤ q ≤ n −
1, and we get F i ∼ = H i for all integers i ≤ n − n + ≥
2, the module M is 2-torsionfree with respect to K , and M ⊗ R K ∼ = Hom R ( M ∗ , K ) by [4, Proposition(2.6)]. Thus there are isomorphismsExt iR ( M ∗ ⊗ R N , K ) = E i ∼ = H i ∼ = F i = Ext iR ( N , Hom R ( M ∗ , K )) ∼ = Ext iR ( N , M ⊗ R K ) for all integers i ≤ n − (cid:4) We obtain the following corollary as a consequence of the above theorem.
Corollary 2.4.
Let R be a local ring, and let M , N be R-modules. Assume that M is ( t + ) -torsionfree and that NF ( M ) ∩ NF ( N ) ⊆ Y t ( R ) . (1) If Ext tR ( Hom R ( M , N ) , R ) = Ext t − R ( N , M ) = , then Hom R ( M , N ) = . (2) Suppose that
Hom ( M , N ) has depth at least and finite G-dimension. Then there exists an isomorphism Ext t − R ( N , M ) ∼ = Ext tR ( Hom R ( M , N ) , R ) . In particular,
Ext t − R ( N , M ) = if and only if Hom R ( M , N ) = .Proof. (1) If t =
0, then Hom R ( Hom R ( M , N ) , R ) =
0, and Hom R ( M , N ) = R ( M , N ) =
0. Let t >
0. Theorem 2.3(1) shows Ext tR ( Hom R ( M , N ) , R ) =
0, while grade R Hom R ( M , N ) ≥ t by[4, Lemma (4.5)] or [8, Proposition 1.2.10(a)]. Hence grade R Hom R ( M , N ) > t . Thus, if Hom R ( M , N ) is nonzero,then its annihilator contains an R -sequence of length more than t = depth R , which cannot occur. We must haveHom R ( M , N ) = iR ( Hom R ( M , N ) , R ) = i = t − , t . We have Gdim R ( Hom R ( M , N )) = depth R − depth R ( Hom R ( M , N )) ≤ t −
2, which implies that Ext iR ( Hom R ( M , N ) , R ) = i > t − (cid:4) Remark 2.5.
In fact, one can generalize Corollary 2.4 to a ( t + ) -torsionfree module M with respect to a module K satisfying NF ( M ) ∩ NF ( N ) ⊆ Y n ( K ) with n = depth R K , assuming that K is semidualizing for (2). We leave it tothe reader as an exercise.Corollary 2.4 immediately recovers Corollary 1.4 in the case where dim R ≥
2, as follows. Thus our Theorem2.3 (Theorem 1.2) recovers the Auslander–Reiten duality theorem in dimension at least two.
Corollary 2.6 (Auslander–Reiten duality in dimension at least two) . Let ( R , m ) be a Gorenstein local ring withd ≥ . Let M , N be maximal Cohen–Macaulay R-modules with NF ( M ) ∩ NF ( N ) ⊆ { m } . Then for each i ∈ Z thereis an isomorphism c Ext iR ( N , M ) ∨ ∼ = c Ext ( d − ) − iR ( M , N ) . Proof. As d ≥
2, one can apply Corollary 2.4(2) to get Ext d − R ( N , M ) ∼ = Ext dR ( Hom R ( M , N ) , R ) , which and [8,Corollary 3.5.9] show Ext d − R ( N , M ) ∨ ∼ = Hom R ( M , N ) . Substituting Ω i − ( d − ) N for N (and using basic facts onTate cohomology [25, Section 7]), we obtain c Ext iR ( N , M ) ∨ ∼ = Ext d − R ( Ω i − ( d − ) N , M ) ∨ ∼ = Hom R ( M , Ω i − ( d − ) N ) ∼ = c Ext ( d − ) − iR ( M , N ) , which give an isomorphism as in the assertion. (cid:4) We prepare a lemma for the proof of the second main result of this paper, which is also be used later. The proofof the lemma is analogous to that of [15, Lemma 2.3].
