arXiv:1308.3071v2 [math.AC] 25 Aug 2015 module. ntemr eea etn fa arbitrary an of for setting Herzog module general and Bruns more by the developed in is rings local Cohen-Macaulay m and eoe h alsda uco Hom functor dual Matlis the denotes oe-aalymdl ftp n n ffiieijciedim injective finite of modu and canonical one a type turn of that module condition namely Cohen-Macaulay this definition, Cohen.Macaulay, Herzog’s is and Bruns’ R where case special ai opeinof completion -adic e od n phrases. and words Key 2010 hogotti paper, this Throughout .Mxmlrltv oe-aalymdls13 modules Cohen-Macaulay References relative Maximal modules 5. dualizing Generalized duality 4. local Generalized Prerequisites 3. and Notation 2. Introduction 1. M ahmtc ujc Classification. Subject Mathematics a Abstract. ulH dual lnerslswt epc otemdl H module s the provide to we respect Also, with results rings. alence local Cohen-Macaulay over modules enpoie vraChnMcua oa igwihadmits which ring Dua local Local Cohen-Macaulay the a as over provided such been results, known some of generalizations san is of R R R a c OEDAIYADEUVLNERESULTS EQUIVALENCE AND DUALITY SOME n set and ( safiieygenerated finitely a is mdl.I h aewhere case the In -module. R ) ∨ e ( Let fthe of c R R, =ht := E( , R m oa ooooy aoia oue ope,rltv Coh relative complex, module, canonical cohomology, Local mdl H -module earltv oe-aalylclrn ihrsett ni an to respect with ring local Cohen-Macaulay relative a be ) R R/ a nti ae,w netgt oepoete fteMatlis the of properties some investigate we paper, this In . sacmuaieNehra ring, Noetherian commutative a is m eoe h netv ulo h eiu field residue the of hull injective the denotes ) AI AR ZARGAR RAHRO MAJID a c ( 1. R R n eso htsc oue ra iecanonical like treat modules such that show we and ) R 31,1D5 13D02. 13D45, 13C14, Introduction ( Contents -module − , R E( 1 n slclwt aia ideal maximal with local is R/ a c dmninllclrn ( ring local -dimensional ( R m C ) ∨ ) h hoyo aoia oue for modules canonical of theory The )). uhthat such n oteerslsla oachieve to lead results these so and iyTerm hc have which Theorem, lity aoia module. canonical a C m ult n equiv- and duality ome u ob qiaetto equivalent be to out s ninta nsm fthe of some in that ension ⊗ ei rcsl maximal a precisely is le n[4 hpe ] But, 3]. Chapter [14, in a R sapoe da of ideal proper a is R b = ∼ R, m H m , m n R ,acanonical a ), ˆ ( R eoe the denotes R/ deal en-Macaulay ) ∨ nthe In . m and 18 R ∨ 7 5 3 1 2 M.R. ZARGAR literature is called a dualizing module. As a remarkable result, Foxby [9], Reiten [22] and Sharp [25] proved that a Cohen-Macaulay local ring R admits a canonical module if and only if it is homomorphic image of a Gorenstein local ring. In particular, if R is complete n ∨ and Cohen-Macaulay, then Hm(R) is a canonical module of R. On the other hand, the Local Duality Theorem provides a fundamental tool for the study of local cohomology modules with respect to the maximal ideal of a local ring. Although it only applies to local rings which can be expressed as homomorphic images of Gorenstein local rings, this is not a great restriction, because the class of such local rings includes the local rings of points on affine and quasi-affine varieties, and, as mentioned above, all complete local rings. This theorem provides a functorial isomorphism between the Ext modules of the canonical module and the local cohomology modules with respect to the maximal ideal of a local ring. Over a Cohen-Macaulay local ring, a canonical module, if exists, plays an important role in the studying of the algebraic and homological properties of ideals and modules. Thus, finding modules which preserve beneficiaries of the canonical modules is the aim of many commutative algebraists. In this direction, the principal aim of this c ∨ paper is to study the properties of the R-module Da := Ha(R) in the case where R is a relative Cohen-Macaulay local ring with respect to a and c = ht Ra (i.e. there is precisely one non-vanishing local cohomology module of R with respect to a) and to provide a c ∨ connection between the module Ha(R) and the local cohomology module with respect to the ideal a of R. Indeed, we show that these modules treat like canonical modules over Cohen-Macaulay local rings. Recently, such modules have been studied by some authors, such as Hellus and Str¨uckrad [13], Hellus [10], Hellus and Schenzel [12], Khashyarmanesh [17] and Schenzel [24], and has led to some interesting results. The organization of this paper is as follows. In section 2, we collect some notations and definitions which will be used in the present paper. In section 3, first as a generalization of the Local Duality Theorem, we provide the following result:
Theorem 1.1. Let a be an ideal of R, E be an injective R-module and let Y be an i arbitrary complex of R-modules such that Ha(Y ) = 0 if and only if i = c for some integer 6 c. Then, for all integer i and for all complexes of R-modules X, there are the following isomorphisms: R c i L (i) Tor c−i(X, Ha(Y )) ∼= Ha(X R Y ). c−i c ⊗ i L (ii) Ext (X, Hom (Ha(Y ), E)) = Hom (Ha(X Y ), E). R R ∼ R ⊗R Also, among of other things as an application of the above theorem in Proposition 3.4, we provide a connection between the module Da and the local cohomology modules with respect to the ideal a. Next, in Theorem 4.3, we state one of the main results which shows that the module Da has some of the basic properties of ordinary canonical modules over Cohen-Macaulay local rings. Indeed, we prove the following result: DUALITYFORLOCALCOHOMOLOGYMODULES 3
Theorem 1.2. Let (R, m) be an a-RCM local ring with ht Ra = c. Then the following statements hold true.
