A study of the cohomological rigidity property
aa r X i v : . [ m a t h . A C ] N ov A STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY
MOHSEN ASGHARZADEH, OLGUR CELIKBAS, ARASH SADEGHIA
BSTRACT . In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we studyvarious aspects of the cohomological rigidity property of tensor product of modules over commutativeNoetherian rings. We determine conditions under which the vanishing of a single local cohomology moduleof a tensor product implies the vanishing of all the lower ones, and obtain new connections between thelocal cohomology modules of tensor products and the Tate homology. Our argument yields bounds for thedepth of tensor products of modules, as well as criteria for freeness of modules over complete intersectionrings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections. C ONTENTS
1. Introduction 12. Notations and preliminary results 33. Tor-rigidity and the vanishing of local cohomology 94. Applications of the main theorem 135. On the depth of tensor powers of modules 186. Applications in prime characteristic 207. A relation between the local cohomology and the Tate homology 248. Splitting criteria for vector bundles 29Acknowledgements 35References 361. I
NTRODUCTION
A vector bundle E on a projective scheme X equipped with a very ample line bundle O ( ) is said tohave the cohomological rigidity property if there is a positive integer i such that the vanishing ofH i ∗ ( X , E ⊗ E ∗ ) : = M n ∈ Z H i ( X , E ⊗ E ∗ ( n )) implies that E is trivial, i.e., isomorphic to a direct sum of line bundles. A paradigm for this rigidityproperty is a result of Luk and Yau [43], which is concerned with ( P n C , O ( )) .Our motivation in this paper comes from a beautiful work of Huneke and Wiegand [36], which re-proves and extends the aforementioned rigidity result of Luk and Yau via the machinery of commutativealgebra. Huneke and Wiegand investigates suitable conditions under which if one local cohomologymodule of a tensor product of finitely generated modules vanishes, then all lower ones vanish. As aconsequence, Huneke and Wiegand [36] obtained remarkable results that relate Serre’s conditions to thevanishing of a single local cohomology module of a tensor product.In this paper, following the work of Huneke and Wiegand [36], we study the (non) vanishing of localcohomology modules, and investigate depth and torsion properties of tensor products of modules. The Mathematics Subject Classification.
Key words and phrases. arithmetically Cohen-Macaulay, complete intersections, complexity, depth formula, local coho-mology, Serre’s condition, specialization-closed subsets, vanishing of Ext and Tor, vector bundle, torsion theory. new advantage we have is that we work with local cohomology functors with respect to specialization-closed subsets of Spec R ; this allows us to generalize the results of Huneke and Wiegand in this direction,as well as various results from the literature, especially those stated in terms of Serre’s conditions.One of our main results is Theorem 3.8, which can be considered as a generalization of rigiditytheorem of Huneke and Wiegand [36, 2.4]. A special case of Theorem 3.8 can be stated as follows; seeProposition 3.11. Theorem 1.1.
Let R be a regular local ring and let M and N be non-zero finitely generated R-modules.Let Z ⊂ Spec
R be a specialization-closed subset and let n ≥ be an integer. Assume the following hold: (i) NF ( M ) ∩ NF ( N ) ⊆ Z . (ii) H n Z ( M ⊗ R N ) = . (iii) grade R ( Z , M ) ≥ n and grade R ( Z , N ) ≥ n.Then it follows that H i Z ( M ⊗ R N ) = for all i = , . . . , n, and Tor Rj ( M , N ) = for all j ≥ . In Theorem 1.1, NF ( M ) denotes the set { p ∈ Spec ( R ) : M p is not free over R p } , i.e., the non-free locus of M . Recall that a subset Z of Spec R is called specialization-closed if every prime ideal of R containingsome prime ideal in Z belongs to Z . Clearly, every closed subset of Spec R in the Zariski topology isspecialization-closed. For an integer n , we denote by H n Z ( − ) the n -th local cohomology functor withrespect to Z . Moreover, the grade of M with respect to Z is denoted by grade R ( Z , M ) , and is definedas the infimum of the set of integers n such that H n Z ( M ) is non-zero; see 2.8 and 2.15 for further details.The torsion in tensor product of modules was initially studied by Auslander in his seminal paper“Modules over unramified regular local rings” [4]. For nonzero finitely generated modules M and N overan unramified (or equi-characteristic) regular local ring, Auslander proved that M and N are torsion-freeand Tor Ri ( M , N ) = i ≥
1, provided that M ⊗ R N is torsion-free; subsequently, the ramified caseof Auslander’s result was established by Lichtenbaum [41]. Theorem 1.1 provides a generalization ofthe aforementioned results of Auslander and Lichtenbaum; this is because the torsion submodule of amodule M can be characterized by the Z -torsion submodule Γ Z ( M ) of M for a suitable choice of Z ;see Proposition 2.19 and Corollary 3.12. To the best of our knowledge, the conclusion of Theorem 1.1 isnew even for the closed subsets of Spec ( R ) .In Section 3 we give a proof of Theorem 1.1. Sections 4 and 5 are devoted to several applications; see,for example, Corollaries 4.6, 4.1 and 5.2. In Section 6 we apply our results and study the vanishing oflocal cohomology modules of the Frobenius powers over local rings of prime characteristic; see Theorem6.6. We make use of the fact that the Frobenius endomorphism ϕ r R is Tor-rigid over such completeintersection rings, and obtain the following as an application of Theorem 6.6; see Corollary 6.10. Theorem 1.2.
Let ( R , m ) be a d-dimensional local complete intersection ring of prime characteristic.Then R is regular provided that at least one of the following conditions hold: (i) R is reduced and H d − m ( ϕ r R ⊗ R ϕ s R ) = for some integers r , s ≥ . (ii) R is normal and H d − m ( ϕ r R ⊗ R ϕ s R ) = for some integers r , s ≥ . In Section 7 we determine a new connection between local cohomology of tensor products of modulesand the Tate homology c Tor; see 2.20 for the definition. The next result follows from Theorem 7.1; it isthe second main theorem of this paper besides Theorem 3.8.
Theorem 1.3.
Let R be a Gorenstein local ring, and let M and N be finitely generated R-modules.Assume Z ⊂ Spec
R is a specialization-closed subset, and that the following conditions hold: (i) NF ( M ) ∩ NF ( N ) ⊆ Z . (ii) depth R p ( M p ) + depth R p ( N p ) ≥ depth ( R p ) + n for each p ∈ Z and for some integer n ≥ .Then it follows that: (1) H i Z ( M ⊗ R N ) ∼ = c Tor R − i ( M , N ) for all i = , . . . , n − . (2) There is an injection c Tor R − n ( M , N ) ֒ → H n Z ( M ⊗ R N ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 3
As far as we know, Theorem 1.3 is new, even for closed subsets of Spec ( R ) . Various applications ofTheorem 1.3 corroborating the literature include Corollaries 7.3, 7.4 and 7.15. Furthermore, Theorem1.3 determines a useful bound on depth of tensor products M ⊗ R N of certain modules M and N overcomplete intersection rings: Corollary 1.4.
Let R be a local complete intersection ring of codimension c, and let M and N benonzero finitely generated R-modules, each of which is locally free on the punctured spectrum of R.If depth R ( M ) + depth R ( N ) − depth ( R ) ≥ c, then depth R ( M ⊗ R N ) ≤ depth R ( M ) + depth R ( N ) − depth ( R ) . The conclusion of Corollary 1.4 seems interesting to us since it does not assume the vanishing of Tormodules; see Corollary 7.5 and cf. [22, 3.1].Our aim in Section 8 is, motivated by the work of Huneke and Wiegand [36], to determine some newcriteria for freeness of modules in terms of the vanishing of local cohomology. More precisely, we areconcerned with the cohomological rigidity property of tensor products in the sense that the freeness ofa module M follows from the vanishing of H i m ( M ⊗ R M ∗ ) for some integer i . Our work extends severalresults from the literature. For example, as a consequence of Theorem 8.1, we proved the followingresult in Corollary 8.14; it extends [18, 3.9] which establishes the case where n = Corollary 1.5.
Let R be a local complete intersection ring of even dimension d, and let M be a maximalCohen-Macaulay R-module which is locally free on the punctured spectrum of R. If H n m ( M ⊗ R M ∗ ) = for some integer n where ≤ n < d, then M is free. The cohomological rigidity property on hypersurfaces of odd dimension was first studied by Dao [28]:a vector bundle E on an odd dimensional hypersurface of dimension at least 3 splits if and only ifH ( X , E ⊗ E ∗ ( i )) = i ∈ Z [28, 1.5]. This result has been recently studied by ˇCesnaviˇcius, whoproved that a vector bundle E on a smooth complete intersection of dimension at least 3 splits into asum of line bundles if and only if H ( X , E ⊗ E ∗ ( i )) = = H ( X , E ⊗ E ∗ ( i )) for all i ∈ Z [24, 1.2]. Asan application of our study, for an arithmetically Cohen–Macaulay vector bundle on a smooth completeintersection of odd dimension we obtain a stronger result; see Corollary 8.15: Corollary 1.6.
Let k be a field and let X ⊂ P nk be a globally complete intersection of odd dimension d.Assume E is an arithmetically Cohen–Macaulay vector bundle and that H i ( X , E ⊗ E ∗ ( j )) = for allj ∈ Z and for some i where < i < d. Then E is a direct sum of powers of O ( ) . Furthermore, for an arithmetically Cohen–Macaulay vector bundle on a hypersurface we have:
Corollary 1.7.
Let k be a field and let X ⊂ P nk be a hypersurface. Assume E is an arithmetically Cohen-Macaulay vector bundle such that H i ( X , E ⊗ E ∗ ( j )) = for all j ∈ Z and for some even integer i where < i < dim X . Then E is direct sum of powers of O ( ) .
2. N
OTATIONS AND PRELIMINARY RESULTS
Throughout, R , Mod ( R ) , mod ( R ) and P ( R ) denote a commutative Noetherian ring, the category of R -modules, the category of finitely generated R -modules, and the subcategory of mod ( R ) of finitelygenerated projective R -modules, respectively.If R is a local ring, i.e., a commutative Noetherian local ring, then m denotes the unique maximal idealof R , and k denotes the residue field of R . ( − ) ∗ stands for the algebraic dual Hom R ( − , R ) , and e M is the natural map M → M ∗∗ . M , N ∈ Mod ( R ) are said to be stably isomorphic , denoted by M ≈ N , provided that M ⊕ P ∼ = N ⊕ Q for some P , Q ∈ P ( R ) . An R -module homomorphism f : X → M (respectively, f : M → X ) with X ∈ P ( R ) is called a right (respectively, left ) projective approximation of M providedthat every R -module homomorphism g : Y → M (respectively, g : M → Y ) with Y ∈ P ( R ) factors through ASGHARZADEH, CELIKBAS, SADEGHI f , that is, g = f ◦ h (respectively, g = h ◦ f ) for some R -module homomorphism h : Y → X (respectively, h : X → Y ).A right (respectively, left) projective approximation f : X → M (resp. f : M → X ) is called minimal ifevery endomorphism g : X → X satisfying f = f ◦ g (resp. f = g ◦ f ) is an automorphism.Note that a right projective approximation (respectively, a minimal right projective approximation) isnothing but a surjective homomorphism from a projective R -module (respectively, a projective cover).Note also that, if M ∈ mod ( R ) and φ : P ։ M ∗ is an epimorphism with P ∈ P ( R ) , then e M ◦ φ ∗ is a leftprojective approximation of M . Let M ∈ mod ( R ) that has a right projective approximation P ∂ → M . Then the kernel of ∂ is called the first syzygy of M ; it is denoted by Ω M and unique up toprojective equivalence. Inductively, we define the n -th syzygy module of M as Ω n M : = Ω ( Ω n − M ) forall n ≥
1. We set, by convention, Ω M = M .Let P ∂ → P ∂ → M → M . Then the transpose of M , denoted byTr M , is coker ∂ ∗ given in the following exact sequence(2.2.1) 0 → M ∗ → P ∗ ∂ ∗ → P ∗ → Tr M → . Note that M ∗ ≈ Ω Tr M . Note also that Tr M is unique, up to projective equivalence, and the minimalprojective presentations of M represent isomorphic transposes of M .For every M ∈ mod ( R ) and N ∈ Mod ( R ) , there exists the following exact sequence:(2.2.2) 0 → Ext R ( Tr M , N ) → M ⊗ R N → Hom R ( M ∗ , N ) → Ext R ( Tr M , N ) → , where the middle map is the evaluation map [5, 2.6]. In particular, setting N = R , we see that thecanonical map M → M ∗∗ is part of the exact sequence(2.2.3) 0 → Ext R ( Tr M , R ) → M → M ∗∗ → Ext R ( Tr M , R ) → . Also, there is a 4-term exact sequence [5, 2.8]:(2.2.4) Tor R ( Tr Ω n M , N ) → Ext nR ( M , R ) ⊗ R N → Ext nR ( M , N ) → Tor R ( Tr Ω n M , N ) → . Suppose M ∈ mod ( R ) equipped with a left projective approximation M ∂ − → P − . Then we call thecokernel of ∂ − the first cosyzygy of M and denote it by Ω − M . Inductively, we define the n -th cosyzygymodule of M as Ω − n M : = Ω − ( Ω − ( n − ) M ) for all n ≥
1. It follows that Ω − i M ≈ Tr Ω i Tr M for all i ≥ M with Tr M in (2.2.4), we obtain the following exact sequence:(2.2.5) Tor R ( Ω − n M , N ) → Ext nR ( Tr M , R ) ⊗ R N → Ext nR ( Tr M , N ) → Tor R ( Ω − n M , N ) → . For each M ∈ mod ( R ) and integer i ≥
1, it follows by the definition that there exists an exact sequence:(2.2.6) 0 → Ext iR ( M , R ) → Tr Ω i − M → X → , where X ≈ Ω Tr Ω i M . By replacing M with Tr M in (2.2.6) and using the fact that Ω − i M ≈ Tr Ω i Tr M forall i ≥