Lemma 2.7.
Let R be a Cohen–Macaulay local ring with canonical module ω . Let M be an R-module and N amaximal Cohen–Macaulay R-module. Let ≤ m ≤ d be an integer such that NF ( M ) ∩ NF ( N ) ⊆ Y m ( R ) . Then Ext iR ( M , N † ) ∼ = Ext iR ( M ⊗ R N , ω ) for all integers ≤ i ≤ m. We are now ready to present our second main result, which includes Theorem 1.3 from the Introduction.
WO GENERALIZATIONS OF AUSLANDER–REITEN DUALITY AND APPLICATIONS 5
Theorem 2.8.
Let ( R , m ) be a Cohen–Macaulay local ring with canonical module ω . Let M , N be R-modules suchthat NF ( M ) ∩ NF ( N ) ⊆ { m } . Assume that N is maximal Cohen–Macaulay. (1) There is an isomorphism
Hom R ( M , N ) ∨ ∼ = Ext d + R ( Tr M , N † ) . Therefore, if d ≥ , then one has Hom R ( M , N ) ∨ ∼ = Ext d − R ( M ∗ , N † ) . (2) Suppose that M is locally totally reflexive on the punctured spectrum of R. Then there is an isomorphism
Ext iR ( M , N ) ∨ ∼ = Ext d + − iR ( Tr M , N † ) for all ≤ i ≤ d. In particular, for all ≤ i ≤ d − one has Ext iR ( M , N ) ∨ ∼ = Ext ( d − ) − iR ( M ∗ , N † ) . Proof.
We claim that E : = Ext jR ( Tr M , N † ) has finite length for all j ≥ d . Indeed, let p be a non-maximal primeideal. As NF ( M ) ∩ NF ( N ) ⊆ { m } , either M p or N p is free. If M p is free, then so is Tr M p , and E p =
0. If N p is free,then ( N † ) p is a direct sum of copies of ω p , which has injective dimension at most dim R p < d , and hence E p = Ri ( Tr M , N ) has finite length for all i >
0. There are isomorphismsHom R ( M , N ) ∼ = Tor R ( Tr M , N ) ∼ = Ext dR ( Tor ( Tr M , N ) , ω ) ∨ ∼ = Ext d + R ( Tr M , N † ) ∨ , where the first two isomorphisms are obtained by [27, Lemma (3.9)] and [8, Corollary 3.5.9], respectively, whilethe last isomorphism is shown to hold by the spectral sequence argument in [2, 2.4]. By the above claim we seethat Hom R ( M , N ) ∨ ∼ = Ext d + R ( Tr M , N † ) ∨∨ ∼ = Ext d + R ( Tr M , N † ) by [8, Proposition 3.2.12(c)].(2) We show the assertion by induction on i . There are isomorphismsExt R ( M , N ) ∼ = Γ m ( Ext R ( M , N )) ∼ = Γ m ( Tr M ⊗ R N ) ∼ = Ext dR ( Tr M ⊗ R N , ω ) ∨ ∼ = Ext dR ( Tr M , N † ) ∨ . Let us explain each of the above isomorphisms. As NF ( M ) ∩ NF ( N ) ⊆ { m } and M is locally totally reflexiveon the punctured spectrum, Ext R ( M , N ) has finite length. Hence the first isomorphism holds. There is an exactsequence 0 → Ext R ( M , N ) → Tr M ⊗ R N → Hom R (( Tr M ) ∗ , N ) by [4, Proposition (2.6)], while Hom R (( Tr M ) ∗ , N ) has positive depth by [8, Exercise 1.4.19]. Thus the second isomorphism holds. The third and fourth isomor-phisms follow from [8, Corollary 3.5.9] and Lemma 2.7, respectively. Consequently, Ext R ( M , N ) ∨ is isomorphicto Ext dR ( Tr M , N † ) ∨∨ , which is isomorphic to Ext dR ( Tr M , N † ) by the claim given at the beginning of the proofand [8, Proposition 3.2.12(c)]. Thus the case i = i ≥