i (i) For all ideals b of R such that a b, Hb(Da) = 0 if and only if i = c. ⊆ 6 (ii) id R(Da)= c. (iii) µc(m, Da) = 1. (iv) For all t = 0, 1,...,c and for all a-RCM R-modules M with cd (a, M)= t, one has i (a) Ext R(M, Da) = 0 if and only if i = c t. i c−t 6 − (b) Ha(Ext (M, Da)) = 0 if and only if i = t R 6 c m On the other hand, in Lemma 4.5, it is shown that Hb(Da(R)) ∼= E(R/ ) for all ideals b such that a b. Also, in Proposition 4.8, we provide a result which leads us to ⊆ determine the endomorphism ring of Da and its non-zero local cohomology module at support a. Next, in section 5, first we introduce the concept of maximal relative Cohen- Macaulay modules with respect to an ideal a of R. Then, in Proposition 5.1, we generalize a result and, in Remark 5.3, we raise a quite long-standing known conjecture related to the maximal Cohen-Macaulay modules. In addition, in Theorem 5.4, we generalize another known result related to ordinary canonical modules of Cohen-Macaulay local rings. Indeed, we show that if R is a relative Cohen-Macaulay local ring with respect to a, then for all a-RCM R-modules M with cd (a, M)= t, there exists a natural isomorphism c−t c−t M Rˆ = Ext (Ext (M, Da), Da). In particular, if M is maximal a-RCM, then M ⊗R ∼ R R ⊗R ˆ R ∼= Hom R(Hom R(M, Da), Da). Also, in Theorem 5.5, over relative Cohen-Macaulay local rings, we establish a characterization of maximal relative Cohen-Macaulay modules. Now, suppose for a moment that R is a Cohen-Macaulay local ring with a dualizing module ΩR and that M is a non-zero finitely generated R-module. Kawasaki, in [16, Theorem 3.1], showed that if M has finite projective dimension, then M is Cohen-Macaulay if and only if M Ω is Cohen-Macaulay. Next, in [18, Theorem 1.11], Khatami and ⊗R R Yassemi generalized this result. Indeed, they showed that the above result hold true whenever M has finite Gorenstein dimension. In Theorem 5.6, we establish another main result which, by using Da instead of ΩR, provides a generalization of the above mentioned result of Khatami and Yassemi. Finally, in Proposition 5.9 as an application of the above theorem, we could find another generalization of the results [16, Theorem 3.1] and [18, Theorem 1.11].
2. Notation and Prerequisites
Hyperhomology 2.1. An R-complex X is a sequence of R-modules (Xv)v∈Z together X with R-linear maps (∂ : X X − ) ∈Z, v v −→ v 1 v X X ∂v+1 ∂v X = X X X − , ··· −→ v+1 −→ v −→ v 1 −→ · · · 4 M.R. ZARGAR
such that ∂X ∂X = 0 for all v Z. For any R-module M, Γa(M) is defined by Γa(M) := v v+1 ∈ x M Supp (Rx) V(a) . The right derived functor of the functor Γa( ) exists in { ∈ | R ⊆ } − D(R) and the complex RΓa(X) is defined by RΓa(X) := Γa(I), where I is any injective resolution of the complex X. Then, for any integer i, the i-th local cohomology module i of X with respect to a is defined by Ha(X) := H−i(RΓa(X)). Also, there is the functorial L isomorphism RΓa(X) Cˇx X in the derived category D(R), where Cˇx denotes the ≃ ¯ ⊗R ¯ Cˇech complex of R with respect to x = x ,...,x and √a = p(x ,...,x ). Hence, this ¯ 1 t 1 t i ˇ L implies that Ha(X) = H−i(Cx X)(for more details refer to [19]). Let X D(R) and/or ∼ ¯ ⊗R ∈ Y D(R). The left-derived tensor product complex of X and Y in D(R) is denoted by ∈ X L Y and is defined by ⊗R L ′ ′ X Y F Y X F F F, ⊗R ≃ ⊗R ≃ ⊗R ≃ ⊗R ′ where F and F are flat resolutions of X and Y , respectively. Also, let X D(R) and/or ∈ Y D(R). The right derived homomorphism complex of X and Y in D(R) is denoted by ∈ RHom R(X,Y ) and is defined by
RHom (X,Y ) Hom (P,Y ) Hom (X,I) Hom (P,I), R ≃ R ≃ R ≃ R where P and I are projective resolution of X and injective resolution of Y , respectively. For i R any two complex X and Y , we set Ext R(X,Y ) = H−i(RHom R(X,Y )) and Tor i (X,Y )= H (X L Y ). For any integer n, the n-fold shift of a complex (X,ξX ) is the complex ΣnX i ⊗R n ΣnX n X n given by (Σ X) = X − and ξ = ( 1) ξ − . Also, we have H (Σ X) = H − (X). v v n v − v n i i n Next, for any contravariant, additive and exact functor T : C(R) C(R) and X C(R), −→ ∈ where C(R) denotes the category of R-complexes, we have Hl(T (X)) ∼= T (H−l(X)). Definition 2.2. We say that a finitely generated R-module M is relative Cohen-Macaulay with respect to a if there is precisely one non-vanishing local cohomology module of M with respect to a. Clearly this is the case if and only if grade (a, M) = cd (a, M), where i cd (a, M) is the largest integer i for which Ha(M) = 0. For the convenance, we use the 6 notation a-RCM, for an R-module which is relative Cohen-Macaulay with respect to a. Furthermore, a non-zero finitely generated R-module M is said to be maximal a-RCM if M is a-RCM with cd (a, M) = cd (a,R).
Observe that the above definition provides a generalization of the concept of Cohen- Macaulay and maximal Cohen-Macaulay modules. Also, notice that the notion of relative Cohen-Macaulay modules is connected with the notion of cohomologically complete in- tersection ideals which has been studied in [11] and has led to some interesting results. Furthermore, recently, such modules have been studied in [12], [20] and [21].
Definition 2.3. For any R-module M and a proper ideal a of R, we set grade (a, M) := inf i Ext i (R/a, M) = 0 . Notice that grade (a, M) is the least integer i such that { | R 6 } DUALITYFORLOCALCOHOMOLOGYMODULES 5
i grade (•,•) Ha(M) = 0. For any local ring (R, m), we define D•( ) := Hom (H• ( ), E(R/m)). 6 • R • In the case where R is a-RCM, for the convenience we set Da := Da(R).
3. Generalized local duality
The starting point of this section is the following result which plays an essential role in the present paper. This result provides a generalization of the local duality theorem. Indeed the generalization is done for ideals a of the given ring R and complexes Y which their local cohomology modules with respect to a have precisely one non-vanishing. The generalization is such that local duality becomes the special case where Y = R and a is the maximal ideal m of the given local ring (R, m).