1, we obtain the following fact that will be used throughout the paper:(2.2.7) If Ext iR ( Tr M , R ) = i ≥ , then Ω − ( i − ) M ≈ ΩΩ − i M . The notion of Gorenstein dimension wasinitially introduced by Auslander [3] and subsequently developed by Auslander and Bridger in [5].An R -module M is called totally reflexive provided that the natural map M → M ∗∗ is an isomorphismand Ext iR ( M , R ) = = Ext iR ( M ∗ , R ) for all i ≥
1. The
Gorenstein dimension of M , denoted G-dim R ( M ) ,is defined to be the infimum of all nonnegative integers n , such that there exists an exact sequence0 → G n → · · · → G → M → , in which each G i is a totally reflexive R -module. STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 5
Every finitely generated module over a Gorenstein ring has finite Gorenstein dimension. Moreover, if R is local and G-dim R ( M ) < ∞ , then it follows that G-dim R ( M ) = depth R − depth R ( M ) ; see [5, 4.13].A quasi-deformation of R is a diagram R → A և Q of local homomorphisms, in which R → A isfaithfully flat, and A և Q is surjective with kernel generated by a regular sequence. The completeintersection dimension of M , introduced by Avramov, Gasharov and Peeva [9], is:CI-dim R ( M ) = inf { pd Q ( M ⊗ R A ) − pd Q ( A ) | R → A և Q is a quasi-deformation } . Therefore, an R -module M has finite complete intersection dimension if there exists a quasi-deformation R → A և Q for which pd Q ( M ⊗ R A ) is finite.Note that, if M is finitely generated R -module, then it follows G-dim R ( M ) ≤ CI-dim R ( M ) ≤ pd R ( M ) ,and CI-dim R ( M ) < ∞ if R is a complete intersection ring. Moreover, if R is local and CI-dim R ( M ) < ∞ ,then one has by [9, 1.4] that(2.3.1) G-dim R ( M ) = CI-dim R ( M ) = depth R − depth R ( M ) . Assume ( R , m ) is local and M , N ∈ mod ( R ) . Then the complexity of the pair ( M , N ) ,defined by Avramov and Buchweitz [8], is:cx R ( M , N ) = inf { b ∈ N | ∃ a ∈ R such that ν R ( Ext nR ( M , N )) ≤ an b − for all n ≫ } , where ν R ( − ) denotes the minimal number of generators. Accordingly, the complexity cx R ( M ) of M ,initially introduced by Avramov [7] in local algebra, can be given as cx R ( M , k ) .Avramov, Gasharov and Peeva [9, 5.3] proved that every module of finite complete intersection di-mension also has finite complexity.Note, cx R ( M , N ) = iR ( M , N ) = i ≫
0. Also, it follows by the definition thatpd R ( M ) < ∞ if and only if cx R ( M ) =
0, and M has bounded Betti numbers if and only if cx R ( M ) ≤ R is a complete intersection, then one has [8, 5.7]:(2.4.1) cx R ( M , N ) = cx R ( N , M ) ≤ min { cx R ( M ) , cx R ( N ) } ≤ codim R . If R is local, then a pair ( M , N ) in mod ( R ) is said to satisfy the depth formula provided that the following equality holds:depth R ( M ) + depth R ( N ) = depth R + depth R ( M ⊗ R N ) . Auslander [4, 1.2] proved that, if ( M , N ) is Tor-independent (i.e., Tor Ri ( M , N ) = i ≥
1) andpd R ( M ) < ∞ , then the depth formula holds for ( M , N ) . Auslander’s result has been extended by Hunekeand Wiegand for complete intersection rings: Tor-independent modules over complete intersection ringssatisfy the depth formula; see [35, 2.5]. More generally, one has: Theorem 2.6. (Araya and Yoshino [1]) Assume R is a local ring and M , N ∈ mod ( R ) . If CI-dim R ( M ) < ∞ and Tor Ri ( M , N ) = for all i ≥ , then ( M , N ) satisfies the depth formula; see [1, 2.5]. Assume R is local and M , N ∈ mod ( R ) . If CI-dim R ( M ) < ∞ and Tor Ri ( M , N ) = i ≫
0, then it follows from [38, 2.2] that:(2.7.1) sup { i | Tor Ri ( M , N ) = } = sup { depth R p − depth R p ( M p ) − depth R p ( N p ) | p ∈ Spec R } . A subset Z ⊂ Spec R is called specialization-closed provided that the following condition holds:If p , q ∈ Spec ( R ) , where p ∈ Z and p ⊆ q , then it follows that q ∈ Z .We collect some examples of specialization-closed subsets of Spec R : Example 2.9. (i) Every closed subset of Spec ( R ) with respect to Zariski toplogy is specialization-closed.(ii) If R is a domain, then Spec ( R ) \ { } is specialization-closed subset of Spec ( R ) .(iii) Let 0 ≤ n ≤ dim R . Then { p ∈ Spec ( R ) | ht ( p ) ≥ n } is a specialization-closed subset of Spec ( R ) . ASGHARZADEH, CELIKBAS, SADEGHI
One of the aims of this paper is to extend the following result of Auslander and Lichtenbaum [4,41]: if R is a regular local ring, M , N ∈ mod ( R ) and M ⊗ R N is torsion-free, then M and N are Tor-independent torsion-free modules. Our technique relies upon using local cohomology theory with respectto specialization-closed subsets [31]. Let Z be a specialization-closed subset of Spec R . For each M ∈ Mod ( R ) , we define the following submodule of M : Γ Z ( M ) = { m ∈ M | Supp R ( Rm ) ⊆ Z } . For i ∈ N , the i -th right derived functor of Γ Z ( − ) , denoted by H i Z ( − ) , is referred to as the i -th localcohomology functor with respect to Z .If Z is a closed subset of Spec R , i.e., Z = V ( a ) for some ideal a of R , then we denote Γ Z ( − ) (respectively, H i Z ( − ) ) by Γ a ( − ) (respectively, H i a ( − ) ). M ∈ Mod ( R ) is called torsion-free with respect to Z provided that Γ Z ( M ) =
0, and is called torsionwith respect to Z precisely when Γ Z ( M ) = M . Note that M is torsion with respect to Z if and only ifSupp R ( M ) ⊆ Z .We set, for a specialization-closed subset Z of Spec R , that Σ = { a ⊳ R | V ( a ) ⊆ Z } , and proceedby collecting some of the basic properties of local cohomology modules with respect to specialization-closed subsets in the following; for details, we refer the reader to [13, 31]. Theorem 2.11.
Let Z and W be specialization-closed subsets of Spec ( R ) . Then, (i) Γ Z ( M ) = S a ∈ Σ Γ a ( M ) . (ii) Each short exact sequence → M → M → M → in Mod ( R ) yields a long exact sequence: · · · → H i Z ( M ) → H i Z ( M ) → H i Z ( M ) → H i + Z ( M ) → · · · . (iii) If M is torsion with respect to Z , i.e., Γ Z ( M ) = M, then H i Z ( M ) = for all i ≥ . (iv) Γ Z ( Γ W ( M )) = Γ Z ∩ W ( M ) = Γ W ( Γ Z ( M )) . The following results are used throughout the paper; although they are well-known, we recall themfor the convenience of the reader: [14, Theorem 3.5.7] Let ( R , m ) be a local ring and let M ∈ mod ( R ) be a module of deptht and dimension d. Then (i) H i m ( M ) = for i < t and i > d. (ii) H t m ( M ) = and H d m ( M ) = . Theorem 2.14. [14, Corollary 3.5.9] Let ( R , m , k ) be a Cohen-Macaulay local ring of dimension d withcanonical module ω R . Then for all M ∈ mod ( R ) and all integers i there exist natural isomorphisms H i m ( M ) ∼ = Hom R ( Ext d − iR ( M , ω R ) , E R ( k )) , where E R ( k ) denotes the injective envelope of the reside field. Let Z ⊂ Spec R be a specialization-closed subset and let M ∈ Mod ( R ) . Then the grade of M withrespect to Z is: grade R ( Z , M ) = inf { i ∈ N | H i Z ( M ) = } . If Z = V ( a ) for some ideal a of R , then we denote grade R ( Z , M ) by grade R ( a , M ) . Note that, if M ∈ mod ( R ) , then grade ( a , M ) is equal to the maximal length of an M -regular sequence contained in a .The next result, for the case where Z is a closed subset of Spec R , was initially proved byGrothendieck; see [31, III.2.9]. This fact can be viewed as a generalization of [51, 4.1]. As the resultplays an important role for our arguments in this paper, we provide the details. STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 7
Proposition 2.16.
Let Z ⊂ Spec
R be specialization-closed and M ∈ mod ( R ) . Then it follows that grade R ( Z , M ) = inf { depth ( M p ) | p ∈ Z } . Proof.
Let 0 → M → E ( M ) ∂ → E ( M ) ∂ → · · · be a minimal injective resolution of M , where we have that E i ( M ) = L q ∈ Spec R E ( R / q ) µ i ( q , M ) . Apply the functor Γ Z ( − ) to the above resolution, we get the complex(2.16.1) 0 −→ Γ Z ( M ) −→ Γ Z ( E ( M )) Γ Z ( ∂ ) −→ Γ Z ( E ( M )) Γ Z ( ∂ ) −→ · · · . Set n : = inf { depth ( M p ) | p ∈ Z } . Hence, there exists a prime ideal p ∈ Z such that depth R p ( M p ) = n .It follows from [48, Theorem 2] that(2.16.2) µ n ( p , M ) = = µ i ( p , M ) for all p ∈ Z and i < n . Note that Supp R ( E ( R / p )) ⊆ V ( p ) for all p ∈ Spec R . As Z is specialization-closed, we see that(2.16.3) Γ Z ( E ( R / q )) = (cid:26) E ( R / q ) if q ∈ Z q / ∈ Z . It follows from (2.16.3) that(2.16.4) Γ Z ( E i ( M )) ∼ = M q ∈ Spec R Γ Z ( E ( R / q )) µ i ( q , M ) ∼ = M q ∈ Z Γ Z ( E ( R / q )) µ i ( q , M ) . Hence, by (2.16.2) and (2.16.4) we have(2.16.5) Γ Z ( E n ( M )) = = Γ Z ( E i ( M )) for all i < n . Therefore, by (2.16.1) and (2.16.5) we see that H i Z ( M ) ∼ = ker ( Γ Z ( ∂ i )) / im ( Γ Z ( ∂ i − )) = i < n .In other words, grade R ( Z , M ) ≥ n = inf { depth ( M p ) | p ∈ Z } . Thus, it is enough to show that H n Z ( M ) = −→ H n Z ( M ) −→ Γ Z ( E n ( M )) Γ Z ( ∂ n ) −→ Γ Z ( E n + ( M )) . Note that E n ( M ) is an essential extension of im ∂ n − . In other words, im ∂ n − ∩ L = L of E n ( M ) . It follows from (2.16.5) that Γ Z ( E n ( M )) is a non-zero submodule of E n ( M ) .Therefore, by the exact sequence (2.16.6) we obtain the following:H n Z ( M ) = ker ( Γ Z ( ∂ n )) = ker ∂ n ∩ ( Γ Z ( E n ( M ))) = im ∂ n − ∩ ( Γ Z ( E n ( M ))) = , as desired. (cid:3) The next result can be found in [14, 1.4.19] when Z is a closed subset of Spec R . Lemma 2.17.
Let Z ⊂ Spec R be a specialization-closed subset, and let M ∈ mod ( R ) and N ∈ Mod ( R ) .Then it follows that grade R ( Z , Hom R ( M , N )) ≥ min { , grade R ( Z , N ) } . Proof.
Let F → F → M → M . Applying the functor Hom R ( − , N ) to theabove exact sequence, we obtain the following exact sequence 0 → Hom R ( M , N ) → Hom R ( F , N ) → Hom R ( F , N ) which induces the following exact sequences:0 → Hom R ( M , N ) → Hom R ( F , N ) → X → → X → Hom R ( F , N ) . By Theorem 2.11(ii), the above exact sequences induce the following exact sequences:(2.17.1) Γ Z ( Hom R ( M , N )) ֒ → Γ Z ( Hom R ( F , N )) → Γ Z ( X ) → H Z ( Hom R ( M , N )) → H Z ( Hom R ( F , N )) , (2.17.2) 0 → Γ Z ( X ) → Γ Z ( Hom R ( F , N )) . Now the assertion follows easily from (2.17.1) and (2.17.2). (cid:3)
ASGHARZADEH, CELIKBAS, SADEGHI If M ∈ mod ( R ) and n ≥ M is saidto satisfy Serre’s condition ( S n ) provided that depth R p ( M p ) ≥ min { n , ht p } for all p ∈ Spec R (recall that,by convention, depth R ( ) = ∞ .)If N ∈ Mod ( R ) , then the torsion submodule of N , denoted by T ( N ) , is the kernel of the natural ho-momorphism N → Q ( R ) ⊗ R N where Q ( R ) is the total quotient ring of R . The module N is said to be torsion-free if T ( N ) =
0, and torsion if T ( N ) = N . Note that, if R is unmixed, i.e., all associated primeideals of R are minimal, e.g., R is reduced, and N ∈ mod R , then T ( N ) = N satisfies ( S ) .If M ∈ Mod ( R ) and n ≥
1, then M is called n - torsion-free if Ext iR ( Tr M , R ) = i = , . . . , n .In the following we investigate the relation between Serre’s condition and the vanishing of local co-homology with respect to specialization-closed subsets of Spec R . Proposition 2.19.
Let M ∈ Mod ( R ) and let N ∈ mod ( R ) . (i) If R is unmixed and Z = { p ∈ Spec ( R ) | ht p ≥ } , then it follows that Γ Z ( M ) = T ( M ) . (ii) Let Z = { p ∈ Spec R | ht p ≥ n } for some integer n ≥ . Then N satisfies ( S n ) if and only if H i Z ( N ) = for all i < n and N p is a maximal Cohen-Macaulay R p -module for all p ∈ Spec R \ Z . (iii) Assume R is local and Cohen-Macaulay, Z is a specialization-closed subset of Spec
R, and n ≥ isan integer. Assume further that Supp R ( Ext iR ( Tr N , R )) ⊆ Z for ≤ i ≤ n (e.g., N p is totally reflexivefor all p ∈ Supp R ( N ) \ Z .) If grade R ( Z , N ) ≥ n, then N is n-torsion-free.Proof. Note that part (i) follows easily form the definition, and part (ii) follows from Proposition 2.16.Hence we proceed to prove part (iii).Set t = inf { i ≥ | Ext iR ( Tr N , R ) = } and let p ∈ Ass R ( Ext tR ( Tr N , R )) . Assume contrarily that t ≤ n .Hence by our assumption p ∈ Z . As Ext iR ( Tr N , R ) p = < i < t , we have by (2.2.7) that Ω − ( i − ) R p N p ≈ Ω R p Ω − iR p N p for all 0 < i < t . Hence, we obtain the following(2.19.1) N p ≈ Ω t − R p Ω − ( t − ) R p N p . It follows from the inclusion Ext tR ( Tr N , R ) ֒ → Ω − ( t − ) N (see (2.2.6)) that p ∈ Ass R ( Ω − ( t − ) N ) . Now itis easy to see that depth R p ( Ω t − R p Ω − ( t − ) R p N p ) = t − R p ( N p ) = t −
1. This is acontradiction because Proposition 2.16 implies that t ≤ n ≤ grade R ( Z , N ) ≤ depth R p ( N p ) . (cid:3) We call a (homologically indexed) complex acyclic if it has zero homology.An acyclic complex T of free R -modules is called totally acyclic if the dual complex Hom R ( T , R ) is alsoacyclic. For M ∈ mod ( R ) , it follows that M is totally reflexive if and only if there is a totally acycliccomplex T with M ∼ = coker ( T → T ) .A complete resolution of M ∈ mod ( R ) is a diagram T τ → P ≃ → M , where P ≃ → M is a projective resolu-tion, T is a totally acyclic complex of free R -modules, and τ i is an isomorphism for all i ≫
0. It is knownthat M has finite Gorenstein dimension if and only if it has a complete resolution [11].Suppose M ∈ mod ( R ) is equipped with a complete resolution T → P → M . For each N ∈ Mod ( R ) andfor each i ∈ Z , the Tate (co)homology of M and N is defined as: c Tor Ri ( M , N ) = H i ( T ⊗ R N ) and c Ext iR ( M , N ) = H i ( Hom R ( T , N )) . In the following we catalog some basic properties of Tate (co)homology; for details, please see [8, 11,26].