2. Applying the induction hypothesis to Ω M , we obtain isomorphismsExt iR ( M , N ) ∨ ∼ = Ext i − R ( Ω M , N ) ∨ ∼ = Ext d + − ( i − ) R ( Tr Ω M , N † ) ∼ = Ext d + − iR ( Ω Tr Ω M , N † ) . There is an exact sequence 0 → Ext R ( M , R ) → Tr M → Ω Tr Ω M → R ( M , R ) has finite length and N † is maximal Cohen–Macaulay, Ext jR ( Ext R ( M , R ) , N † ) = j < d . We obtain anisomorphism Ext d + − iR ( Ω Tr Ω M , N † ) ∼ = Ext d + − iR ( Tr M , N † ) , which completes the proof. (cid:4) Theorem 2.8 not only recovers but also refines a main result of [10] (more precisely, [10, Theorem 1.1]), where M is assumed to be locally free on the punctured spectrum. Corollary 2.9 (Celikbas–Takahashi, improved) . Let ( R , m ) be a Cohen–Macaulay local ring with a canonicalmodule. Let X be a totally reflexive R-module, and let M be a maximal Cohen–Macaulay R-module. Assume that NF ( X ) ∩ NF ( M ) ⊆ { m } . Then for each i ∈ Z there is an isomorphism c Ext iR ( X , M ) ∨ ∼ = c Ext ( d − ) − iR ( X ∗ , M † ) . Proof.
We have the following isomorphisms, where the second one follows from Theorem 2.8(1) and the fact thatthe Ext module has finite length (use [8, Proposition 3.2.12(c)]). c Ext iR ( X , M ) ∨ ∼ = Hom R ( Ω i X , M ) ∨ ∼ = Ext d + R ( Tr Ω i X , M † ) ∼ = Hom R ( Ω d + Tr Ω i X , M † ) , c Ext ( d − ) − iR ( X ∗ , M † ) ∼ = Hom R ( Ω ( d − ) − i Ω Tr X , M † ) ∼ = Hom R ( Ω d + − i Tr X , M † ) . It suffices to show Tr Ω i X ∼ = Ω − i Tr X up to free summands, which follows from [25, Proposition 7.1]. (cid:4) It is evident that Corollary 2.9 implies Corollary 1.4. Thus our Theorem 2.8 (Theorem 1.3) also recovers theAuslander–Reiten duality theorem (for any dimension).
ARASH SADEGHI AND RYO TAKAHASHI
3. T
ESTING FREENESS BY VANISHING OF E XT MODULES
In this section, we present applications of our results obtained in the previous section to give criteria for freenessof modules over local rings and to recover various known results about the Auslander–Reiten conjecture. To giveour first application, we establish a lemma, which will also be used later.
Lemma 3.1.
Let M, N be R-modules. Suppose that M is reflexive (i.e. -torsionfree) and that NF ( M ) ∩ NF ( N ) ⊆ Y ( R ) . Then one has an isomorphism Hom R ( M , N ) ∗ ∼ = Hom R ( N , M ) .Proof. Let n = i = K = R in Lemma 2.2(2), and then use the tensor-hom adjunction. (cid:4) Corollary 3.2.