Theorem 3.1. Let a be an ideal of R, E be an injective R-module and let Y be an i arbitrary complex of R-modules such that Ha(Y ) = 0 if and only if i = c for some integer 6 c. Then, for all integer i and for all complexes of R-modules X, there are the following isomorphisms: R c i L (i) Tor c−i(X, Ha(Y )) ∼= Ha(X R Y ). c−i c ⊗ i L (ii) Ext (X, Hom (Ha(Y ), E)) = Hom (Ha(X Y ), E). R R ∼ R ⊗R Proof. (i). Let x = x ,...,x be elements of R such that √a = p(x ,...,x ) and let ¯ 1 t 1 t ˇ ˇ i Cx denotes the Cech complex of R with respect to x1,...,xt. Now notice that Ha(Y ) ∼= ¯ ˇ L ˇ L H−i(Cx R Y ) for all i. Thus, by our assumption we have Hj(Cx R Y ) = 0 for all j = c. ¯ ⊗ ¯ ⊗L L 6 − Hence, one can deduce that there exists an isomorphism Cˇx Y H(Cˇx Y ) in ⊗R ≃ ⊗R −c c L ¯ ¯ derived category D(R), and so Σ Ha(Y ) Cˇx Y . Therefore, we can use the following ≃ ¯ ⊗R isomorphisms R c L c Tor − (X, Ha(Y )) = H − (X Ha(Y )) c i ∼ c i ⊗R −c L c = H− (Σ (X Ha(Y ))) ∼ i ⊗R L −c c = H− (X Σ Ha(Y )) ∼ i ⊗R L ˇ L ∼= H−i(X R Cx R Y ) i ⊗L ¯ ⊗ = Ha(X Y ), ∼ ⊗R where the third isomorphism follows from [5, Lemma 2.4.9], to compte the proof. c c (ii). Note that since E is an injective R-module, RHom (Ha(Y ), E) Hom (Ha(Y ), E). R ≃ R It therefore follows, by the following isomorphisms c−i c c − R Ext R (X, Hom R(Ha(Y ), E)) ∼= Hi c( Hom R(X, Hom R(Ha(Y ), E))) c ∼= Hi−c(RHom R(X, RHom R(Ha(Y ), E))) L c − R ∼= Hi c( Hom R(X R Ha(Y ), E)) L⊗ c = Hom (H − (X Ha(Y )), E) ∼ R c i ⊗R R c ∼= Hom R(Tor c−i(X, Ha(Y )), E) i L = Hom (Ha(X Y ), E), ∼ R ⊗R as required. Here notice that third isomorphism follows from [5, Theorem 4.4.2]. 6 M.R. ZARGAR
The following result is the special case of the above theorem for category of R-modules.
Proposition 3.2. Let a be an ideal of R, E be an injective R-module and let N be an i arbitrary R-module such that Ha(N) = 0 if and only if i = c for some integer c. Then, 6 R for all integers i and for all R-modules M such that Tor j (M,N) = 0 for all j > 0, there are the following isomorphisms: R c i (i) Tor c−i(M, Ha(N)) ∼= Ha(M R N). c−i c ⊗ i (ii) Ext (M, Hom (Ha(N), E)) = Hom (Ha(M N), E). R R ∼ R ⊗R Proof. Notice that since Tor R(M,N) = 0 for all j > 0, there is the isomorphism M N j ⊗R ≃ M L N in the derived category. Hence, it immediately follows from Theorem 3.1. ⊗R As an consequence of the above result, we have the following result.
Corollary 3.3. Let (R, m) be d-dimensional Cohen-Macaulay local ring with a dualizing
R-module ΩR. Then, for all R-modules M with finite Gorenstein flat dimension (See [6, Definition 4.5]) there are the following isomorphisms: R m i (i) Tor d−i(M, E(R/ )) ∼= Hm(M R ΩR). d−i i ⊗∨ (ii) Ext (M, R) = Hm(M Ω ) . R b ∼ ⊗R R Proof. First of all notice that ΩR is a maximal Cohen-Macaulay R-module such that d m i Hm(ΩR) ∼= E(R/ ) and also, in view of [8, Proposition 10.4.17], Tor R(M, ΩR) = 0 for all i > 0. Hence, one can use the known fact that R = Hom (E(R/m), E(R/m)) and b ∼ R Proposition 3.2 to complete the proof.
The following result, which is a generalization of local duality theorem, is a consequence of Proposition 3.2 and recovers the [12, Theorem 3.1].
Proposition 3.4. Let (R, m) be an a-RCM local ring with ht a = c. Then, for all R- c modules M and for all flat R-modules F such that Ha(F ) = 0 (e.g. faithfully flat R- 6 modules), there are the following isomorphisms: R c i (i) Tor c−i(M, Ha(F )) ∼= Ha(F R M). c−i i ⊗ ∨ (ii) Ext (M, Da(F )) = Ha(F M) . R ∼ ⊗R Proof. First notice that, by [23, Theorem 5.40], there is a directed index set I and a family of finitely generated free R-modules F ∈ such that F = limF . Notice that { i}i I i −→i∈I each Fi is relative Cohen-Macaulay with respect to a and that cd (a, Fi) = c. Therefore, j j Ha(F ) = limHa(F ) = 0 for all j = c. Hence, one can use the Proposition 3.2 to complete i 6 −→i∈I the proof.