Theorem 2.21.
Let M ∈ mod ( R ) , N ∈ Mod ( R ) , and assume G-dim R ( M ) < ∞ . (i) If pd R ( M ) < ∞ , then c Tor Ri ( M , N ) = = c Ext iR ( M , N ) for all i ∈ Z . (ii) If G-dim R ( M ) = , i.e., if M is totally reflexive, then c Tor Ri ( M , N ) ∼ = c Ext − i − R ( M ∗ , N ) for all i ∈ Z . (iii) c Tor Ri + n ( M , N ) ∼ = c Tor Ri ( Ω n M , N ) and c Ext i + nR ( M , N ) ∼ = c Ext iR ( Ω n M , N ) for all i ∈ Z and for all n ≥ . (iv) c Tor Ri ( M , N ) ∼ = Tor Ri ( M , N ) and c Ext iR ( M , N ) ∼ = Ext iR ( M , N ) for all i > G-dim R ( M ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 9
Next is a property for the vanishing of Tate homology for modules of finite complete intersectiondimension:
Theorem 2.22. ( [8, 4.9]) Let M ∈ mod ( R ) and let N ∈ Mod ( R ) . Assume CI-dim R ( M ) < ∞ . Then, c Tor Ri ( M , N ) = for all i ∈ Z ⇐⇒ c Tor Ri ( M , N ) = for all i ≪ ⇐⇒ Tor Ri ( M , N ) = for all i ≫ .
3. T OR - RIGIDITY AND THE VANISHING OF LOCAL COHOMOLOGY
In this section we prove one of the main results of this paper, namely Theorem 3.8, and extend thefollowing celebrated result of Auslander and Lichtenbaum:
Theorem 3.1. (Auslander and Lichtenbaum [4,41]) Let R be a regular local ring and let M , N ∈ mod ( R ) .If M ⊗ R N is nonzero and torsion-free, then M and N are torsion-free and Tor-independent.
Besides the generalization we obtain in Theorem 3.8, a consequence of our argument yields a newproof of Theorem 3.1, which we record at the end of this section.The proof of Theorem 3.8 requires substantial preparation; we proceed and prove several lemmasprior to giving a proof for Theorem 3.8.The first item of the following lemma is to be compared with [15, 4.1].
Lemma 3.2.
Let Z ⊂ Spec R be a specialization-closed subset, N ∈ Mod ( R ) and let M ∈ mod ( R ) .Assume t is an integer such that 2 ≤ t ≤ grade R ( Z , N ) and Supp R ( Ext iR ( M , N )) ⊆ Z for all i = , . . . , t −
1. Then the following hold:(i) Ext iR ( M , N ) ∼ = H i + Z ( Hom R ( M , N )) for all i = , . . . , t − t − R ( M , N ) ֒ → H t Z ( Hom R ( M , N )) . Proof.
We prove both part (i) and part (ii) simultaneously.Recall that M admits a free resolution · · · −→ F ∂ −→ F ∂ −→ M −→ , where F i is a finitely generated free R -module for all i ≥
0. Applying the functor ( − ) ▽ : = Hom R ( − , N ) to the above resolution we obtain the following complex:0 −→ M ▽ ∂ ▽ −→ ( F ) ▽ ∂ ▽ −→ · · · ∂ ▽ i − −→ ( F i − ) ▽ ∂ ▽ i −→ ( F i ) ▽ ∂ ▽ i + −→ ( F i + ) ▽ −→ · · · , which induces the following exact sequences:0 −→ Ext iR ( M , N ) −→ T i −→ X i −→ , −→ X i −→ ( F i + ) ▽ −→ T i + −→ , where T i : = ( F i ) ▽ im ( ∂ ▽ i ) and X i : = im ( ∂ ▽ i + ) for all i >
0. For each i ≥
0, we note that grade R ( Z , ( F i ) ▽ ) = grade R ( Z , N ) ≥ t . By our assumption, Ext iR ( M , N ) is torsion with respect to Z for all 0 < i < t . ThenExt iR ( M , N ) = Γ Z ( Ext iR ( M , N )) and H j Z ( Ext iR ( M , N )) = j > < i < t (see Theorem2.11(iii)). Applying the functor Γ Z ( − ) to the above exact sequences, we get the isomorphisms(3.2.1) Ext iR ( M , N ) = Γ Z ( Ext iR ( M , N )) ∼ = Γ Z ( T i ) for all 0 < i < t , (3.2.2) H j Z ( T i ) ∼ = H j Z ( X i ) for all 0 < i < t and j > , (3.2.3) H j Z ( T i + ) ∼ = H j + Z ( X i ) for all 0 ≤ j < n − and i ≥ . It follows from (3.2.1), (3.2.2) and (3.2.3) thatExt iR ( M , N ) ∼ = Γ Z ( T i ) ∼ = H Z ( X i − ) ∼ = H Z ( T i − ) (3.2.4) ∼ = H Z ( X i − ) ∼ = H Z ( T i − ) ∼ = · · · ∼ = H i − Z ( T ) , for each 0 < i < t . Applying the functor Γ Z ( − ) to the exact sequences, 0 → M ▽ → ( P ) ▽ → X → → X → ( P ) ▽ → T →
0, we obtain the isomorphism(3.2.5) H i − Z ( T ) ∼ = H i + Z ( M ▽ ) for all 0 < i < n − . Also, there is an injection(3.2.6) 0 −→ H t − Z ( X ) −→ H t Z ( M ▽ ) . The first assertion follows from (3.2.4) and (3.2.5). Note that by (3.2.3) and (3.2.4)Ext t − R ( M , N ) ∼ = H t − Z ( T ) ∼ = H t − Z ( X ) . (3.2.7)The second assertion follows from (3.2.6) and (3.2.7). (cid:3) Remark 3.3.
The injection Ext t − R ( M , N ) ֒ → H t Z ( Hom R ( M , N )) from Lemma 3.2 is not necessarily anisomorphism: to see this, we consider R = k [[ x , y ]] , m = ( x , y ) , M = N = R and Z = V ( m ) . Then itfollows t = = Ext R ( R , R ) = Ext t − R ( M , N ) ֒ → H t m ( Hom R ( M , N )) = H m ( R ) = (cid:3) For M ∈ mod ( R ) , we denote by NF ( M ) the non-free locus of M . It is well-known that NF ( M ) is aclosed subset of Spec R . In passing, we record an immediate consequence of Lemma 3.2: Corollary 3.4.
Let M ∈ mod ( R ) , N ∈ Mod ( R ) , and let a be an ideal of R. Assume NF ( M ) ⊆ V ( a ) , e.g., a is a defining ideal of NF ( M ) . If n is an integer such that ≤ n ≤ grade R ( a , N ) , then the following hold: (i) Ext iR ( M , N ) ≃ H i + a ( Hom R ( M , N )) for all i = , . . . , n − . (ii) There is an injection
Ext n − R ( M , N ) ֒ → H n a ( Hom R ( M , N )) . Next we use Lemma 3.2 and obtain:
Lemma 3.5.
Let Z ⊂ Spec R be a specialization-closed subset, M ∈ mod ( R ) and let N ∈ Mod ( R ) .Assume n is an integer such that 0 ≤ n ≤ grade R ( Z , N ) and Supp R ( Ext iR ( Tr M , N )) ⊆ Z for all i = , . . . , n +
1. Then the following hold:(i) H i Z ( M ⊗ R N ) ∼ = Ext i + R ( Tr M , N ) for all i = , . . . , n − n + R ( Tr M , N ) ֒ → H n Z ( M ⊗ R N ) . Proof.
Consider the exact sequence (2.2.2):(3.5.1) 0 → Ext R ( Tr M , N ) → M ⊗ R N → Hom R ( M ∗ , N ) → Ext R ( Tr M , N ) → . By applying the functor Γ Z ( − ) to the 4-term exact sequence (3.5.1) and noting that Ext R ( Tr M , N ) istorsion with respect to Z we get the injection Ext R ( Tr M , N ) ֒ → Γ Z ( M ⊗ R N ) . Hence, from now on wemay assume that n >
0. The exact sequence (3.5.1) induces the following short exact sequences:(3.5.2) 0 → Ext R ( Tr M , N ) → M ⊗ R N → X → , (3.5.3) 0 → X → Hom R ( M ∗ , N ) → Ext R ( Tr M , N ) → . As Ext iR ( Tr M , N ) is torsion with respect to Z for all 1 ≤ i ≤ n , by Theorem 2.11(ii) we have(3.5.4) Γ Z ( Ext iR ( Tr M , N )) = Ext iR ( Tr M , N ) and H j Z ( Ext iR ( Tr M , N )) = j > ≤ i ≤ n . By Lemma 2.17, grade R ( Z , Hom R ( M ∗ , N )) ≥ min { , grade R ( Z , N ) } >
0. It follows from the exactsequence (3.5.3) that grade R ( Z , X ) >
0. In other words, Γ Z ( X ) =
0. Applying the functor Γ Z ( − ) to the exact sequences (3.5.2), (3.5.3) and using (3.5.4) we get the following isomorphisms and exactsequence:(3.5.5) Γ Z ( M ⊗ R N ) ∼ = Γ Z ( Ext R ( Tr M , N )) = Ext R ( Tr M , N ) and H Z ( X ) ∼ = H Z ( M ⊗ R N ) , (3.5.6) Ext R ( Tr M , N ) = Γ Z ( Ext R ( Tr M , N )) ֒ → H Z ( X ) → H Z ( Hom R ( M ∗ , N )) , (3.5.7) H i Z ( M ⊗ R N ) ∼ = H i Z ( Hom R ( M ∗ , N )) for all i > . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 11
The case n = n >
1. Therefore, grade R ( Z , Hom R ( M ∗ , N )) ≥ Z ( Hom R ( M ∗ , N )) =
0. It follows from (3.5.5) and (3.5.6) that(3.5.8) H Z ( M ⊗ R N ) ∼ = Ext R ( Tr M , N ) . Note that M ∗ ≈ Ω Tr M . Hence, we have the following exact sequence(3.5.9) Ext R ( Tr M , N ) ∼ = Ext R ( M ∗ , N ) ֒ → H Z ( Hom R ( M ∗ , N )) ∼ = H Z ( M ⊗ R N ) , where the injection follows from Lemma 3.2 and the last isomorphism follows from (3.5.7). Now theassertion is clear by (3.5.5), (3.5.8) and (3.5.9) for the case n =
2. Finally, let n >
2. By (3.5.7) andLemma 3.2, we get the following isomorphisms:H i Z ( M ⊗ R N ) ∼ = H i Z ( Hom R ( M ∗ , N )) ∼ = Ext i − R ( M ∗ , N ) ∼ = Ext i + R ( Tr M , N ) for all 1 < i < n and also we obtain an injection Ext n + R ( Tr M , N ) ֒ → H n Z ( M ⊗ R N ) as desired. (cid:3) The next result is used throughout the paper.
Lemma 3.6.
Let M , N ∈ mod ( R ) , and let X ⊆ Spec R . Assume M p is totally reflexive over R p for all p ∈ X. Then the following conditions are equivalent:(i) c Tor R p i ( M p , N p ) = i ∈ Z and for all p ∈ X.(ii) Supp R ( Ext iR ( Tr M , N )) S Supp R ( Tor Ri ( M , N )) ⊆ Spec R \ X for all i ≥ Proof.
Let p ∈ X. Note that M ∗ ≈ Ω Tr M . We apply this along with parts (ii),(iii) and (iv) of Theorem2.21 to deduce, for all i ≤
0, that c Tor R p i ( M p , N p ) ∼ = c Ext − i − R p ( M ∗ p , N p ) ∼ = c Ext − i + R p ( Tr R p M p , N p ) ∼ = Ext − i + R p ( Tr R p M p , N p ) . Also, since M p is totally reflexive over R p , we know c Tor R p j ( M p , N p ) ∼ = Tor R p j ( M p , N p ) for all j ≥
1. Hencethe assertion follows. (cid:3)
Recall that M ∈ Mod ( R ) is called Tor-rigid provided that the vanishing of a single Tor Rj ( M , N ) forsome N ∈ Mod ( R ) and for some j ≥ Ri ( M , N ) for all i ≥ j . Clearly, over alocal ring, every syzygy module of the residue field is Tor-rigid. Note that, it follows from Theorem 3.1that each finitely generated module over a regular local ring is Tor-rigid. For more examples of Tor-rigidmodules, see, for example, [30].The following is the key for our proof of Theorem 3.8. Lemma 3.7.
Assume R is local, M , N ∈ mod ( R ) , and N is Tor-rigid. Assume further, for a specialization-closed subset Z ⊆ Spec R , that the following conditions hold:(i) M p is a totally reflexive R p -module for all p ∈ Supp R ( M ) \ Z (e.g., NF ( M ) ⊆ Z ).(ii) c Tor R p i ( M p , N p ) = p ∈ Spec R \ Z and for i ∈ Z (e.g., NF ( M ) ⊆ Z ).(iii) H n Z ( M ⊗ R N ) = n , where 0 ≤ n ≤ grade R ( Z , N ) .Then the following hold:(a) Ext n + R ( Tr M , R ) =
0. Moreover, if grade R ( Z , R ) ≥ n +
1, then H n Z ( M ) = Rj ( Ω − n M , N ) = j ≥ n ≥
1, then H n − Z ( M ⊗ R N ) ∼ = Ext nR ( Tr M , R ) ⊗ R N . Therefore, if n ≤ grade R ( Z , R ) , thenH n − Z ( M ⊗ R N ) ∼ = H n − Z ( M ) ⊗ R N . Proof.
We prove the statements simultaneously.