Let R be a local ring, and let M be a ( t + ) st syzygy R-module. Assume that M is locally free on X n ( R ) for some < n < t. Then M is free, provided that one of the following three conditions is satisfied: (1) Ext iR ( Hom R ( M , M ) , R ) = Ext jR ( M , M ) = for all n + ≤ i ≤ t and n ≤ j ≤ t − . (2) Ext jR ( M , M ) = for all n ≤ j ≤ t − and Gdim R Hom R ( M , M ) < ∞ . (3) Ext jR ( M , M ) = for all n ≤ j ≤ t − and Hom R ( M , M ) is ( t + ) -torsionfree.Proof. First of all, note that the inequalities 0 < n < t especially say that t ≥ s = t − n . If s =
1, then n = t −
1. Note from Example 2.1(3) that M is ( t + ) -torsionfree.Corollary 2.4(1) implies Hom R ( M , M ) =
0, which is equivalent to saying that M is free. Let s ≥ M is locally free on X t − ( R ) . Indeed, pick any prime ideal p ∈ X t − ( R ) and set t ′ = depth R p . If t ′ ≤ n , then p ∈ X n ( R ) , and M p is R p -free by assumption. If t ′ > n , then 0 < n < t ′ and M p is locally free on X n ( R p ) .As t ′ ≤ t , the module M p is a ( t ′ + ) st syzygy R p -module, and Ext iR p ( Hom R p ( M p , M p ) , R p ) = Ext jR p ( M p , M p ) = n + ≤ i ≤ t ′ and n ≤ j ≤ t ′ −
1. Since t ′ − n < s , we can apply the induction hypothesis to deduce that M p is R p -free. Thus the claim follows.By Example 2.1(3) the module M is ( t + ) -torsionfree. Corollary 2.4(1) shows Hom R ( M , M ) =
0, so M is free.(2) Similarly to the proof of (1), one can prove that M is ( t + ) -torsionfree and locally free on X t − ( R ) . As M is a ( t + ) st syzygy module and t ≥
2, the depth lemma shows that M has depth at least two, and so doesHom R ( M , M ) by [8, Exercise 1.4.19]. Now the assertion follows from Corollary 2.4(2).(3) Again, one can show that M is ( t + ) -torsionfree. In particular, M is 2-torsionfree. Set H = Hom R ( M , M ) .Lemma 3.1 implies H ∼ = H ∗ . As H is ( t + ) -torsionfree, we have Ext iR ( H , R ) ∼ = Ext iR ( H ∗ , R ) ∼ = Ext i + R ( Tr H , R ) = ≤ i ≤ t . Now the assertion follows from (1). (cid:4) We record various consequences of Corollary 3.2. First of all, Corollary 3.2(2) immediately recovers the theo-rems of Araya [1, Corollary 10], Ono and Yoshino [23, Theorem], and the following result due to Araya, Celikbas,Sadeghi and Takahashi [2, Corollary 1.6].
Corollary 3.3 (Araya–Celikbas–Sadeghi–Takahashi) . Let R be a Gorenstein local ring. Let M be a maximalCohen–Macaulay R-module, and < n < d an integer. Suppose that M is locally free on X n ( R ) and Ext iR ( M , M ) = for all n ≤ i ≤ d − . Then M is free. Recall that an R -module M is called torsion-free if the natural map M → M ⊗ R Q ( R ) is injective, or in otherwords, each non-zerodivisor of R is also a non-zerodivisor of M . Corollary 3.4.
Let R be a Cohen–Macaulay local ring of dimension d with a canonical module ω . Assumethat M is a ( d + ) st syzygy R-module that is locally free in codimension one. Suppose that Ext iR ( M , M ) = for ≤ i ≤ d − and Hom R ( M , M ) ⊗ R ω R is torsion-free. Then M is free.Proof. First of all, since R satisfies ( S ) , an R -module is torsion-free if and only if it satisfies ( S ) . Hence thetorsion-free property localizes. We argue by induction on d . The assertion is evident if d ≤
1, so we may assumethat d > M is locally free on the punctured spectrum of R . Set H = Hom R ( M , M ) . As H is locally free onthe punctured spectrum, by Lemma 2.7 we have Ext iR ( ω ⊗ R H , ω ) ∼ = Ext iR ( H , R ) for all 0 ≤ i ≤ d . Since ω ⊗ R H has positive depth, Ext dR ( H , R ) = (cid:4) The following two results are also consequences of Corollary 3.2. The first one is nothing but Corollary 1.5from the Introduction. Both should be compared with the theorems of Huneke and Leuschke [17, Theorem 1.3],Goto and Takahashi [15, Theorem 1.5(2)], and Dao, Eghbali and Lyle [13, Theorem 3.16].