As an application of the Theorem 3.1, we prove the known Local Duality Theorem which has already been prove in [2, Theorem 11.2.8]. DUALITYFORLOCALCOHOMOLOGYMODULES 7
Corollary 3.5. Let (R, m) be a d-dimensional Cohen-macaulay local ring with a dualizing
R-module ΩR. Then, for all R-modules M and all integers i, one has
i d−i ∨ Hm(M) ∼= Ext R (M, ΩR) . i Proof. First notice that for all i, Hm(M) is Artinian, and so there is the Rˆ-isomorphism i i and also R-isomorphism Hm(M) = Hm(M) Rˆ. Therefore, using the Flat Base Change ∼ ⊗R Theorem, the Matlis Duality Theorem [2, Theorem 10.2.12], and the fact that DmRˆ ∼= Ω Rˆ, implies that following isomorphisms: R ⊗R i i Hm(M) H (Mˆ ) ∼= mRˆ = Hom (Hom (Hi (Mˆ ), E (R/ˆ mRˆ)), E (R/ˆ mRˆ)) ∼ Rˆ Rˆ mRˆ Rˆ Rˆ − = Hom (Ext d i(M,ˆ D ), E (R/ˆ mRˆ)) ∼ Rˆ Rˆ mRˆ Rˆ − = Hom (Ext d i(M,ˆ Ω Rˆ), E (R/ˆ mRˆ)) ∼ Rˆ Rˆ R R Rˆ − ⊗ = Hom (Ext d i(M, Ω ) R,ˆ E (R/ˆ mRˆ)) ∼ Rˆ R R ⊗R Rˆ d−i ˆ ˆ m ˆ ∼= Hom R(Ext R (M, ΩR), Hom Rˆ(R, ERˆ(R/ R)) d−i ˆ m ˆ ∼= Hom R(Ext R (M, ΩR), ERˆ(R/ R)) d−i ∨ ∼= Ext R (M, ΩR) . Notice that the third isomorphism follows from Proposition 3.4 and the last isomorphism follows from [2, 10.2.10].
Notice that for a local Cohen-Macaulay ring R with a dualizing R-module ΩR and a finitely generated R-module M, we have sup i Ext i (M, Ω ) = 0 = depth R depth M. { | R R 6 } − So, as a generalization of this result, we provide the following result which is an immediate consequence of the Proposition 3.4.
Corollary 3.6. Let (R, m) be an a-RCM local ring. Then, for all R-modules M, one has
i sup i Ext (M, Da(R)) = 0 = grade (a,R) grade (a, M). { | R 6 } − 4. Generalized dualizing modules
Notice that if (R, m) is a local complete ring with respect to m-adic topology, then Dm coincides to the ordinary dualizing module ΩR of R. So the main aim in this section is to determine some properties of the module Da and to show that how much such modules have behavior similar to the dualizing modules. In this direction, we need to the following two lemmas will play essential role in the proof of some our results.
Lemma 4.1. Let M be an R-module and n be a non-negative integer. Then the following conditions are equivalent: i (i) Ha(M) = 0 for all i Proof. (i) (ii). Let N be an a-torsion R-module. Then, it is straightforward to see that ⇒ Hom R(N, M) = Hom R(N, Γa(M)). Hence, in view of [16, Theorem 10.47], we obtain the Grothendieck third quadrant spectral sequence with p,q p q p+q E = Ext (N, Ha(M)) = Ext (N, M). 2 R ⇒p R i−q,q i−q,q Note that E = 0 for all 0 i The following lemma, which is needed to prove some next results, was proved in [21, Propositon 3.1] by using the spectral sequence tools. For the convenience, we provide a new proof by using the derived category tools. i Lemma 4.2. Let a be an ideal of R, M be an R-module such that Ha(M) = 0 if and only if i = c for some non-negative integer n. Then, for all a-torsion R-modules N and for 6 R c R any integer i, one has Tor i (N, Ha(M)) ∼= Tor i−c(N, M). Proof. Let N be an a-torsion R-module and let x = x ,...,x be elements of R such ¯ 1 t ˇ ˇ that √a = p(x1,...,xt) and let Cx denotes the Cech complex of R with respect to i ¯ ˇ L x1,...,xt. Now notice that Ha(M) = H−i(Cx M) for all i. Then, by our assumption ∼ ⊗R ˇ L ¯ we have Hi(Cx M) = 0 for all i = c. Hence, one can deduce that there exists an ⊗R 6 − ˇ¯ L ˇ L −c c ˇ L isomorphism Cx R M H(Cx R M) in derived category, and so Σ Ha(M) Cx R M. ¯ ⊗ ≃ ¯ ⊗ L ≃ ¯ ⊗ Also, by similar argument, one has N Cˇx N. Therefore, we can use the following ≃ ¯ ⊗R isomorphisms: R c L c Tor (N, Ha(M)) = H (N Ha(M)) i ∼ i ⊗R −c L c = H − (Σ (N Ha(M))) ∼ i c ⊗R L −c c = H − (N Σ Ha(M)) ∼ i c ⊗R L ˇ L = Hi−c(N Cx M) ∼ ⊗R ⊗R i ˇ L ¯ L ∼= Hi−c(Cx R N R M) i ¯ ⊗L ⊗ = H − (N M) ∼ i c ⊗R R ∼= Tor i−c(N, M), to complete the proof. As a first main result in this section, we provide the following result which determines some basic properties of the module Da. DUALITYFORLOCALCOHOMOLOGYMODULES 9 Theorem 4.3. Let (R, m) be an a-RCM local ring with ht Ra = c. Then the following statements hold true. i (i) For all ideals b of R such that a b, Hb(Da) = 0 if and only if i = c. ⊆ 6 (ii) id R(Da)= c. (iii) µc(m, Da) = 1. (iv) For all t = 0, 1,...,c and for all a-RCM R-modules M with cd (a, M)= t, one has i (a) Ext R(M, Da) = 0 if and only if i = c t. i c−t 6 − (b) Ha(Ext (M, Da)) = 0 if and only if i = t R 6 Proof. (ii). First notice that since R is a-RCM, one can use [21, Theorem 3.1] to see that c fd R(Ha(R)) = c, and so id R(Da)= c. i (i). Let b be an ideal of R such that a b. Then, by part(ii), Hb(Da) = 0 for all i > c. i ⊆ Now, we show that Hb(Da) = 0 for all i < c. To this end, by Lemma 4.1, it is enough to i b show that Ext R(R/ , Da) = 0 for all i < c. But, by Proposition 3.4, we have i b c−i b ∨ Ext R(R/ , Da) ∼= Ha (R/ ) . Hence the assertion is done because of a-torsionness of the module R/b. c m c m (iii). Since µ ( , Da) = vdimExt R(R/ , Da), it easily follows from Proposition 3.4. (iv)(a). It follows from Proposition 3.4 and the our assumption. c−t (iv)(b). First notice that Supp (Ext R (M, Da)) Supp(M) and that there exists a di- ⊆ c−t rected index set I and a family of finitely generated submodules Ni i∈I of Ext R (M, Da) c−t { } such that Ext (M, Da) = limN . Now, since Supp (N ) Supp(M) for all i I, by [7, R ∼ i i ⊆ ∈ −→i∈I a i c−t Theorem 2.2], cd ( ,Ni) t for all i. Therefore, Ha(Ext R (M, Da)) = 0 for all i > t. i ≤ c−t Next, we show that Ha(Ext R (M, Da)) = 0 for all i < t. To this end, by using Lemma 4.1, it is enough to show that i a c−t Ext R(R/ , Ext R (M, Da)) = 0 a R a for all i < t. Now, since M is -RCM, in view of Lemma 4.2 we have Tor i−t(R/ , M) ∼= R a t Tor i (R/ , Ha(M)) for all i. Therefore, one can use Proposition 3.4 to get the following isomorphisms i a c−t R a t ∨ Ext R(R/ , Ext R (M, Da)) ∼= Tor i (R/ , Ha(M)) R a ∨ ∼= Tor i−t(R/ , M) , which complete the proof. The following corollary which is some known results about a dualizing module is a consequence of the previous theorem. Corollary 4.4. Let (R, m) be d-dimensional Cohen-Macaulay local ring with a dualizing R-module ΩR. Then the following statements hold true. 10 M.R. ZARGAR i (i) Hm(Ω ) = 0 if and only if i = d. R 6 (ii) id R(ΩR)= d. d (iii) µ (m, ΩR) = 1. (iv) For all t = 0, 1,...,d and for all Cohen-Macaulay R-modules M with dim M = t, one has i (a) Ext R(M, ΩR) = 0 if and only if i = d t. i d−t 6 − (b) Hm(Ext (M, Ω )) = 0 if and only if i = t R R 6 Proof. It is straightforward to see that one may assume that R is a complete local ring. Therefore, one has ΩR ∼= Dm; and so by using the Theorem 4.3 there nothing to prove. The following lemma, which is needed in the proof of the next proposition, recovers [17, Corollary 2.6]. Lemma 4.5. Let the situation be as in Theorem 4.3. Then, for all ideals b of R such that c a b, there exists the isomorphism Hb(Da) = E(R/m). ⊆ ∼ c Proof. First, one can use Theorem 4.3(i)-(ii) and [20, Theorem 2.5(i)] to see that Hb(Da) is injective. Now, in view of [20, Proposition 2.1] and Theorem 4.3(i), we get the following isomorphisms b c c b Hom R(R/ , Hb(Da)) ∼= Ext R(R/ , Da) b ∨ ∼= (R/ ) , where the last isomorphism follows from Proposition 3.4(ii). Therefore, by Melkerson c Lemma [2, Theorem 7.2.1], Hb(Da) is an Artinian R-module; and hence there exists a c m t non-negative integer t such that Hb(Da) ∼= E(R/ ) . On the other hand, by [20, Corollary 0 c c 2.2], we have µ (m, Hb(Da)) = µ (m, Da). It therefore follows form Theorem 4.3(iii) that t = 1. As an immediate consequence of the previous lemma we have the next result. Corollary 4.6. Let (R, m) be a d-dimensional local ring with t := depth R. Let x1,...,xd be a system of parameters for R which x ,...,x is an R-sequence. Then, for all 0 i t 1 t ≤ ≤ and for all 0 s d i, we have Hi (D ) = Hi (D ) = ≤ ≤ − (x1,...,xi) (x1,...,xi) ∼ (x1,...,xi+s) (x1,...,xi) ∼ E(R/m). Next, we provide an example to justify Theorem 4.3 and Lemma 4.5. Example 4.7. Let R = k[[x,y,z]] be the ring of power series, where k is a field. Set a = (x). Notice that R is an a-RCM local ring with the maximal ideal m = (x,y,z) and ht Ra = 1. Consider the following exact sequence 1 0 R R Ha(R) 0. −→ −→ x −→ −→ DUALITYFORLOCALCOHOMOLOGYMODULES 11 Now, by applying the functor ∨ on the above exact sequence, one gets the exact sequence ∨ (4.1) 0 Da R E(R/m) 0. −→ −→ x −→ −→ Notice that Hom (R , E(R/m)) is an injective R-module; and hence id (Da) 1. Now, R x R ≤ we show that id R(Da) = 1. To this end, it is enough to show that Da is not injective. Assume the contrary that it is injective. Since Da = 0, there exists a non-zero element 1 6 f Da and a non-zero element y Ha(R) such that 0 = f(y) E(R/m); and hence there ∈ ∈ 6 ∈ exists an element r in R such that 0 = rf(y) R/m. On the other hand, there exists 6 ∈ an element t N such that xty = 0. Now, since Da is divisible, there exists an element ∈ h Da such that xth = f, which is a contradiction. Hence id (Da) = 1. Also, by (4.1), ∈ R µ1(m, Da) = 1. Next, applying the functor Γa( ) on (4.1) we obtain the exact sequence − ∨ 1 (4.2) 0 Γa(Da) Γa(R ) Γa(E(R/m)) Ha(Da) 0. → → x → → → ∨ i 1 Therefore, since Γa(R ) = 0, we have Ha(Da) = 0 for all i = 1 and Ha(Da) = E(R/m). Let x 6 ∼ i i M = R/(y, z). Then, Ha(M) = 0 for all i = 1. Since, id (Da) = 1, Ext (M, Da)=0 for 6 R R all i> 1, Also, one can use (4.1) together with the fact that the natural map M M −→ x 1 1 ∨ is one to one, to see that Ext (M, Da) = 0. But, Hom (M, Da) = Ha(M) = 0. Now, R R ∼ 6 a 1 a ∨ since Hom R(R/ , Hom R(M, Da)) ∼= Ha(M/ M) = 0, one has Γa(Hom R(M, Da)) = 0. i On the other hand, one has Ha(Hom