Note that by Lemma 3.6, Supp R ( Ext iR ( Tr M , N )) ⊆ Z for all i >
0. It follows from Lemma 3.5 andassumption (iii) that Ext n + R ( Tr M , N ) =
0. By (2.2.5), there exists the following exact sequence(3.7.1) Tor R ( Ω − ( n + ) M , N ) → Ext n + R ( Tr M , R ) ⊗ R N → Ext n + R ( Tr M , N ) ։ Tor R ( Ω − ( n + ) M , N ) . It follows from (3.7.1) that Tor R ( Ω − ( n + ) M , N ) =
0. As N is Tor-rigid, we have(3.7.2) Tor Rj ( Ω − ( n + ) M , N ) = j ≥ . Again, by the exact sequence (3.7.1), we get that Ext n + R ( Tr M , R ) ⊗ R N = n + R ( Tr M , R ) =
0. Note that by assumption (i) we have Supp R ( Ext iR ( Tr M , R )) ⊆ Z for all i ≥ R ( Z , R ) ≥ n +
1, then H n Z ( M ) ∼ = Ext n + R ( Tr M , R ) = Ω − n M ≈ ΩΩ − ( n + ) M . Therefore, by (3.7.2), we getTor Rj ( Ω − n M , N ) = j > n >
0. By part (b) and (3.7.1), we get the following isomorphism:(3.7.3) Ext nR ( Tr M , R ) ⊗ R N ∼ = Ext nR ( Tr M , N ) . Now the assertion (c) follows from (3.7.3) and Lemma 3.5. (cid:3)
Now we are ready to state and prove the main result of this section:
Theorem 3.8.
Assume R is local, M , N ∈ mod ( R ) , and N is Tor-rigid. Assume further, for an integern ≥ and a specialization-closed subset Z ⊆ Spec
R, that the following conditions hold: (i) M p is totally reflexive over R p for all p ∈ Supp R ( M ) \ Z . (ii) c Tor R p i ( M p , N p ) = for all i ∈ Z and for all p ∈ Spec R \ Z . (iii) grade R ( Z , M ) ≥ n and grade R ( Z , N ) ≥ n. (iv) H n Z ( M ⊗ R N ) = .If, either n ≤ grade R ( Z , R ) , or R is Cohen-Macaulay, then H i Z ( M ⊗ R N ) = for all i ≤ n, and M and Nare Tor-independent, i.e., Tor Rj ( M , N ) = for all j ≥ .Proof. We proceed by induction on the integer n .If n =
0, then the required result follows from the assumption H n Z ( M ⊗ R N ) = n ≥
1. Note that it suffices to show H n − Z ( M ⊗ R N ) =
0; as then the induction hypothesisimplies H i Z ( M ⊗ R N ) = i ≤ n − Rj ( M , N ) = j ≥
1, as required.First suppose n ≤ grade R ( Z , R ) . Then, by Lemma 3.7(c), we have the following isomorphism:(3.8.1) H n − Z ( M ⊗ R N ) ∼ = H n − Z ( M ) ⊗ R N . As grade R ( Z , M ) ≥ n , we conclude from (3.8.1) that H n − Z ( M ⊗ R N ) =
0; see 2.15.Next suppose R is Cohen-Macaulay. Then it follows from Lemma 3.7(c) that the following holds:(3.8.2) H n − Z ( M ⊗ R N ) ∼ = Ext nR ( Tr M , R ) ⊗ R N . As M p is totally reflexive over R p for all p ∈ Supp R ( M ) \ Z and grade R ( Z , M ) ≥ n , Proposition 2.19(iii)shows that M is n -torsion-free. Thus Ext nR ( Tr M , R ) = n − Z ( M ⊗ R N ) = (cid:3) The following example shows that the Tor-rigidity assumption in Theorem 3.8 is necessary:
Example 3.9.
Let R = k [[ x , y , u , v ]] / ( xy − uv ) , M = ( x , u ) , N = M ∗ , and Z = V ( m ) . It has been estab-lished in [36, 1.8] that M is a maximal Cohen-Macaulay R -module which is locally free on the puncturedspectrum of R . Also, one can check that there is an isomorphism M ⊗ R N ≃ m .We set n =
2. Then all of the hypotheses of Theorem 3.8, except Tor-rigidity, hold: Tor R ( M , N ) = R ( M , N ) ≃ Tor R ( M , N ) ≃ k =
0. The desired claim is not true because H m ( M ⊗ R N ) ≃ H m ( m ) = STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 13
Lemma 3.10.
Let M , N ∈ mod ( R ) and let X ⊆ Spec R be a subset. Assume CI-dim R p ( M p ) = p ∈ X . Then the following conditions are equivalent:(i) c Tor R p i ( M p , N p ) = i ∈ Z and for all p ∈ X.(ii) Supp R ( Tor Ri ( M , N )) ⊆ Spec R \ X for all i ≫ R ( Tor Ri ( M , N )) ⊆ Spec R \ X for all i ≥ Proof.
The equivalence of parts (i) and (ii) is due to Theorem 2.22. Moreover, the equivalence of parts(ii) and (iii) follows from (2.3.1) and the dependency formula 2.7. (cid:3)
Theorem 1.1, advertised in the introduction, is a special case of the following proposition: thanks toTheorem 3.8, it can now be easily established.
Proposition 3.11.
Assume R is local and a complete intersection, M , N ∈ mod ( R ) are nonzero, and Z ⊂ Spec
R is a specialization-closed subset. Assume further N is Tor-rigid and, for an integer n ≥ ,the following conditions hold: (i) M p is maximal Cohen-Macaulay over R p for all p ∈ Supp R ( M ) \ Z . (ii) Supp R ( Tor Ri ( M , N )) ⊆ Z for all i ≫ . (iii) H n Z ( M ⊗ R N ) = . (iv) grade R ( Z , M ) ≥ n and grade R ( Z , N ) ≥ n.Then H i Z ( M ⊗ R N ) = for all i ≤ n, and M and N are Tor-independent.Proof. Note, by (2.3.1) and assumption (i), we have that CI-dim R p ( M p ) = p ∈ Supp R ( M ) \ Z .Thus, letting X = Supp R ( M ) \ Z , we see that M p is totally reflexive over R p for all p ∈ Supp R ( M ) \ Z .Moreover, Lemma 3.10 yields the vanishing of Tor R p i ( M p , N p ) for all i ∈ Z and for all p ∈ X. So theassertion is clear by Theorem 3.8. (cid:3)
We can now make use of Proposition 3.11, which is a vast generalization of Theorem 3.1, and prove:
Theorem 3.12. (Auslander and Lichtenbaum) Assume R is local and regular, and let M , N ∈ mod ( R ) . IfM ⊗ R N is nonzero and torsion-free, then M and N are torsion-free and Tor-independent.Proof.
Recall that finitely generated modules are Tor-rigid over regular local rings [4, 41]. Set Z : = Spec ( R ) \ { } . By Proposition 2.19, Γ Z ( L ) = T ( L ) for every L ∈ mod ( R ) . Now the assertion followsfrom Lemma 3.7(a), (b) and Proposition 3.11. (cid:3)
4. A
PPLICATIONS OF THE MAIN THEOREM
In this section we give several examples, and applications of the results we obtain from Section 3. Thefirst result, Corollary 4.4, we aim to prove is an application of Theorem 3.8.When R is local, we denote by mod ( R ) , the subcategory of finitely generated R -modules which arelocally free on the punctured spectrum of R , namelymod ( R ) = { M ∈ mod ( R ) | M p is free over R p for each p ∈ Spec ( R ) − { m }} . Proposition 4.1.
Assume R is local such that depth ( R ) ≥ , and assume I is an m -primary ideal of Rsatisfying the following condition:If X ∈ mod ( R ) and Tor Rt ( X , R / I ) = for some integer t ≥ , then it follows that pd R ( X ) < t . Then H m ( M ⊗ R I ) = for each M ∈ mod ( R ) such that = M and depth R ( M ) ≥ .Proof. We assume contrary that H m ( M ⊗ R I ) =
0, and seek a contradiction.Letting N = I , we obtain from Theorem 3.8 that Tor Ri ( M , I ) = i ≥
1, and H m ( M ⊗ R I ) = R ( M ⊗ R I ) ≥
2. Now it follows by our assumption on the ideal I that pd R ( M ) < ∞ . ThereforeTheorem 2.6 yields:(4.1.1) pd R ( M ) = depth R − depth R ( M ) = depth R ( I ) − depth R ( M ⊗ R I ) . As I is m -primary, we have that depth R ( I ) =
1. Thus (4.1) shows pd R ( M ) <
0, or equivalently, M = m ( M ⊗ R I ) = (cid:3) Next we recall a class of ideals that enjoys the rigidity property stated in Proposition 4.1: ( [19, 2.10]) Assume R is local, and let I and J be ideals of R . Assume depth ( R ) ≥
1, and thefollowing conditions hold:(i) ( I : R J ) = ( m I : R m J ) .(ii) I is m -primary and 0 = I ⊆ m J .If Tor Rt ( M , R / I ) = M ∈ mod ( R ) and some integer t ≥
0, then it follows that pd R ( M ) < t .When R is local ring and depth ( R ) ≥
1, an integrally closed m -primary ideal I satisfies the conditions(i) and (ii) of 4.2 for the case where J = R ; see [19, the paragraph following 2.1 and 2.4], [17, 2.1 and2.10] and also [27, 3.3]. On the other hand, if I is as in 4.2, then it does not need to be integrally closed,in general; see [17, 2.7 and 2.8] and [19, 4.3]. Although it is difficult to check whether or not a givenideal is integrally closed, in view of 4.2, one can easily construct examples of ideals that have the rigidityproperty stated in Proposition 4.1: Remark 4.3.
Assume R is local and depth ( R ) ≥
1, and let A be an m -primary ideal of R such that A = m .Set I = ( A : R m ) . Then I satisfies the conditions (i) and (ii) of 4.2 for the case where J = R . Therefore,if Tor Rt ( M , R / I ) = M ∈ mod ( R ) and some integer t ≥
0, then it follows that pd R ( M ) < t ;see [17, 2.2] and [19, 2.9].As a consequence of Proposition 4.1 and Remark 4.3, we conclude: Corollary 4.4.
Assume R is local and I is an m -primary ideal of R. Set J = ( I : R m ) . If depth ( R ) ≥ and I = m , then H m ( J ⊗ R J ) = . When the module considered in Proposition 4.1 has depth at least two, we can say more aboutH m ( M ⊗ R I ) . Proposition 4.5.
Assume R is local, I is an m -primary ideal of R, and M ∈ mod ( R ) is such that depth R ( M ) ≥ . Then it follows H i m ( M ⊗ R I ) ≃ Tor R ( M , R / I ) if i = M / IM if i = i m ( M ) if i > Proof.
Applying − ⊗ M to the exact sequence 0 → I → R → R / I →
0, we get the exact sequence0 −→ Tor R ( M , R / I ) −→ I ⊗ R M −→ M −→ R / I ⊗ R M −→ . We break it down into the following exact sequences: ( a ) : 0 → Tor R ( M , R / I ) → I ⊗ R M → X → , ( b ) : 0 → X → M → R / I ⊗ R M → . As Tor R ( M , R / I ) has finite length, H m ( Tor R ( M , R / I )) = Tor R ( M , R / I ) and H j m ( Tor R ( M , R / I )) = j >
0. It follows from the exact sequence ( b ) that depth ( X ) >
0. The exact sequence ( a ) inducesthe isomorphisms H m ( M ⊗ R I ) ≃ Tor R ( M , R / I ) and H j m ( I ⊗ R M ) ≃ H j m ( X ) for all j >
0. Applying thefunctor Γ m ( − ) to the exact sequence ( b ) and noting that depth R ( M ) > = H m ( M ) −→ H m ( R / I ⊗ R M ) −→ H m ( X ) −→ H m ( M ) = , and H i m ( X ) ∼ = H i m ( M ) for all i >
1. Therefore, we obtain the following isomorphisms:H m ( I ⊗ R M ) ≃ H m ( X ) ≃ H m ( R / I ⊗ R M ) = R / I ⊗ R M ≃ M / IM . Also, H i m ( I ⊗ R M ) ≃ H i m ( M ) for all i > (cid:3) Next our aim is to make use of Theorem 3.8 and prove the following:
STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 15
Theorem 4.6.
If R is regular of dimension d, then, for nonzero M , N ∈ mod ( R ) , the following hold: (i) depth R ( M ⊗ R N ) ≤ min { depth R ( M ) , depth R ( N ) } . Furthermore, it follows that H i m ( M ⊗ R N ) = foreach i, where depth R ( M ⊗ R N ) ≤ i ≤ min { depth R ( M ) , depth R ( N ) } . (ii) Assume depth R ( M ) + depth R ( N ) = d + n for some n ≥ . Then, (a) H n m ( M ⊗ R N ) = so that depth R ( M ⊗ R N ) ≤ n. (b) If H i m ( M ⊗ R N ) = for some i, where ≤ i < n, then depth R ( M ⊗ R N ) = n. (iii) If depth R ( M )+ depth R ( N ) ≤ d, then H i m ( M ⊗ R N ) = for each i = , . . . , min { depth R ( M ) , depth R ( N ) } . (iv) If depth R ( M ) ≤ d, then H i m ( M ⊗ R M ) = for each i = , . . . , depth R ( M ) . The proof of Theorem 4.6 requires some preparation; we start with:
Proposition 4.7.
Assume R is a local complete intersection and assume Z ⊂ Spec
R is a specialization-closed subset. Assume further that M , N ∈ mod ( R ) are nonzero such that NF ( M ) ⊆ Z ⊆ Supp R ( M ) and grade R ( Z , N ) ≤ grade R ( Z , M ) . If N is Tor-rigid, then grade R ( Z , M ⊗ R N ) ≤ grade R ( Z , N ) . Moreover, H i Z ( M ⊗ R N ) = for all i, where grade R ( Z , M ⊗ R N ) ≤ i ≤ grade R ( Z , N ) .Proof. Set n = grade R ( Z , N ) . Assume that H j Z ( M ⊗ R N ) = ≤ j ≤ n . By Corollary 3.11,(4.7.1) grade R ( Z , M ⊗ R N ) > j and also Tor Rk ( M , N ) = k ≥ . By Proposition 2.16, depth R p ( N p ) = n for some p ∈ Z ⊆ Supp R ( M ) . As grade R ( Z , M ⊗ R N ) > j , againby Proposition 2.16, we conclude that depth R p ( M p ⊗ R p N p ) > j . Therefore, in view of Theorem 2.6,(2.3.1) and (4.7.1) we have0 ≤ CI-dim R p ( M p ) = depth R p − depth R p ( M p ) = depth R p ( N p ) − depth R p ( M p ⊗ R p N p ) < n − j . It follows from the above inequality that j < n . In other words, H n Z ( M ⊗ R N ) =
0. In particular,grade R ( Z , M ⊗ R N ) ≤ grade R ( Z , N ) . The second assertion follows from (4.7.1). (cid:3) In passing we record:
Corollary 4.8.