Corollary 3.5.
Let R be a generically Gorenstein local ring, and let < n < t be an integer. Let M be an R-modulewhich is locally free on X n ( R ) . Then M is free, if the following three conditions hold. WO GENERALIZATIONS OF AUSLANDER–REITEN DUALITY AND APPLICATIONS 7 (1) Ext iR ( M , M ) = for all n ≤ i ≤ t − . (2) Ext iR ( M , R ) = for all i > . (3) Ext iR ( Hom R ( M , M ) , R ) = for all n + ≤ i ≤ t, or Hom R ( M , M ) has finite G-dimension.Proof. It follows from generic Gorensteinness, (2) and [28, Corollary 1.3] that M is a totally reflexive module, anda j th syzygy module for all j ≥
0. The assertion follows from the first and second assertions of Corollary 3.2. (cid:4)
Note that every normal ring R is generically Gorenstein and every syzygy R -module is locally free on X ( R ) .Thus the following is immediate from Corollary 3.5, which is nothing but Corollary 1.6 from the Introduction. Corollary 3.6.
Let R be a normal local ring. An R-module M is free if either (1) Ext iR ( M , M ) = Ext jR ( M , R ) = Ext hR ( Hom R ( M , M ) , R ) = for all ≤ i ≤ t − , j ≥ and ≤ h ≤ t, or (2) Ext iR ( M , M ) = Ext jR ( M , R ) = for all ≤ i ≤ t − and j ≥ , and that Hom R ( M , M ) has finite G-dimension. Corollary 3.6 recovers the following result due to Huneke and Leuschke [17, Theorem 0.1] and Araya [1,Corollary 4], which asserts that the Auslander–Reiten conjecture holds for normal Gorenstein rings.
Corollary 3.7 (Huneke–Leuschke, Araya) . Let R be a (not necessarily local) Gorenstein normal ring. Let M bean R-module. If
Ext iR ( M , M ) = = Ext iR ( M , R ) for all i > , then M is projective. Now we deal with reflexive modules M such that Hom R ( M , M ) is a direct sum of copies of M . Such moduleswere first studied by Auslander [3] to give a new proof of a theorem on the purity of the branch locus for regularlocal rings; see [3, Proposition 1.2 and Theorem 1.3]. The following result especially says that the Auslander–Reiten conjecture holds true for such modules over normal rings. Corollary 3.8.
Let R be a local ring with t ≥ , and let M be a reflexive R-module that is locally free on X ( R ) .Assume that Hom R ( M , M ) ∼ = M ⊕ n for some n > . If Ext iR ( M , R ) = Ext jR ( M , M ) = for all ≤ i ≤ t and ≤ j ≤ t − , then M is free.Proof. We use induction on t . We may assume that M is locally free on X t − ( R ) . By Lemma 3.1, we have M ⊕ n ∼ = Hom R ( M , M ) ∼ = Hom R ( M , M ) ∗ ∼ = ( M ∗ ) ⊕ n . Since M is reflexive, so is Hom R ( M , M ) andExt iR ( Tr Hom R ( M , M ) , R ) ∼ = Ext i − R ( Hom R ( M , M ) ∗ , R ) ∼ = Ext i − R ( M , R ) ⊕ n = ≤ i ≤ t +
2. Hence Hom R ( M , M ) is ( t + ) -torsionfree, and so is M . In particular, M is a ( t + ) nd syzygymodule. Now the assertion follows from Corollary 3.2(3). (cid:4) Here we state an application of Theorem 2.8.
Corollary 3.9.