R(M, Da)) = 0 for all i > 1. Now, we must prove 1 that Ha(Hom (M, Da)) = 0. To this end, by Lemma 4.1, it is enough to show that R 6 Ext 1 (R/a, Hom (M, Da)) = 0. Since R R 6 1 a R a 1 ∨ Ext R(R/ , Hom R(M, Da)) ∼= Tor 1 (R/ , Ha(M)) , R a 1 we need only to show that Tor 1 (R/ , Ha(M)) = 0. To this end, consider the exact 1 6 sequence 0 M M Ha(M) 0 to induce the exact sequence −→ −→ x −→ −→ R 1 ϕ 1 Tor (R/a, Ha(M)) M/aM M /aM Ha(M) R/a 0, 1 → −→ x x → ⊗R → and use the fact that ϕ = 0 to complete the proof. Proposition 4.8. Let (R, m, k) be an a-RCM local ring with ht Ra = c and let b a be i ⊆ an ideal of R. Set t := inf i Hb(Da) = 0 and consider the following statements: { | 6 } i (i) Hb(Da) = 0 for all i = t. c−t t 6 i t (ii) Ha (Hb(Da)) = E (k) and Ha(Hb(Da)) = 0 for i = c t . ∼ R 6 − c m c a c−t c t i c t (iii) For = and = , Ext R (R/ , Hb(Da)) ∼= ER/c(k) and Ext R(R/ , Hb(Da)) = 0 for all i = c t . 6 − Then, the implications (i) (ii) and (ii) (iii) hold true. Moreover, if condition (i) is ⇒ ⇔ i t t t t satisfied, then Ext (Hb(Da), Hb(Da)) = 0 for all i = 0 and R = Hom (Hb(Da), Hb(Da)). R 6 b ∼ R 12 M.R. ZARGAR Proof. First, notice that one can use Lemma 4.1 and Proposition 3.4 to deduce that t = c cd (a, R/b); and hence it is a finite number. − (i) (ii). In view of [20, Proposition 2.8] and the assumption, one has the isomorphism i ⇒t i+t Ha(Hb(Da)) = Ha (Da) for all i 0. Therefore, one can use Theorem 4.3(i) and Lemma ∼ ≥ 4.5 to complete the proof. (ii) (iii): Let c a, m . Since R/c is an a-torsion R-module, Lemma 4.1(i) (ii) im- ⇒ ∈ { } ⇒ i t c−t t plies that Ext (R/c, Hb(Da)) = 0 for all i < c t. On the other hand, since Ha (Hb(Da)) R − i c t is injective, one can use [20, Proposition 2.1] to see that Ext R(R/ , Hb(Da))) = 0 for all c−t t c−t t c t < i and that Ext (R/c, Hb(Da)) = Hom (R/c, Ha (Hb(Da))). Hence the proof is − R ∼ R complete. (iii) (ii): First, in view of Lemma 4.1(iii) (i) and our assumption for c = a, we deduce ⇒i t ⇒ that Ha(Hb(Da)) = 0 for all i < c t. Notice that, in view of Proposition 3.4, we have − c−i t i t ∨ Ext R (Hb(Da), Da) ∼= Ha(Hb(Da)) i i for all i. Now, since Hb(Da) = 0 for all i i t t i+t t Ext R(Hb(Da), Hb(Da)) ∼= Ext R (Hb(Da), Da) c−i−t t ∨ ∼= Ha (Hb(Da)) . Now, one can use (ii) to complete the proof. The following corollary is an immediate consequence of the above proposition and The- orem 4.3. Notice that the first part of the next corollary has been proved in [24, Theorem 1.3(b)]. Corollary 4.9. Let (R, m) be an a-RCM local ring with ht Ra = c. Then the following statements hold true: ˆ c c (i) R ∼= Hom R(Ha(Da), Ha(Da)) ∼= Hom R(Da, Da). i i c c (ii) Ext R(Da, Da) = Ext R(Ha(Da), Ha(Da)) = 0 for all i> 0. DUALITYFORLOCALCOHOMOLOGYMODULES 13 5. Maximal relative Cohen-Macaulay modules Given an R-module M and a finite free resolution F = F , λF of M, we define { i i } F F Ωi (M) = ker λi−1. The starting point of this section is the following proposition which re- covers the well known fact that, for a finitely generated module M over a Cohen-Macaulay local ring, ΩF (M) is zero or maximal Cohen-Macaulay for all n dim R. n ≥ Proposition 5.1. Let M be a finitely generated R-module and let F be a finite free resolution for M. Then the following hold true: (i) grade (a, ΩF (M)) min n, grade (a,R) . n ≥ { } (ii) If R is a-RCM, then ΩF (M) is zero or maximal a-RCM for all n cd (a,R). n ≥ Proof. (i). We proceed by induction on n. If n = 0, then there is nothing to prove. Let n> 0 and suppose that the result has been proved for n 1. Let − F = F F − F F M 0 ··· −→ n −→ n 1 −→··· −→ 1 −→ 0 −→ −→ be a finite free resolution for M. Then, there are the following exact sequences F F 0 Ω (M) F − Ω − (M) 0 −→ n −→ n 1 −→ n 1 −→ F 0 Ω − (M) F − F F M 0. −→ n 1 −→ n 2 −→··· −→ 1 −→ 0 −→ −→ Hence, one can use [3, Proposition 1.2.9] and inductive hypothesis to see that F F grade (a, Ω (M)) min grade (a, F − ), grade (a, Ω − (M)) + 1 n ≥ { n 1 n 1 } min grade (a, F − ), min n 1, grade (a,R) + 1 ≥ { n 1 { − } } min grade (a,R),n . ≥ { } (ii). Let R be a-RCM and n cd (a,R). Then, cd (a,R) = grade (a,R). Next, since ≥ Supp(ΩF (M)) Spec (R), by [7, Theorem 2.2], we have cd (a, ΩF (M)) cd (a,R). There- n ⊆ n ≤ fore, one can use (i) to complete the proof. An immediate application of the previous proposition is the next corollary. Corollary 5.2. Let (R, m) be a local ring, x1,...,xn be an R-sequence and let M be a non-zero finitely generated R-module. Then, for all i n, x ,...,x is an ΩF M-sequence, ≥ 1 n i F whenever Ωi M is non-zero. Remark 5.3. It is an important conjecture which indicates that any local ring admits a maximal Cohen-Macaulay module. We can raise the above conjecture for maximal relative Cohen-Macaulay modules. Indeed, we have the following conjecture. Conjecture. Let a be a proper ideal of a local