Assume R is local and regular, and Z ⊂ Spec
R is a specialization-closed subset. Assumefurther that M , N ∈ mod ( R ) are nonzero such that NF ( M ) ∪ NF ( N ) ⊆ Z ⊆ Supp R ( M ⊗ R N ) . Then grade R ( Z , M ⊗ R N ) ≤ min { grade R ( Z , M ) , grade R ( Z , N ) } . Moreover, H i Z ( M ⊗ R N ) = for all i, where grade R ( Z , M ⊗ R N ) ≤ i ≤ min { grade R ( Z , M ) , grade R ( Z , N ) } .Proof. Recall that over regular local rings each finitely generated module is Tor-rigid [4, 41]. Hence theassertion is clear by Proposition 4.7. (cid:3)
The non-vanishing results we prove in the next theorem are, to the best of our knowledge, new evenover regular local rings.
Theorem 4.9.
Assume R is a local complete intersection of dimension d, M ∈ mod ( R ) and N ∈ mod ( R ) is Tor-rigid. Then, for a nonnegative integer n, the following hold: (i) If depth R ( M ) + depth R ( N ) = d + n, then H n m ( M ⊗ R N ) = . In particular, depth R ( M ⊗ R N ) ≤ n. Inaddition, if H i m ( M ⊗ R N ) = for some ≤ i < n, then depth R ( M ⊗ R N ) = n. (ii) If depth R ( M ) + depth R ( N ) ≤ d, then H i m ( M ⊗ R N ) = for all ≤ i ≤ min { depth R ( M ) , depth R ( N ) } . (iii) depth R ( M ⊗ R N ) = max { , depth R ( M ) + depth R ( N ) − d } . Proof. (i). As depth R ( M ) + depth R ( N ) = d + n , we conclude that depth R ( N ) ≥ n . Assume contrarily thatH n m ( M ⊗ R N ) =
0. It follows from Theorem 3.8 that Tor Ri ( M , N ) = i > R ( M ⊗ R N ) > n .By Theorem 2.6, depth R ( M ) + depth R ( N ) = d + depth R ( M ⊗ R N ) > d + n , which is a contradiction. Thesecond assertion can be proved similarly.(ii). Set n : = min { depth R ( M ) , depth R ( N ) } . Assume contrarily that H i m ( M ⊗ R N ) = ≤ i ≤ n . By Theorem 3.8 Tor Ri ( M , N ) = i > R ( M ⊗ R N ) > i . Due to Theorem 2.6depth R ( M ) + depth R ( N ) = d + depth R ( M ⊗ R N ) > d which is a contradiction.(iii). This is an immediate consequence of parts (i) and (ii). (cid:3) We are now ready to give a proof of Proposition 4.6:
Proof of Proposition 4.6.
Recall that every finitely generated module over a regular local ring is Tor-rigid [4, 41]. Therefore part (i) follows from Proposition 4.7 by setting Z = V ( m ) . Moreover, parts (ii)and (iii) are immediate consequence of Theorem 4.9, and part (iv) is a special case of part (iii). (cid:3) We finish this section by giving two examples. Example 4.10 corroborates Theorem 4.9 and showthat our results are sharp. On the other hand, Example 4.11 provides a detailed computation of the localcohomology of Ω k ⊗ R Ω k over a regular local ring of dimension four; cf. Theorem 4.6.In the following, we adopt the notations of Theorem 4.9 and collect some examples of specialization-closed subsets of Spec R : Example 4.10. (i) Assume R is a three-dimensional complete intersection ring and x ∈ m is a non zero-divisor on R .Let N = M = R / xR . Then pd R ( M ) = M is Tor-rigid. We consider Theorem 4.9(i) for thecase n =
1; depth R ( M )+ depth R ( N ) = d + n =
4. However, H m ( M ⊗ R N ) = M ⊗ R M ≃ M anddepth R ( M ) =
2. Note that M / ∈ mod ( R ) . Hence the example shows that the locally free assumptionis necessary for Theorem 4.9(i).(ii) Let R = k [[ x , y , u , v ]] / ( xy − uv ) , M = ( x , u ) and let N = M ∗ . Recall from Example 3.9 that M islocally free and maximal Cohen–Macaulay, and that M ⊗ R N ∼ = m . We consider Theorem 4.9(i) forthe cases where n = i =
2. Then it follows that depth R ( M ) + depth R ( N ) = d + n =
6. Notethat, although H m ( M ⊗ R N ) ≃ H m ( m ) =
0, we have that depth R ( M ⊗ R N ) = = n =
3. As N is notTor-rigid, this example shows that the Tor-rigidity assumption is crucial for Theorem 4.9(i).(iii) Assume R is a five-dimensional regular local ring, and let M = Ω k and N = Ω k . We considerTheorem 4.9(ii). Note that M and N are locally free on the punctured spectrum of R , N is Tor-rigid,and depth R ( M ) + depth R ( N ) = + > = dim R . Moreover, in view of [2, 5.5(ii)], we have thatH m ( M ⊗ R N ) =
0. This example shows that the depth condition in Theorem 4.9(ii) is necessary.In the following, β n ( M ) denotes the n -th Betti number of a given module M . Example 4.11.
Assume R is a four-dimensional regular local ring and let M = Ω k . Then M is locallyfree on the punctured spectrum of R , has depth two and ℓ ( H i m ( M ⊗ R M )) = β ( k ) if i = β ( k ) if i = β ( k ) if i = β ( k ) if i = i m ( M ⊗ R M ) = i ≤ Proof.
It follows from the Theorem 2.13(i) that H m ( M ⊗ R M ) =
0. Therefore, we proceed to show theother claims of the example.Consider the following exact sequences: ( a ) : 0 → Ω k → R β → Ω k → , ( b ) : 0 → Ω k → R → k → . The exact sequence (a) induces the following exact sequence:0 → Tor R ( Ω k , Ω k ) → Ω k ⊗ R Ω k → R β ⊗ R Ω k → Ω k ⊗ R Ω k → . We break it down into ( a . ) : 0 → Tor R ( Ω k , Ω k ) → Ω k ⊗ R Ω k → X → , ( a . ) : 0 → X → R β ⊗ R Ω k → Ω k ⊗ R Ω k → . It follows from ( a . ) that depth ( X ) >
0. Hence the exact sequence ( a . ) induces the isomorphisms(4.11.1) H m ( Ω k ⊗ R Ω k ) ≃ H m ( Tor R ( Ω k , Ω k )) ≃ Tor R ( Ω k , Ω k ) ≃ Tor R ( k , k ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 17
Therefore, ℓ ( H m ( M ⊗ R M )) = β ( k ) . Since Tor R ( Ω k , Ω k ) has finite length, H i m ( Tor R ( Ω k , Ω k )) = i >
0. Thus, the exact sequence ( a . ) induces the isomorphism H j m ( Ω k ⊗ R Ω k ) ≃ H j m ( X ) forall j >
0. Note that H m ( Ω k ) = H m ( Ω k ) =
0. Therefore, the exact sequence ( a . ) induces the exactsequence: 0 = H m ( R β ⊗ R Ω k ) → H m ( Ω k ⊗ R Ω k ) → H m ( X ) → H m ( R β ⊗ R Ω k ) = . Hence, we get the following isomorphisms(4.11.2) H m ( Ω k ⊗ R Ω k ) ≃ H m ( X ) ≃ H m ( Ω k ⊗ R Ω k ) . The exact sequence ( b ) induces the following exact sequence:0 −→ Tor R ( k , Ω k ) → Ω k ⊗ R Ω k −→ Ω k → k ⊗ R Ω k −→ . We break it down into the following short exact sequences: ( b . ) : 0 → Tor R ( k , Ω k ) → Ω k ⊗ R Ω k → Y → , ( b . ) : 0 → Y → Ω k → k ⊗ R Ω k → . It follows from ( b . ) that depth ( Y ) >
0. Hence, the exact sequence ( b . ) induces the isomorphism(4.11.3) H m ( Ω k ⊗ Ω k ) ≃ H m ( Tor R ( k , Ω k )) ≃ Tor R ( k , Ω k ) ≃ Tor R ( k , k ) . By (4.11.2) and (4.11.3), we have(4.11.4) H m ( Ω k ⊗ R Ω k ) ≃ H m ( Ω k ⊗ R Ω k ) ≃ Tor R ( k , k ) . Therefore, ℓ ( H m ( M ⊗ R M )) = β ( k ) . Recall that H m ( Ω k ) = H m ( Ω k ) = m ( Ω k ) = k (see [32, 3.3(2)]). Therefore, applying the functor Γ m ( − ) to the exact sequence ( a . ) , we get the longexact sequence:(4.11.5) 0 → H m ( Ω k ⊗ R Ω k ) → H m ( X ) → H m ( R β ⊗ R Ω k ) → H m ( Ω k ⊗ R Ω k ) → H m ( X ) → . Also, the exact sequences ( b . ) and ( b . ) induce the following isomorphisms:(4.11.6) H i m ( Ω k ⊗ R Ω k ) ≃ H i m ( Y ) and H j m ( Y ) ≃ H j m ( Ω k ) for all i > j > , (4.11.7) k ⊕ β ≃ k ⊗ R Ω k ≃ H m ( k ⊗ R Ω k ) ≃ H m ( Y ) ≃ H m ( Ω k ⊗ R Ω k ) . Therefore, by using (4.11.6) and (4.11.7) one can rewrite the exact sequence (4.11.5) as follows:(4.11.8) 0 → k ⊕ β → H m ( X ) → k ⊕ β f → k g → H m ( X ) → . Claim. H m ( M ⊗ M ) ∼ = H m ( X ) = m ( X ) v ≃ H m ( M ⊗ M ) v ≃ H m ( M ∗ ⊗ M ∗ ) ≃ H m ( Ω k ⊗ Ω k ) , where ( − ) v is the Matlis dual. Recall that depth ( Ω k ) + depth ( Ω k ) = (cid:0) + +
1. In view of [36, 2.4]we see that H m ( Ω k ⊗ Ω k ) =
0. Now the assertion follows from (4.11.9) and the proof of the claim iscomplete.Since H m ( X ) is non-zero and a homomorphic image of k , we conclude that H m ( X ) ≃ k . In particular, ℓ ( H m ( M ⊗ M )) = ℓ ( H m ( X )) = β ( k ) . It follows from the exact sequence (4.11.8) that g is an isomorphismand f = ℓ ( H m ( M ⊗ M )) = ℓ ( H m ( X )) = β ( k ) + β ( k ) = β ( k ) , as required. (cid:3)
5. O
N THE DEPTH OF TENSOR POWERS OF MODULES
Auslander studied the torsion-freeness of tensor powers of modules and obtained a criteria for free-ness. More precisely, over an unramified regular local ring R of dimension d , Auslander [4] proved thatan R -module M is free provided that the d -fold tensor product M ⊗ d of M is torsion-free. The aim ofthis section is to generalize Auslander’s result over more general rings, and find some new criteria forfreeness of modules in terms of the local cohomology; see Corollary 5.4.We call M ∈ Mod ( R ) a self-test module provided that the following condition holds:Tor Ri ( M , M ) = i ≫ ⇐⇒ pd R ( M ) < ∞ . It is an open question whether or not each module is self-test over local rings. However, it is knownthat, if R is a complete intersection or Golod, then each module in mod ( R ) is self-test; see [8, TheoremIV] and [38, 3.6]. We refer the reader to [23] for further details about the self-test modules.The next theorem is the main result of this section; to the best of our knowledge, it is new, even if thering in question is regular. Theorem 5.1.
Assume R is local, Z ⊂ Spec
R is a specialization-closed subset, and M ∈ mod ( R ) isnonzero and Tor-rigid such that NF ( M ) ⊆ Z . (i) If grade R ( Z , M ) < grade R ( Z , R ) , then H grade R ( Z , M ) Z ( ⊗ nR M ) = for all n ≥ . (ii) Assume, for some n ≥ , and for some i, where i < min { grade R ( Z , M ) , grade R ( Z , R ) } , that wehave H i Z ( ⊗ nR M ) = . Then: (a) Tor Rk ( M , M ) = for all k ≥ , and H j Z ( ⊗ mR M ) = for all m = , . . . , n and for all j = , . . . , i. (b) If M is self-test, then pd R ( M ) < ( depth R − i ) / n and also dim R ( NF ( M )) < dim R − n − i.Proof. We prove the statements simultaneously. For each j ≥
1, set M j : = j ⊗ M . Assume H i Z ( n ⊗ M ) = i ≤ grade R ( Z , M ) and n ≥
2. If i < grade R ( Z , R ) , then by Lemma 3.7(a), we get H i Z ( M n − ) =
0. By repeating this argument, inductively we get the following:(5.1.1) H i Z ( M k ) = ≤ k ≤ n . In particular, H i Z ( M ) =
0. Hence, i < grade R ( Z , M ) . This prove part (i). Next we claim the following: Claim: grade R ( Z , M j + ) > i and Tor Rk ( M j , M ) = ≤ j ≤ n − k > j . By (5.1.1), H i Z ( M ⊗ R M ) =
0. It follows from Theorem 3.8that Tor Rk ( M , M ) = k > R ( Z , M ) > i . Hence the case j = j > R ( Z , M j ) > i and Tor Rk ( M j − , M ) = k >
0. By (5.1.1), H i Z ( M j ⊗ R M ) =
0. It follows from Theorem 3.8 that Tor Rk ( M j , M ) = k > R ( Z , M j + ) > i . Thus,the proof of part (a), as well as that of the claim is completed.Now assume M is self-test. By the Claim Tor Rk ( M j , M ) = ≤ j ≤ n − k >
0. Hence,pd R ( M j ) < ∞ for all 1 ≤ j ≤ n and so by [3, 1.3] we get the following equality:(5.1.2) pd R ( M j + ) = pd R ( M ) + pd R ( M j ) for all 1 ≤ j ≤ n − . It follows from Auslander–Buchsbaum formula and (5.1.2) thatdepth R − depth R ( M n ) = pd R ( M n ) = n . pd R ( M ) . By the claim, grade R ( Z , M n ) > i . In particular, by Proposition 2.16, depth R ( M n ) > i . Therefore,pd R ( M ) < ( depth R − i ) / n .Let dim R ( NF ( M )) = t = dim ( R / p ) for some p ∈ NF ( M ) . By the Claim Tor Rk ( M j , M ) p = k > ≤ j ≤ n −
1. Thus, we get the equality pd R p (( M j + ) p ) = pd R p (( M j ) p ) + pd R p ( M p ) for all 1 ≤ j ≤ n − . Therefore, we obtain the following equality:(5.1.3) n . pd R p ( M p ) = pd R p (( M n ) p ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 19
Assume contrary that dim ( R / p ) = dim R ( NF ( M )) ≥ dim R − n − i . Hence, we obtain the inequality:(5.1.4) depth R p ≤ dim R p ≤ dim R − dim ( R / p ) ≤ n + i . By the Claim and Proposition 2.16, depth R p (( M n ) p ) > i and so by (5.1.4) we have(5.1.5) pd R p (( M n ) p ) = depth R p − depth R p (( M n ) p ) < n . It follows from (5.1.3) and (5.1.5) that p / ∈ NF ( M ) which is a contradiction. (cid:3) Next is a consequence of Theorem 5.1; part (iii) of Corollary 5.2 is a generalization of [4, 3.2].