Let R be a Cohen–Macaulay local ring with canonical module ω . Let < n < d be an integer, andlet M be an R-module that is locally free on X n ( R ) . Then M is free, if it satisfies one of the following conditions. (1) M satisfies ( S ) , M ∗ is maximal Cohen–Macaulay, and Ext iR ( M , ( M ∗ ) † ) = for all n ≤ i < d. (2) Min M ⊆ Ass
R, M ∗ is maximal Cohen–Macaulay, and Ext iR ( M , ( M ∗ ) † ) = for all n ≤ i ≤ d. (3) M is a syzygy module, and
Ext iR ( M , R ) = Ext jR ( M , M ⊗ R ω ) = for all < i ≤ d and n ≤ j < d.Proof. (1) First of all, note that the inequalities 0 < n < d especially says d ≥ M is locally free on the punctured spectrum of R . In this case, Theorem 2.8(1) implies Hom R ( M ∗ , M ∗ ) ∨ ∼ = Ext d − R ( M ∗∗ , M ∗ † ) . Let f : M → M ∗∗ be the naturalhomomorphism. As f is locally an isomorphism on the punctured spectrum, K = Ker f and C = Coker f have finitelength. Since M ∗ † is maximal Cohen–Macaulay, we have Ext jR ( K , M ∗ † ) = = Ext jR ( C , M ∗ † ) = j < d . Itis easy to observe that the homomorphism Ext d − R ( M ∗∗ , M ∗ † ) → Ext d − R ( M , M ∗ † ) induced from f is injective. Byassumption, the latter Ext module vanishes. Hence Hom R ( M ∗ , M ∗ ) =
0, which implies that M ∗ is free.Consider the case M ∗ =
0. Then f = M = K . Hence M has finite length, while M has positive depth as itsatisfies ( S ) . Therefore M =
0. In particular M is free, and we are done. Thus we may assume M ∗ =
0. Theorem2.8(2) implies Ext d − R ( M , M ∗ † ) ∨ ∼ = Ext R ( Tr M , M ∗ ) , and the former Ext module vanishes. As M ∗ is a nonzero freemodule, Ext R ( Tr M , R ) =
0. This means C =
0, and we have an exact sequence 0 → K → M → M ∗∗ →
0, whoselast term is free. Hence this exact sequence splits, and we get an isomorphism M ∼ = K ⊕ M ∗∗ . As M has positivedepth and K has finite length, we have K =
0. Consequently, M is free. ARASH SADEGHI AND RYO TAKAHASHI (ii) Now let us consider the general case. We handle this by induction on d . When d =
2, we have n = = d − n ( R ) coincides with the punctured spectrum of R . Case (i) shows the assertion. Let d >
2, and pick any non-maximal prime ideal p . Then one can apply the induction hypothesis to the localizations R p and M p to deduce that M p is free. This means that M is locally free on the punctured spectrum, and again case (i) implies the assertion.(2) We use the same argument as the proof of (1). Using the assumption that Min M ⊆ Ass R , we easily deducethat M ∗ p = p ∈ Supp M . When M is locally free on the punctured spectrum, M ∗ is a free module, and Itfollows from Theorem 2.8(2) that Ext R ( Tr M , M ∗ ) ∼ = Ext dR ( M , M ∗ † ) ∨ =
0. As M ∗ =
0, we have Ext R ( Tr M , R ) = M is a syzygy, and in particular it satisfies ( S ) .(3) Dualizing a free resolution F of M by R , we obtain an exact sequence 0 → M ∗ → ( F ) ∗ → · · · → ( F d + ) ∗ .This especially says that M ∗ is maximal Cohen–Macaulay. By [7, Corollary B4(3)] we have ( M ∗ ) † ∼ = M ⊗ R ω . As M is a syzygy module, it satisfies ( S ) . Now the assertion follows from (1). (cid:4) The above result is actually a refinement of [2, Theorem 1.4 and Corollary 2.7].