ring R. Then there exists a maximal relative Cohen-Macaulay module with respect to a over R. 14 M.R. ZARGAR Let R be a Cohen-Macaulay local ring with a dualizing R-module ΩR. Then, there d−t d−t exists the known isomorphism M ∼= Ext R (Ext R (M, ΩR), ΩR), whenever M is a Cohen- Macaulay R-module of dimension t. In the following result, we use Da instead of ΩR to provide a generalization of the above result. Theorem 5.4. Let (R, m) be a local a-RCM local ring with grade (a,R) = c. Then, for all nonzero a-RCM R-modules M with cd (a, M)= t, there exists a natural isomorphism c−t c−t M R = Ext (Ext (M, Da), Da). ⊗R b ∼ R R In particular, if M is maximal a-RCM, then M R = Hom (Hom (M, Da), Da). ⊗R b ∼ R R Proof. Let M be an a-RCM R-module with cd (a, M) = t. Then, in view of Theorem j 4.3(iv), one has Ext (M, Da) = 0 for all j = c t. Therefore, for all i, one can use the R 6 − following isomorphisms i c−t c−t a − R a Ext R(M, Σ D ) ∼= H i( Hom R(M, Σ D )) c−t ∼= H−i(Σ RHom R(M, Da)) ∼= H−i+t−c(RHom R(M, Da)) i+c−t ∼= Ext R (M, Da), c−t c−t i c−t to deduce that Hom R(M, Σ Da) ∼= Ext R (M, Da) and Ext R(M, Σ Da) = 0 for all i> 0. Notice that the second isomorphism follows from [5, Lemma 2.3.10]. Hence we obtain − − the isomorphism Hom (M, Σc tDa) RHom (M, Σc tDa) in the derived category. On R ≃ R the other hand, in view of Corollary 4.9, we have RHom (Da, Da) Hom (Da, Da) = R. R ≃ R ∼ b Hence, one can use Theorem 4.3(ii) and [4, A.4.24] to obtain the following isomorphism L RHom (RHom (M, Da), Da) M RHom (Da, Da) R R ≃ ⊗R R M L R. ≃ ⊗R b Therefore, we can use the following isomorphisms c−t c−t c−t a a − R a a Ext R (Ext R (M, D ), D ) ∼= Ht c( Hom R(Ext R (M, D ), D )) c−t ∼= Ht−c(RHom R(Hom R(M, Σ Da), Da)) c−t ∼= Ht−c(RHom R(RHom R(M, Σ Da), Da)) c−t ∼= Ht−c(RHom R(Σ RHom R(M, Da), Da)) t−c ∼= Ht−c(Σ RHom R(RHom R(M, Da), Da)) ∼= H0(RHom R(RHom R(M, Da), Da)) = M R, ∼ ⊗R b to complete the proof. The forth isomorphism follows from [5, Lemma 2.3.10] and the fifth isomorphism follows from [5, Lemma 2.3.16]. The following theorem provides a characterization of maximal relative Cohen-Macaulay modules over a relative Cohen-Macaulay ring. DUALITYFORLOCALCOHOMOLOGYMODULES 15 Theorem 5.5. Let (R, m) be an a-RCM local ring and let M be a non-zero finitely gen- erated R-module. Then the following statements are equivalent: (i) M is a maximal a-RCM R-module. i (ii) Ext R(M, Da) = 0 for all i> 0. (iii) The following conditions hold true: (a) M Rˆ = Hom (Hom (M, Da), Da). ⊗R ∼ R R i (b) Ext R(Hom R(M, Da), Da) = 0 for all i> 0. Proof. The implications (i) (ii) follows from Corollary 3.6 and the fact that cd (a, M) ⇔ ≤ cd (a,R). Also, the implication (i) (iii)(a) follows from Theorem 5.4. ⇒ i (i) (iii)(b). In view of Theorem 4.3(iv)(b) we have Ha(Hom (M, Da)) = 0 for all ⇒ R i = cd (a, M). Therefore, one can use Proposition 3.4 to complete the proof. 6 (iii) (i). In view of Proposition 3.4, for all i, one has ⇒ i c−i ∨ Ext R(Hom R(M, Da), Da) ∼= Ha (Hom R(M, Da)) , j where c = cd (a,R). Therefore, by the assumption (iii)(b), one gets Ha(Hom R(M, Da)) = 0 for all j < c. On the other hand, by the same argument as in the proof Theorem 4.3(iv)(b), j one can see that cd (a, Hom (M, Da)) cd (a,R); and hence Ha(Hom (M, Da)) = 0 for R ≤ R all j = c. Therefore, one can use Proposition 3.4, the assumption (iii)(a) and Lemma 4.2 6 to obtain the following isomorphisms i i c ∨ Ext (R/a, M Rˆ) = Ext (R/a, Ha(Hom (M, Da)) ) R ⊗R ∼ R R R a c ∨ ∼= Tor i (R/ , Ha(Hom R(M, Da))) R a ∨ ∼= Tor i−c(R/ , Hom R(M, Da)) . i i Hence, Ext (R/a, M Rˆ) = 0 for all i < c, and so by Lemma 4.1, Ha(M Rˆ)=0 for R ⊗R ⊗R all i < c. Therefore, using the Flat Base Change Theorem and Independence Theorem i implies that Ha(M) = 0 for all i < c. Hence c grade (a, M); and so M is maximal ≤ a-RCM. As we described in the introduction, the following theorem, which is one of the main results, provides a generalization of the result [18, Theorem 1.11] of Khatami and Yassemi. In the next theorem, we shall use the notion of grade of M which is defined by grade (M)= inf i Ext i (M,R) = 0 . { | R 6 } Theorem 5.6. Let (R, m) be an a-RCM local ring of dimension d with ht Ra = c and let R M be a non-zero finitely generated R-module such that Tor i (M, Da) = 0 for all i > 0. Then the following statements hold true: (i) Ext i (M Da, Da) = Ext i (M, Rˆ) for all i and M Da = 0. R ⊗R ∼ R ⊗R 6 16 M.R. ZARGAR i (ii) Suppose that sup i Ext R(M,R) = 0 = depth R depth M and that grade (M)= { | 6 } − i depth R dim M. Then M is Cohen-Macaulay if and only if Ha(M Da) = 0 − ⊗R for all i = c grade (M). 