Corollary 5.2.
Assume R is a local complete intersection of dimension d and M ∈ mod ( R ) is Tor-rigid(e.g., R is regular). Then, for n ≥ , the following hold: (i) If a is an ideal of R, NF ( M ) ⊆ V ( a ) and H i a ( ⊗ nR M ) = for some ≤ i < grade R ( a , M ) , then pd R ( M ) < ( d − i ) / n. Therefore, if d ≤ i + n, then it follows that M is free. (ii) If NF ( M ) ⊆ V ( m ) and H i m ( ⊗ nR M ) = for some ≤ i < depth R ( M ) and n ≥ d − i, then M is free. (iii) If M p is free for each p ∈ Ass ( R ) and ⊗ nR M is torsion-free for some n ≥ d, then M is free.Proof. First we note, by [8, Theorem IV], that M is a self-test module.(i) The assertion follows from Theorem 5.1 by setting Z = V ( a ) .(ii) This is a special case of part (i).(iii) We set Z = { p ∈ Spec R | ht ( p ) > } . Then, by Proposition 2.19(i) and our assumption, it followsthat Γ Z ( ⊗ nR M ) = T ( ⊗ nR M ) =
0. Note by Proposition 2.16 that grade R ( Z , R ) >
0. Hence, by Theorem5.1(i), we may assume that grade R ( Z , M ) >
0. Now the assertion follows from Theorem 5.1(ii). (cid:3)
We give several examples and show that the conclusions of Corollary 5.2 are sharp:
Example 5.3. (i) Assume R is a three-dimensional regular local ring and let M = Ω R k . Then M has depth two, is notfree but is locally free on the punctured spectrum of R , and is Tor-rigid. In view of [2, 5.3] one hasthat H m ( M ⊗ R M ) =
0. This example shows that the bound on n in parts (ii) and (iii) of Corollary5.2 is necessary and sharp.(ii) Let R = C [[ x , y , z ]] and let M = R / xR . Then it follows that M is not free, M ⊗ R M ∼ = M anddepth R ( M ) =
2. Moreover, we have that H ≤ m ( ⊗ nR M ) = H ≤ m ( M ⊗ R M ) ∼ = H ≤ m ( M ) =
0. This showsthat, letting i = n =
2, the assumption NF ( M ) ⊆ V ( m ) in Corollary 5.2(ii) cannot be removed.(iii) Let R = k [[ x , y ]] / ( xy ) and let M = R / xR . Then R is reduced, dim ( R ) = = depth R ( M ) , M is locallyfree on the punctured spectrum of R , and H m ( ⊗ nR M ) = n ≥
2. However, M is not Tor-rigid: Tor R ( M , N ) = = Tor R ( M , N ) , where N = R / yR . This example shows that the Tor-rigidityassumption in Corollary 5.2(iii) is needed.We proceed with another consequence of Theorem 5.1. Corollary 5.4.
Assume R is a local ring of positive depth. Set d = dim ( R ) . Then, for an integer n withn ≥ max { , d } , the following hold: (i) depth R ( ⊗ nR ( Ω i k )) = for all i = , . . . , d − . (ii) If R is not regular, then depth R ( ⊗ nR ( Ω i k )) = for all i ≥ . (iii) The sequence f ( n ) = depth R ( ⊗ nR ( Ω i k )) is eventually constant for all i ≥ .Proof. (i). Without loss of generality, we may assume that 0 < i < d . Assume that we havedepth R ( ⊗ nR M ) ≥
1, where M = Ω i k . Then H m ( ⊗ nR M ) =
0. Since M is Tor-rigid, locally free on thepunctured spectrum of R , and self-test, we conclude from Theorem 5.1(ii) that pd R ( M ) < depth R / n ≤ d / n ≤
1. As d / n ≤
1, we see M is free, which is a contradiction. Because the freeness of M implies theregularity of R , in this case, M is a module of depth i and hence it cannot be free.The proof of part (ii) is similar to that of part (i). Also, part (iii) is a combination of parts (i) and (ii). (cid:3) We finish this section by noting that Corollary 5.4 relaxes the regularity hypothesis in [36, 3.2]. Fur-thermore, it removes the restriction on Ω k in [2, 6.9] and gives an answer to [2, Problem 1.6] in thenontrivial case. 6. A PPLICATIONS IN PRIME CHARACTERISTIC
The aim of this section is to apply our previous results to the study of the vanishing of local cohomol-ogy modules of the Frobenius powers. We prove two main results, namely Theorems 6.1 and 6.6. As weproceed, we state these two theorems, establish several corollaries of them, give examples, and defer theproofs of Theorems 6.1 and 6.6 to the end of this section.In this section, R denotes a local ring of prime characteristic p >
0, and ϕ : R → R denotes the Frobe-nius endomorphism given by ϕ ( a ) = a p for a ∈ R . Each iteration ϕ n of ϕ defines a new R -modulestructure on the set R , and this R -module is denoted by ϕ n R , where a · b = a p n b for a , b ∈ R . More gen-erally, given M ∈ mod ( R ) , we denote by ϕ n M the finitely generated R -module M with the R -action givenby r · x = ϕ n ( r ) x for r ∈ R and x ∈ M . For the proof of several results, we make use of the followingwell-known result of Kunz [40] without reference: R is regular if and only if ϕ r R is a flat R -module.Recall that R is said to be F-finite if the Frobenius endomorphism makes R into a module-finite R -algebra, i.e., if ϕ R is a finitely generated R -module. It is known that, if R is a local complete intersection,then the Frobenius endomorphism ϕ r R (not necessarily a finitely generated R -module) is Tor-rigid for all r ≥
1; see, for example, [44, 5.1.1].Next is the statement of our first theorem:
Theorem 6.1.
Assume R is a complete intersection ring, Z ⊆ Spec
R is a specialization-closed subset,and M ∈ mod ( R ) is such that NF ( M ) ⊆ Z . Then, for an integer n ≥ , the following hold: (i) If H i + Z ( M ⊗ R ϕ n R ) = for some i with i < grade R ( Z , R ) , then H i Z ( M ⊗ R ϕ n R ) ∼ = H i Z ( M ) ⊗ R ϕ n R. (ii) H grade ( Z , M ) Z ( M ⊗ R ϕ n R ) = . Therefore, it follows that grade R ( Z , M ⊗ R ϕ n R ) ≤ grade R ( Z , M ) . (iii) If H i Z ( M ⊗ R ϕ n R ) = for some i with ≤ i < grade R ( Z , M ) , then it follows that pd R ( M ) < ∞ .Therefore, one has that grade R ( Z , M ) = grade R ( Z , M ⊗ R ϕ n R ) . (iv) Either grade R ( Z , M ⊗ R ϕ n R ) = or grade R ( Z , M ⊗ R ϕ n R ) = grade R ( Z , M ) . The first corollary of Theorem 6.1 determines a new freeness criteria in terms of the vanishing of localcohomology modules. Recall that mod ( R ) denotes the category of all finitely generated R -modules thatare locally free on the punctured spectrum of R . Corollary 6.2.
Assume R is a d-dimensional complete intersection, and M ∈ mod ( R ) is a maximalCohen-Macaulay R-module. If H i m ( M ⊗ R ϕ n R ) = for some integer n ≥ and some i with ≤ i < d, thenM is free.Proof. This claim is an immediate consequence of Theorem 6.1(iii). (cid:3)
We now give an example and show that the conclusion of Corollary 6.2 is sharp.
Example 6.3. (i) Let R = F [[ x , y ]] / ( x ) and let M = R / xR . Then M is maximal Cohen–Macaulay. Also, it followsthat M ⊗ R ϕ R ∼ = Rx R = R . Therefore, we have that H d − m ( M ⊗ R ϕ R ) = H m ( R ) =
0. As M is not free,this example shows that the locally free hypothesis in Corollary 6.2 is necessary.(ii) Let R = k [[ x , y ]] and let M = k . Then it follows that M is locally free over punctured spectrumof R , and H m ( M ⊗ R ϕ R ) = H m ( R / m [ p ] ) =
0. However, M is not maximal Cohen-Macaulay. Thisexample shows that the conclusion of Corollary 6.2 may not be true if the module M in question isnot maximal Cohen-Macaulay.If X is a subset of Spec R (where R is a Noetherian ring, not necessarily local or of prime characteristic p ) and M ∈ mod ( R ) , we say M is locally free on X provided that M p is a free R p -module for all p ∈ X. STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 21
For an integer i ≥
0, we set X i ( R ) = { p ∈ Spec ( R ) | ht ( p ) ≤ i } . Next, in passing, we make use of Theorem 6.1 and obtain a new proof of a result of Celikbas, Iyengar,Piepmeyer and Wiegand [21].
Corollary 6.4. ( [21, 3.4]) Assume R is a complete intersection and M ∈ mod ( R ) is locally free on X ( R ) .Then, for an integer n ≥ , the following conditions are equivalent: (i) M ⊗ R ϕ n R is a torsion-free R-module. (ii)
M is a torsion-free R-module such that pd R ( M ) < ∞ .Proof. Set Z : = { p ∈ Spec R | ht ( p ) > } . Note by Proposition 2.19(i) that Γ Z ( L ) = T ( L ) for each L ∈ Mod ( R ) . Therefore, L is torsion-free if and only if grade R ( Z , L ) > ⇒ (ii). By our assumption grade R ( Z , M ⊗ R ϕ n R ) >
0. It follows from parts (iii) and (iv) of Theorem6.1 that grade R ( Z , M ) = grade R ( Z , M ⊗ R ϕ n R ) > R ( M ) < ∞ . In particular, M is torsion-free.(ii) ⇒ (i). Since M has finite projective dimension, as we have seen in the proof of Theorem 6.1(iii),grade R ( Z , M ⊗ R ϕ n R ) = grade R ( Z , M ) . As M is torsion-free, grade R ( Z , M ) >
0. Therefore M ⊗ R ϕ n R has a positive grade with respect to Z . This, in turn, is equivalent to the torsion-free property of themodule. The proof is now complete. (cid:3) Recall that the singular locus Sing ( R ) of R is Sing ( R ) = { p ∈ Spec R | R p is not regular } . Note thatSing ( R ) is a Zariski closed set provided that R is excellent. Corollary 6.5.
Assume R is an F -finite complete intersection and let a be a proper ideal of R such that Sing ( R ) ⊆ V ( a ) . If H i a ( ϕ r M ⊗ R ϕ s R ) = for some integers r ≥ , s ≥ and i with ≤ i < grade R ( a , M ) ,then R is regular.Proof. This is an immediate consequence of Theorem 6.1(iii) and [10, Theorem 1.1]. (cid:3)
Here is the second main result of this section; see 2.18 for the definition of Serre’s condition.
Theorem 6.6.
Assume R is a d-dimensional complete intersection, where d ≥ , and M ∈ mod ( R ) islocally free on X d − n − ( R ) for some n with ≤ n ≤ d − . Assume further that the following hold: (i) H n m ( M ⊗ R ϕ r R ) = for some integer r ≥ . (ii) M satisfies Serre’s condition ( S n ) .Then it follows that pd R ( M ) < d − n. The following example corroborates Theorem 6.6 and show that the result is sharp.
Example 6.7. (i) Let R = k [[ x , y , z , w ]] and let M = Ω R k . Then M is locally free on the punctured spectrum of R ,depth R ( M ) = M satisfies ( S ) , and H m ( M ) =
0. Note, as R is regular, ϕ r R is a flat R -module.Hence we conclude that H m ( M ⊗ R ϕ r R ) ∼ = H m ( M ) ⊗ R ϕ r R =
0. Hence, since pd ( M ) =
2, letting d = dim ( R ) = n =
2, we deduce from this example that the conclusion of Theorem 6.6 maynot hold if the condition H n m ( M ⊗ R ϕ r R ) = R = k [[ x , y ]] and let M = R ⊕ k . Letting n = d = dim ( R ) =
2, we see that M is locally freeover X d − n − ( R ) . Then it follows thatH m ( M ⊗ R ϕ R ) = H m ( ϕ R ) ⊕ H m ( R / m [ p ] ) = . Note that depth R ( M ) = M M doest not satisfy ( S ) . As pd ( M ) = = d > d − n , we deducefrom this example that the conclusion of Theorem 6.6 may not hold if the module M in questiondoes not satisfy ( S n ) .(iii) Let R = k [[ x , y ]] and let M = R . Then, letting d = dim ( R ) = n =
1, we see that M satisfiesall the hypotheses of Theorem 6.6. Furthermore, pd R ( M ) = d − n −
1. Hence we deduce from thisexample that the bound for the projective dimension of M stated in the theorem is sharp. Next we give two corollaries of Theorem 6.6. The first one, Corollary 6.8, is an immediate conse-quence of the theorem. We should note that the hypothesis that R is reduced in Corollary 6.8 is necessary,even in the complete case; see Example 6.3. Corollary 6.8.
If R is a d-dimensional complete intersection, and M ∈ mod ( R ) is a maximal Cohen-Macaulay R-module such that M is locally free on X ( R ) (e.g., R is reduced) and H d − m ( M ⊗ R ϕ r R ) = for some integer r ≥ , then M is free. Next we prove the second corollary of Theorem 6.6, namely Corollary 6.10; the corollary determinesa new criteria for regularity in terms of the Frobenius endomorphism. Furthermore, Theorem 1.2 adver-tised in the introduction is a special case of Corollary 6.10. First we recall:Given an integer n ≥
0, a ring R (where R is a Noetherian ring, not necessarily local, or of characteristic p ) satisfies Serre’s condition ( R n ) provided that R p is a regular local ring for all p ∈ X n ( R ) . Note that R is reduced if and only if R satisfies Serre’s conditions ( R ) and ( S ) , and R is normal if and only if R satisfies Serre’s conditions ( R ) and ( S ) .The following remark is well-known, but we record it here as we use it for the proof of Corollary 6.10,as well as for the proofs of Theorems 6.1 and 6.6, to reduce the argument to the F-finite case. Remark 6.9.
There is a local ring extension ( S , n ) of ( R , m ) such that m S = n , S is F -finite, S is faithfullyflat over R , and S has infinite residue field. For example, letting b R ∼ = k [[ x , · · · , x m ]] / I for some ideal I ,we can pick S = ¯ k [[ x , · · · , x m ]] / I ¯ k [[ x , · · · , x m ]] , where ¯ k denotes the algebraic closure of k . Note, if M isan R -module, then it follows that ( M ⊗ R ϕ n R ) ⊗ R S ∼ = ( M ⊗ R S ) ⊗ S ϕ n S . Corollary 6.10.