Corollary 3.10 (Araya–Celikbas–Sadeghi–Takahashi) . Let R , ω , n be as in Corollary 3.9. (1) An R-module M is free if it satisfies the following conditions. (i)
M is locally of finite projective dimension on X n ( R ) . (ii) M satisfies ( S ) and M ∗ is maximal Cohen–Macaulay. (iii) Ext iR ( M , ( M ∗ ) † ) = for all n ≤ i < d. (2) An R-module M is free if it satisfies the following conditions. (i)
M is locally of finite projective dimension on X n ( R ) . (ii) M is reflexive and
Ext iR ( M , R ) = for all ≤ i ≤ d. (iii) Ext iR ( M , M ⊗ R ω ) = for all n ≤ i < d.Proof. (1) Similarly to the beginning of the proof of [2, Theorem 1.4], the module M is locally free on X n ( R ) .Corollary 3.9(1) implies the assertion.(2) Since M satisfies ( S ) , it is reflexive. As in the proof of Corollary 3.9(3) the module M ∗ is maximal Cohen–Macaulay. Thus the assertion follows from (1). (cid:4) From here to the end of this section, we consider a rational normal surface singularity , that is, a completelocal normal domain of dimension two over an algebraically closed field of characteristic zero which has a rationalsingularity. Recall that an R -module M is called rigid if Ext R ( M , M ) =
0. The following corollary is deduced fromour Theorem 2.3 and Corollary 3.2.
Corollary 3.11.
Let R be a rational normal surface singularity. Let M be a third syzygy R-module. If M is rigid,then M is free.Proof.
Set H = Hom R ( M , M ) . Note that H is maximal Cohen–Macaulay (see [8, Exercise 1.4.19]). The module M is 3-torsionfree by Example 2.1(3). It follows from Theorem 2.3(1) and the rigidity of M that Ext R ( H , R ) = H ∼ = H ∗ . By [20, Theorem 2.2 and Proposition 3.1(2)] the module H is free. Finally, ourCorollary 3.2(1) deduces that M is free. (cid:4) In view of Corollary 3.2(3), we are interested in when a given module is ( d + ) -torsionfree. Any totallyreflexive module is ( d + ) -torsionfree. If R is Gorenstein, then it is equivalent to maximal Cohen–Macaulayness.Thus we are interested in when a non-totally reflexive module over a non-Gorenstein ring is ( d + ) -torsionfree.We discuss the triviality of such modules. Proposition 3.12.
Let R be a non-Gorenstein complete local ring with d ≥ such that k is algebraically closedand has characteristic . Assume that there exists an R-sequence x = x , . . . , x d − such that R / ( x ) is a rationalnormal surface singularity. Let M be an R-module such that Ext iR ( M , R ) = for all ≤ i ≤ d. Then M is R-free.Proof. Set N = Ω d − R M . It is seen from [19, Proposition 1.1.1] that N is a ( d − ) -torsionfree module withExt iR ( N , R ) = ≤ i ≤ d +
2. As N is a ( d − ) nd syzygy module, x is an N -sequence. Hence the Koszulcomplex of x on N induces an exact sequence0 → N → N ⊕ ( d − ) → · · · → N ⊕ ( d − ) → N → N / xN → , which shows that Ext iR / ( x ) ( N / xN , R / ( x )) ∼ = Ext i + d − R ( N / xN , R ) = ≤ i ≤ Ω R / ( x ) ( N / xN ) is free over R / ( x ) . Thus WO GENERALIZATIONS OF AUSLANDER–REITEN DUALITY AND APPLICATIONS 9 pd R N = pd R / ( x ) N / xN is finite (see [8, Lemma 1.3.5]), and so is pd R M . The vanishing assumption on Ext modulesforces M to be free over R . (cid:4) Corollary 3.13.
Let R be as in Proposition 3.12. Let M be a ( d + ) -torsionfree R-module such that Ext iR ( M , R ) = for all ≤ i ≤ d − . Then M is free. In particular: (1) When d = , every ( d + ) -torsionfree R-module is free. (2) R is a G-regular local ring in the sense of [26].Proof.
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