6 − Proof. (i). Let P be a projective R-module. Then, by [23, Theorem 5.40], there exists a directed index set I and a family of finitely generated free R-modules Fj j∈I such that c−i { } P = limF . Therefore, in view of Theorem 4.3(i), we have Ha (F Da) = 0 for all j I j j ⊗R ∈ −→j∈I c−i c−i and for all i = 0, and so Ha (P Da) = limHa (F Da) = 0 for all i = 0. Hence, one 6 ⊗R ∼ i ⊗R 6 −→i∈I can use Proposition 3.4 to see that Ext i (P Da, Da) = 0 for all i = 0. Hence, by using R ⊗R 6 [23, Theorem 10.49] and Corollary 4.9(i), there is a third quadrant spectral sequence with p,q p R p+q E := Ext (Tor (M, Da), Da)= Ext (M, Rˆ). 2 R q p⇒ R p,q Now, since Tor R(M, Da) = 0 for all i> 0, E = 0 for all q = 0. Therefore, this spectral i 2 6 sequence collapses in the column q = 0; and hence one gets, for all i, the isomorphism i i Ext (M Da, Da) = Ext (M, Rˆ), R ⊗R ∼ R as required. Next, notice that since R is a flat R-module, by [8, Theorem 3.2.15], b Ext i (M, Rˆ) = Ext i (M,R) Rˆ for all i. Therefore, M Da = 0. R ∼ R ⊗R ⊗R 6 (ii). Suppose that M is Cohen-Macaulay. Then, Ext i (M,R) = 0 for all i = grade (M). R 6 Hence, one can use (i) and Proposition 3.4 to complete the proof. The converse follows again by (i) and Proposition 3.4. Next, we recall the concept of Gorenstein dimension which was introduced by Auslander in [1]. Definition 5.7. A finite R-module M is said to be of Gorenstein dimension zero and we write G-dim (M) = 0, if and only if i (i) Ext R(M,R) = 0 for all i> 0. i (ii) Ext R(Hom R(M,R),R) = 0 for all i> 0. (iii) The canonical map M Hom (Hom (M,R),R) is an isomorphism. → R R For a non-negative integer n, the R-module M is said to be of Gorenstein dimension at most n, if and only if there exists an exact sequence 0 G G − G M 0, −→ n −→ n 1 −→··· −→ 0 −→ −→ where G-dim (G ) = 0 for 0 i n. If such a sequence does not exist, then we write i ≤ ≤ G-dim (N)= . ∞ Let R be local and M be a non-zero finitely generated R-module of finite G-dimension. Then [6, Corollary 1.10] and [8, Proposition 10.4.17] implies that sup i Ext i (M,R) = { | R 6 DUALITYFORLOCALCOHOMOLOGYMODULES 17 0 = depth R depth M and Tor R(M, Ω ) = 0 for all i> 0, where Ω is a dualizing R- } − i R R module. Next, we provide a remark which shows that the converse of this fact is no longer true, that is, if sup i Ext i (M,R) = 0 = depth R depth M and Tor R(M, Ω )=0 for { | R 6 } − i R all i> 0, then it is not necessary that M has finite G-dimension. Remark 5.8. Let (R, m) be a Cohen-Macaulay local ring which admits a dualizing module Ω and let M be a finitely generated R-module such that sup i Ext i (M,R) = 0 = R { | R 6 } depth R depth M. Next, suppose, in contrary, that the converse of the above mentioned − fact is true. In this case, we prove that M has finite G-dimension. To this end, we may assume that pd (M) = . Let n dim R and let ΩF (M) be the n-th syzygy of R ∞ ≥ n F a finite free resolution F for M. Then, in view of Proposition 5.1, Ωn (M) is maximal i F i+n Cohen-Macaulay. Also, since Ext R(Ωn (M),R) ∼= Ext R (M,R) for all i > 0, one has i F F Ext R(Ωn (M),R) = 0 for all i > 0. Therefore, Hom R(Ωn (M),R) is maximal Cohen- Macaulay. Next, in view of [4, A.4.24], we have the following isomorphisms: L L (5.1) RHom (RHom (ΩF (M),R), Ω ) ΩF (M) RHom (R, Ω ) ΩF (M) Ω R R n R ≃ n ⊗R R R ≃ n ⊗R R i F in derived category. Since Ext R(Ωn (M),R) = 0 for all i > 0, we can deduce that RHom (ΩF (M),R) Hom (ΩF (M),R). On the other hand, in view of [3, Corol- R n ≃ R n i F lary 3.5.11], Ext R(Hom R(Ωn (M),R), ΩR) = 0 for all i > 0. Therefore, by (5.1), i F Tor R(Ωn (M), ΩR) = 0 for all i > 0. Thus, by using the contrary assumption and F F [4, Lemma 1.2.6] for the R-module Ωn (M) , we get G-dim (Ωn (M)) = 0; and hence G-dim (M) < . Indeed, by [4, Corollary 2.4.8], G-dim (M) dim R. Therefore, ∞ ≤ we could prove that every finitely generated R-modules M satisfying the condition sup i Ext i (M,R) = 0 = depth R depth M has finite G-dimension. But, this is { | R 6 } − a contradiction. Because, there exists a local Artinian ring R which admits a finitely i generated R-module M satisfying the condition Ext R(M,R) = 0 for all i > 0, but G-dim (M)= (see [15]). ∞ Considering the above mentioned remark, in the following proposition, we generalize the result [18, Theorem 1.11] which indicates that, over a Cohen-Macaulay local ring R with a dualizing module ΩR, if M is a finitely generated R-module of finite G-dimension, then M is Cohen-Macaulay if and only if M Ω is Cohen-Macaulay. ⊗R R Proposition 5.9. Let (R, m) be a Cohen-Macaulay local ring with a dualizing R-module Ω . Suppose that M is a finitely generated R-module such that sup i Ext i (M,R) = R { | R 6 0 = depth R depth M and that Tor R(M, Ω ) = 0 for all i > 0, then M Ω is } − i R ⊗R R Cohen-Macaulay if and only if M is Cohen-Macauly. Proof. First notice that, we may assume that R is complete. Then ΩR ∼= Dm(R). Also, since R is Cohen-Macaulay, one gets grade (M) = depth R dim M. 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