If R is a d-dimensional complete intersection such that R satisfies Serre’s condition ( R n − ) and H d − n m ( ϕ r R ⊗ R ϕ s R ) = for some integers r ≥ , s ≥ and n ≥ , then R is regular.Proof. It suffices to prove that we may assume R is F-finite: in that case, ϕ r R is a (finitely generated)maximal Cohen-Macaulay R -module, and hence the assertion follows from Theorem 6.6 and [40, 2.1];see also [49, Theorem 2].We consider the local ring extension ( S , n ) of ( R , m ) that follows from Remark 6.9. Then, by the flatbase change theorem along with the independence theorem for local cohomology modules, we haveH d − n m ( ϕ r R ⊗ R ϕ s R ) ⊗ R S ∼ = H d − n m (( ϕ r R ⊗ R ϕ s R ) ⊗ R S ) ∼ = H d − n m ( ϕ r S ⊗ S ϕ s S ) ∼ = H d − n n ( ϕ r S ⊗ S ϕ s S ) . Now let P ∈ X n − ( S ) and set p = P ∩ R . Then it follows that p ∈ X n − ( R ) . Therefore, R p is regularand hence ϕ ( R p ) is flat as an R p -module. As the localization commutes with the Frobenius map, we see: ϕ ( S P ) ∼ = ( ϕ S ) P ∼ = ( ϕ R ⊗ R S ) P ∼ = ( ϕ R ) p ⊗ R p S P ∼ = ϕ ( R p ) ⊗ R p S P . Hence, since R p → S P is flat, we conclude that ϕ ( S P ) is flat as an S P -module. Furthermore, if S is regular,then so is R . Consequently, as the hypotheses and the conclusion do not change by passing to S , we mayassume R is F-finite, as claimed. This finishes the proof. (cid:3) We now proceed and give proofs of Theorems 6.1 and 6.6. First we note:
Remark 6.11.
We say N ∈ Mod ( R ) satisfies in the Nakayama’s property if Supp R ( M ) ∩ Supp R ( N ) = Supp R ( M ⊗ R N ) for each M ∈ mod ( R ) . Clearly, every finitely generated module over a local ring (notnecessarily of prime characteristic) satisfies in the Nakayama’s property. We should note that Lemma3.7 and Theorem 3.8 holds even if N is not finitely generated, but satisfies in the Nakayama’s property. Proof of Theorem 6.1.
First note, by [47, I.1.5], we have that Supp R ( M ⊗ R ϕ n R ) = Supp R ( M ) . In par-ticular, ϕ n R satisfies in the Nakayama’s property; see Remark 6.11. Also, we have grade R ( Z , ϕ n R ) = grade R ( Z , R ) , and that ϕ n R is Tor-rigid.(i). This follows from Lemma 3.7(c) and Remark 6.11.(ii) and (iii). Assume that H i Z ( M ⊗ R ϕ n R ) = i with 0 ≤ i ≤ grade R ( Z , M ) . Then, by Theorem3.8 and Remark 6.11, we have Tor Rj ( M , ϕ n R ) = j ≥
1. In view of [12, Theorem], pd R ( M ) < ∞ ,as claimed by (iii). Now, we proceed to give a proof for part (ii). STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 23
We are going to reduce to the F -finite case. To this end, we pass to the local ring S that follows fromRemark 6.9 and, without losing of generality, we may assume that R is F -finite. We know, by [47, 1.7],the following equality holds for each p ∈ Spec ( R ) :(6.1.2) pd R p ( M p ) = pd R p ( M p ⊗ R p ϕ n R p ) . Therefore, it follows from (6.1.2) and Proposition 2.16 that grade R ( Z , M ) = grade R ( Z , M ⊗ R ϕ n R ) andso the assertion follows.(iv). This is clear by parts (ii) and (iii). (cid:3) To prove Theorem 6.6, we need:
Definition 6.12. ( [5]) If R is Noetherian (not necessarily local, or of prime characteristic) and if M , N ∈ Mod ( R ) , then the grade of the pair ( M , N ) is defined as:grade R ( M , N ) = inf { i | Ext iR ( M , N ) = } . Note that the grade of ( M , N ) is not necessarily finite, in general. If R is local and M , N ∈ mod ( R ) , thenwe have grade R ( M , N ) = grade R ( ann ( M ) , N ) , which is equal to the length of maximal regular sequenceon N in ann R ( M ) ; in this case the grade is finite. Furthermore, by [5, 4.5], we have:(6.12.1) grade R ( M , N ) = inf { depth R p ( N p ) | p ∈ Supp R ( M ) } . For simplicity, we denote the grade of ( M , R ) by grade R ( M ) .The following lemma plays a crucial role in the sequel. Lemma 6.13.
Let M , N , K ∈ Mod ( R ) and let ℓ ≥ jR ( N , K ) = j with1 ≤ j ≤ ℓ −
1, and grade R ( Tor Ri ( M , N ) , K ) > ℓ − i for all i with 1 ≤ i ≤ ℓ . Then the following holds:(i) Ext iR ( M , Hom R ( N , K )) ∼ = Ext iR ( M ⊗ R N , K ) for all i with 0 ≤ i < ℓ .(ii) There is an injection Ext ℓ R ( M , Hom R ( N , K )) ֒ → Ext ℓ R ( M ⊗ R N , K ) . Proof.
There are two spectral sequences converging to the same point: E pq : = Ext pR ( Tor Rq ( M , N ) , K ) = ⇒ H p + q and F pq : = Ext pR ( M , Ext qR ( N , K )) = ⇒ H p + q .As grade R ( Tor Ri ( M , N ) , K ) > ℓ − i for all 1 ≤ i ≤ ℓ , we have that E pq = ≤ q ≤ ℓ and p ≤ ℓ − q , andobtain an isomorphism E i ∼ = H i for all integers i ≤ ℓ . Since Ext iR ( N , K ) = ≤ i ≤ ℓ −
1, we have F pq = ≤ q ≤ ℓ −
1. Hence, we get F i ∼ = H i for all integers i ≤ ℓ − F ℓ ֒ → H ℓ .Thus there are isomorphisms Ext iR ( M ⊗ R N , K ) = E i ∼ = H i ∼ = F i = Ext iR ( M , Hom R ( N , K )) for all integers i ≤ ℓ −
1, and also an injection Ext ℓ R ( M , Hom R ( N , K )) ∼ = F ℓ ֒ → H ℓ ∼ = Ext ℓ R ( M ⊗ R N , K ) . (cid:3) We are now ready to prove Theorem 6.6.
Proof of Theorem 6.6.
We can pass to the local ring S that exists by Remark 6.9 and assume R is F-finite,i.e., ϕ n R ∈ mod ( R ) . Set ℓ = d − n . Then ℓ ≥
1. Also, by (ii) and the Theorem 2.14, we have(6.6.1) Ext ℓ R ( M ⊗ R ϕ r R , R ) = . Let p ∈ Supp R ( Tor Ri ( M , ϕ r R )) for some i ≥
1. It follows from (i) that(6.6.2) depth ( R p ) = dim ( R p ) ≥ d − n > d − n − i . Hence, by (6.12.1) and (6.6.2), we deduce that(6.6.3) grade R ( Tor Ri ( M , ϕ r R )) > d − n − i = ℓ − i for all i with 1 ≤ i ≤ ℓ. As R is Gorenstein, it follows ϕ r R ∼ = Hom R ( ϕ r R , R ) [33, Theorem 1.1]. Also, since ϕ r R is a maximalCohen-Macaulay R -module, we see Ext jR ( ϕ r R , R ) = j ≥
1. Hence, in view of Lemma 6.13(ii),(6.6.1) and (6.6.3) we conclude thatExt ℓ R ( M , ϕ r R ) = Ext ℓ R ( M , Hom R ( ϕ r R , R )) = . Now, by the four-term exact sequence (2.2.4) we obtain Tor R ( Tr Ω ℓ M , ϕ r R ) =
0. So [12, Theorem]implies that Tor Ri ( Tr Ω ℓ M , ϕ r R ) = i ≥ R ( Tr Ω ℓ M ) < ∞ . Another use of (2.2.4) impliesthat Ext ℓ R ( M , R ) =
0. On the other hand, as M satisfies ( S n ) , it is an n -th syzygy module; see [5, 4.25].Therefore, Ω ℓ M is a d -th syzygy module, or equivalently, is a maximal Cohen-Macaulay module. Thisimplies that Tr Ω ℓ M is also maximal Cohen-Macaulay. As Tr Tr ( − ) ≈ ( − ) , we deduce that Ω ℓ M is freeand pd R ( M ) ≤ ℓ . Consequently, the fact that Ext ℓ R ( M , R ) = R ( M ) < ℓ = d − n , as required. (cid:3)
7. A
RELATION BETWEEN THE LOCAL COHOMOLOGY AND THE T ATE HOMOLOGY
In this section, we determine a new relation between the local cohomology of tensor products ofmodules and the Tate homology over Gorenstein rings. Our main result is Theorem 7.1, which will be anew tool in the study of the depth of tensor products of modules. To the best of our knowledge, Theorem7.1 is new, even if the specialization-closed subset Z considered is a closed subset of Spec ( R ) .Theorem 7.1 has various applications that contribute to the literature; we state and prove these appli-cations following the proof of the theorem in this section. Several of the applications we give shouldbe of independent interest. For example, Theorem 7.1 improves a result that has been initially provedby Dao [29, 7.7], and subsequently studied by Celikbas [16, 3.4] and Celikbas, Iyengar, Piepmeyer andWiegand [20, 3.14]; see Corollary 7.4.Recall that R denotes a commutative Noetherian ring throughout. Theorem 7.1.
Let Z ⊂ Spec
R be a specialization-closed subset and let M , N ∈ mod ( R ) . Assume G-dim R ( M ) < ∞ . Assume further, for an integer n ≥ , the following hold: (i) depth R p ( M p ) + depth R p ( N p ) ≥ depth R p + n for each p ∈ Z . (ii) Supp R (cid:0) Tor Ri ( M , N ) (cid:1) T Supp R (cid:0) c Tor Rj ( M , N ) (cid:1) ⊆ Z for all i ≥ and for all j ∈ Z (e.g., NF ( M ) ⊆ Z ).Then it follows: (a) H i Z ( M ⊗ R N ) ∼ = c Tor R − i ( M , N ) for each i = , . . . , n − . (b) There is an injection c Tor R − n ( M , N ) ֒ → H n Z ( M ⊗ R N ) .Proof. Set W : = Supp R ( M ) and Z ′ : = Z ∩ W . Clearly, Z ′ is specialization-closed. First we prove: Claim I. H i Z ( M ⊗ R N ) ∼ = H i Z ′ ( M ⊗ R N ) for all i ≥ L = M ⊗ R N and let 0 → L → E ( L ) → E ( L ) → · · · be the minimal injectiveresolution of X . Note that Supp R ( E i ( L )) ⊆ Supp R ( L ) ⊆ W for all i ≥
0. Therefore, Γ W ( L ) = L and also Γ W ( E i ( L )) = E i ( L ) for all i ≥
0. It follows from Theorem 2.11(iv) that Γ Z ′ ( L ) = Γ Z ( Γ W ( L )) = Γ Z ( L ) .Similarly, Γ Z ′ ( E i ( L )) = Γ Z ( Γ W ( E i ( L ))) = Γ Z ( E i ( L )) for all i ≥ i Z ( L ) ∼ = H i Z ′ ( L ) for all i ≥
0. Thus, the proof of the claim is completed.By Claim I, without loss of generality, by replacing Z with Z ′ , we may assume that Z ⊆ Supp R ( M ) .Consider the following exact sequence(7.1.1) 0 → M → X → G → R ( X ) < ∞ and G-dim R ( G ) = · · · → c Tor Rj + ( G , N ) → c Tor Rj ( M , N ) → c Tor Rj ( X , N ) → c Tor Rj ( G , N ) → · · · , of stable homology modules. Also by Theorem 2.21(i), c Tor Rj ( X , N ) = j ∈ Z . Hence we get thefollowing isomorphism(7.1.2) c Tor Rj + ( G , N ) ∼ = c Tor Rj ( M , N ) for all j ∈ Z . As G is totally reflexive, by Theorem 2.21(iv), (7.1.2) and assumption (ii), we have(7.1.3) Supp R ( Tor Ri ( G , N )) = Supp R ( c Tor Ri ( G , N )) = Supp R ( c Tor Ri − ( M , N )) ⊆ Z for all i > . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 25
Applying the functor − ⊗ R N to the exact sequence (7.1.1), we get the following exact sequence(7.1.4) · · · → Tor Ri ( M , N ) → Tor Ri ( X , N ) → Tor Ri ( G , N ) → · · · . In view of assumption (ii), (7.1.3) and (7.1.4) we have(7.1.5) Supp R ( Tor Ri ( X , N )) ⊆ Supp R ( Tor Ri ( M , N )) ∪ Supp R ( Tor Ri ( G , N )) ⊆ Z for all i > . On the other hand, by the exact sequence (7.1.1) and Auslander–Bridger formula, we see thatdepth R p ( M p ) = depth R p ( X p ) for all p ∈ Z ⊆ Supp R ( M ) . Therefore, by assumption (i)(7.1.6) depth R p ( X p ) + depth R p ( N p ) ≥ depth R p + n ∀ p ∈ Z . Next we claim the following.
Claim II.
Tor Ri ( X , N ) = i > t = sup { j | Tor Rj ( X , N )) = } . Assume contrarily that t > p ∈ Ass R ( Tor Rt ( X , N )) . By (7.1.5), p ⊆ Z . As depth R p ( Tor Rt ( X , N ) p ) =
0, by [4, Theorem 1.2] we havethe equality depth R p ( X p ) + depth R p ( N p ) = depth R p − t , which is a contradiction by (7.1.6). Therefore t = → Tor R ( G , N ) → M ⊗ R N → X ⊗ R N → G ⊗ R N → . Applying the functor Γ Z ( − ) to the exact sequence (7.1.7), we obtain the following injection:(7.1.8) H Z ( Tor R ( G , N )) ֒ → H Z ( M ⊗ R N ) . Note by (7.1.3) that Tor R ( G , N ) is torsion with respect to Z . Hence, by (7.1.2), Theorem 2.21(iv) and(7.1.8) we get the following injection: c Tor R ( M , N ) ∼ = c Tor R ( G , N ) ∼ = Tor R ( G , N ) ∼ = H Z ( Tor R ( G , N )) ֒ → H Z ( M ⊗ R N ) . Therefore, from now on we may assume that n >
0. By Theorem 2.6 and Claim II,(7.1.9) depth R p ( X p ) + depth R p ( N p ) = depth R p + depth R p (( X ⊗ R N ) p ) ∀ p ∈ Supp R ( X ⊗ R N ) . In view of (7.1.6) and (7.1.9), we have depth R p (( X ⊗ R N ) p ) ≥ n for all p ∈ Z . By Proposition 2.16(7.1.10) H i Z ( X ⊗ R N ) = ≤ i ≤ n − . Consider the following two exact sequences, induce from (7.1.7):(7.1.11) 0 → Tor R ( G , N ) → M ⊗ R N → Y → → Y → X ⊗ R N → G ⊗ R N → . Applying the functor Γ Z ( − ) to the above exact sequences and using Theorem 2.11(ii), (iii), (7.1.10) and(7.1.2), one can easily obtain the following isomorphisms:(7.1.12) H Z ( M ⊗ R N ) ∼ = H Z ( Tor R ( G , N )) ∼ = Tor R ( G , N ) ∼ = c Tor R ( G , N ) ∼ = c Tor R ( M , N ) , (7.1.13) H i Z ( M ⊗ R N ) ∼ = H i Z ( Y ) ∼ = H i − Z ( G ⊗ R N ) for all 0 < i < n . Also, we have the following injection(7.1.14) H n − Z ( G ⊗ R N ) ֒ → H n Z ( Y ) ∼ = H n Z ( M ⊗ R N ) . Note by (7.1.2) and assumption (ii) that Supp R ( c Tor Ri ( G , N )) ⊆ Z for all i ∈ Z . As G is totally reflexive,by Lemma 3.6 we have Supp R ( Ext iR ( Tr G , N )) ⊆ Z for all i >
0. As M has finite Gorenstein dimension,by assumption (i) and Auslander–Bridger formula, we have depth R p ( N p ) ≥ G-dim R p ( M p ) + n ≥ n for all p ∈ Z ⊆ Supp R ( M ) . Therefore, by Proposition 2.16 we conclude that grade R ( Z , N ) ≥ n . Since G is totally reflexive, so is Tr G by [5, Lemma 4.9]. Hence, by using Theorem 2.21 and Lemma 3.5 and notingthat G ∗ ≈ Ω Tr G , we get the following isomorphisms:(7.1.15) H j − Z ( G ⊗ R N ) ∼ = Ext jR ( Tr G , N ) ∼ = c Ext jR ( Tr G , N ) ∼ = c Ext j − R ( G ∗ , N ) ∼ = c Tor R − j + ( G , N ) , for all 1 ≤ j ≤ n . Now the first assertion is clear by (7.1.12), (7.1.13), (7.1.15) and (7.1.2). The secondassertion follows from (7.1.14), (7.1.15) and (7.1.2). (cid:3) Next we start proving several corollaries of Theorem7.1. For the first corollary, see 2.4 for the defini-tion of the complexity.
Corollary 7.2.
Assume R is a local complete intersection and let Z ⊂ Spec
R be a specialization-closed.Let M , N ∈ mod ( R ) , and let c and n be integers such that c > cx R ( M , N ) and n ≥ c − . Assume thefollowing hold: (i) depth R p ( M p ) + depth R p ( N p ) ≥ depth R p + n for all p ∈ Z . (ii) H i Z ( M ⊗ R N ) = for all i = n − c + , . . . , n. (iii) Supp R ( Tor Ri ( M , N )) ⊆ Z for all i ≥ (e.g., NF ( M ) ∩ NF ( N ) ⊆ Z ).Then it follows grade R ( Z , M ⊗ R N ) ≥ n + , and Tor Ri ( M , N ) = for all i ≥ .Proof. We first note, by Theorem 2.22 and assumption (iii), that we have Supp R ( c Tor Ri ( M , N )) ⊆ Z forall i ∈ Z . Hence, by Theorem 7.1 and assumption (ii), we get c Tor R − i ( M , N ) = n − c + ≤ i ≤ n .Consider to the following exact sequence 0 → L → X → M → X is totally reflexive and pd R ( L ) < ∞ . By Theorem 2.21(i) we have c Tor Ri ( L , N ) = i ∈ Z . Therefore, the above exact sequenceinduces the following isomorphism c Tor Ri ( M , N ) ∼ = c Tor Ri ( X , N ) for all i ∈ Z (see for example [26, 2.9]).Set Y : = Ω − ( n + ) X . Note that Y is totally reflexive and X ≈ Ω n + Y . In view of Theorem 2.21 we obtainthe following isomorphisms:(7.2.1) Tor Ri ( Y , N ) ∼ = c Tor Ri ( Y , N ) ∼ = c Tor Ri − n − ( X , N ) ∼ = c Tor Ri − n − ( M , N ) = ≤ i ≤ c . It follows from (7.2.1) and [22, 3.5] that c Tor Ri ( Y , N ) = i ∈ Z . Equivalently, it follows that c Tor Ri ( M , N ) ∼ = c Tor Ri ( X , N ) = i ∈ Z . Thus, by Theorem 7.1(i) and assumption (ii), we see thatH i Z ( M ⊗ R N ) = ≤ i ≤ n . In other words, grade R ( Z , M ⊗ R N ) > n . Also, by Theorem 2.21(iv)we have Tor Ri ( M , N ) ∼ = c Tor Ri ( M , N ) = i ≫
0. Now the last assertion follows from assumptions(i), (iii) and the dependency formula (2.7.1). (cid:3)
The following is an immediate consequence of Corollary 7.2.
Corollary 7.3.
Assume R is a local complete intersection, M , N ∈ mod ( R ) , and let c and n be integers.Assume NF ( M ) ∩ NF ( N ) ⊆ { m } , c > cx R ( M , N ) and that n ≥ c − . Assume further the following hold: (i) depth R ( M ) + depth R ( N ) ≥ depth R + n. (ii) H i m ( M ⊗ R N ) = for all i = n − c + , . . . , n.Then it follows depth R ( M ⊗ R N ) ≥ n + and Tor Ri ( M , N ) = for all i ≥ . Next, in Corollary 7.4, we improve a result of Celikbas, Iyengar, Piepmeyer and Wiegand [20, 3.14].Note that the conclusion of Corollary 7.4 was obtained in [20] for the case where c is at least the codi-mension of the ring in question; see also [16, 3.4] and [29, 7.7]. Corollary 7.4. ( [20]) Assume R is a local complete intersection, and let M , N ∈ mod ( R ) . Assume furtherhat the following conditions hold for some integer c such that c > cx R ( M , N ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 27 (i)
M and N satisfy ( S c − ) . (ii) M ⊗ R N satisfies ( S c ) . (iii) Tor Ri ( M , N ) p = for all i ≥ and all p ∈ X c − ( R ) .Then it follows that Tor Ri ( M , N ) = for all i ≥ .Proof. We consider to the specialization-closed subset Z = { p ∈ Spec R | ht p ≥ c } of Spec R . Then itfollows from assumption (iii) that Supp R ( Tor Ri ( M , N )) ⊆ Z for all i ≥
1. Now the assertion follows fromProposition 2.19(ii) and Corollary 7.2 by letting n equal to c − (cid:3) Corollary 7.5 is another application of Theorem 7.1 which determines a new bound on depth of tensorproducts of modules satisfying the depth formula; see 2.5. As Corollary 7.5 does not assume any Tor-vanishing and as the depth formula is associated with the Tor-vanishing, the conclusion of Corollary7.5 seems quite interesting to us, cf., [22, 3.1]. Note that Corollary 1.4, advertised in the introduction,follows from Corollary 7.5 and (2.4.1).
Corollary 7.5.
Assume R is a local complete intersection. Assume further NF ( M ) ∩ NF ( N ) ⊆ { m } for some M , N ∈ mod ( R ) . If depth R ( M ) + depth R ( N ) − depth R ≥ cx R ( M , N ) , then it follows that depth R ( M ⊗ R N ) + depth R ≤ depth R ( M ) + depth R ( N ) .Proof. Set c = cx R ( M , N ) + n = depth R ( M ) + depth R ( N ) − depth R . Then we have n ≥ c − R ( M ⊗ R N ) > n . Therefore, H i m ( M ⊗ R N ) = n − c + ≤ i ≤ n . ByCorollary 7.2, it follows Tor Ri ( M , N ) = i ≥
1. In view of Theorem 2.6, we have depth R ( M ) + depth R ( N ) = depth R + depth R ( M ⊗ R N ) . Thus, depth R ( M ⊗ R N ) = n which is a contradiction. (cid:3) Example 7.6.
The first item says that the bound depth R ( M ⊗ R N ) + depth R ≤ depth R ( M ) + depth R ( N ) presented in Corollary 7.5 can be achieved. The second item shows that the locally free assumption isnecessary in Corollary 7.5.(i) Let R = k [[ x , y ]] and let M = N = m . Then it follows cx R ( M , N ) =
0. So we have depth R ( M ) = R ( M ) + depth R ( N ) = depth R + n . In particular, the assumptions of Corollary 7.5 hold.Moreover, in view of [2, 3.1], we see H m ( M ⊗ M ) ≃ k =
0. Therefore, it follows thatdepth R ( M ⊗ R N ) + depth R = + = + = depth R ( M ) + depth R ( N ) . (ii) Let R be a 3-dimensional complete intersection and x be a non zero-divisor on R . We look at M = N = R / xR . Then we have depth R ( M ) + depth R ( N ) − depth R = + − = > = cx R ( M , N ) .The following depth R ( M ⊗ R N ) + depth R = + (cid:10) + = depth R ( M ) + depth R ( N ) shows that thelocally free assumption in Corollary 7.5 is really needed.Recall that R is said to be an isolated singularity if R is local and R p is regular for each p ∈ X d − ( R ) ,where d = dim ( R ) . The following result is immediate from Corollary 7.5; cf., [22, 4.7]. Corollary 7.7.
Assume R is a local hypersurface singularity, and let M , N ∈ mod ( R ) such that Mis maximal Cohen-Macaulay. If depth R ( N ) ≥ , then depth R ( N ) ≥ depth R ( M ⊗ R N ) . Therefore, if depth R ( N ) = , then M ⊗ R N cannot be reflexive.
Corollary 7.8.
Assume R is a Gorenstein local ring, and let M , N ∈ mod ( R ) be maximal Cohen-Macaulay. Assume further that NF ( M ) ∩ NF ( N ) ⊆ { m } (e.g., R is an isolated singularity.) Then atleast one of the following conditions holds: (i) M ⊗ R N is a maximal Cohen-Macaulay R-module. (ii) depth R ( M ⊗ R N ) = inf { i ≥ | c Tor R − i ( M , N ) = } . Proof.
Assume that M ⊗ R N is not maximal Cohen–Macaulay and set n : = depth ( M ⊗ R N ) + ≤ depth R .Then depth R ( M ) + depth R ( N ) ≥ depth R + n . In view of Theorem 7.1, we have(7.8.1) H i m ( M ⊗ R N ) ∼ = c Tor R − i ( M , N ) for all 0 ≤ i < n . Note depth R ( M ⊗ R N ) = inf { i ≥ | H i m ( M ⊗ R N ) = } . Therefore, by (7.8.1) we have c Tor R − i ( M , N ) = ≤ i < n − c Tor R − ( n − ) ( M , N ) =
0, which completes the proof. (cid:3)
Corollary 7.9.
Assume R is a d-dimensional hypersurface that has an isolated singularity, where d ≥ ,and M , N ∈ mod ( R ) are nonfree and maximal Cohen-Macaulay. Then at least one of the following holds: (i) M ⊗ R N is torsion-free, depth R ( M ⊗ R N ) = , and Tor R i − ( M , N ) = = Tor R i ( M , N ) for all i ≥ . (ii) M ⊗ R N has torsion, depth R ( M ⊗ R N ) = , and Tor R i ( M , N ) = for all i ≥ .Proof. It follows that depth R ( M ⊗ R N ) ≤
1; see [22, 4.7]. Therefore, since d ≥ M ⊗ R N is not maximalCohen-Macaulay. Hence, Corollary 7.8 shows that depth R ( M ⊗ R N ) = inf { i ≥ | c Tor R − i ( M , N ) = } .If depth R ( M ⊗ R N ) =
0, then M ⊗ R N has torsion. Moreover, we have that c Tor R ( M , N ) = M ∼ = Ω R ( M ) .Next assume depth R ( M ⊗ R N ) =
1. In this case, as 1 = depth R ( M ⊗ R N ) = inf { i ≥ | c Tor R − i ( M , N ) = } ,it follows that c Tor R ( M , N ) = = c Tor R − ( M , N ) . Furthermore, since M and N are locally free on thepunctured spectrum of R , it follows that M ⊗ R N is torsion-free. (cid:3) For some special cases, we have the following variant of Corollary 7.8.
Corollary 7.10.
Assume R is a local complete intersection. Assume further M , N ∈ mod ( R ) such that NF ( M ) ∩ NF ( N ) ⊆ { m } . If depth R ( M ) + depth R ( N ) ≥ depth R + cx R ( M , N ) , then one of the followingconditions holds: (i) depth R ( M ) + depth R ( N ) = depth R + depth R ( M ⊗ R N ) . (ii) depth R ( M ⊗ R N ) = inf { i ≥ | c Tor R − i ( M , N ) = } . Proof.
Set n = depth R ( M ) + depth R ( N ) − depth R . By our assumption, we have n ≥ cx R ( M , N ) . It followsfrom Corollary 7.5 that depth R ( M ⊗ R N ) ≤ n . If depth R ( M ⊗ R N ) = n , then (i) holds and we have nothingto prove. So let depth R ( M ⊗ R N ) < n . Then, by setting Z : = V ( m ) and using Theorem 7.1(i), we seeH i m ( M ⊗ R N ) ∼ = c Tor R − i ( M , N ) for all 0 ≤ i < n . Now it is clear that (ii) holds. (cid:3) As another application, we have the following non-vanishing result.
Corollary 7.11.
Assume R is a local complete intersection, and M , N ∈ mod ( R ) such that M is max-imal Cohen-Macaulay and NF ( M ) ∩ NF ( N ) ⊆ { m } . Assume further cx R ( M , N ) < depth R ( N ) . Then itfollows that H i m ( M ⊗ R N ) = for some i, where depth R ( N ) − cx R ( M , N ) ≤ i ≤ depth R ( N ) . Therefore, depth R ( M ⊗ R N ) ≤ depth R ( N ) .Proof. Set c = cx R ( M , N ) + n : = depth R ( N ) . Assume contrarily that H i m ( M ⊗ R N ) = i with n − c + ≤ i ≤ n . By Corollary 7.2, Tor Ri ( M , N ) = i > R ( M ⊗ R N ) = depth R ( N ) = n which is a contradiction, because H n m ( M ⊗ R N ) = (cid:3) We proceed by giving some further applications of Theorem 7.1 on local cohomology modules oftensor product of modules over hypersurface rings.
Proposition 7.12.
Let Z ⊂ Spec
R be a specialization-closed subset and let M , N ∈ mod ( R ) . Assume CI-dim R ( M ) < ∞ and cx R ( M ) ≤ (e.g., R is a hypersurface.) Assume, for an integer n ≥ , the followingconditions hold: (i) depth R p ( M p ) + depth R p ( N p ) ≥ depth R p + n for all p ∈ Z . (ii) Supp R ( Tor Ri ( M , N )) ⊆ Z for all i ≥ (e.g. NF ( M ) ∩ NF ( N ) ⊆ Z ).Then the following statements hold: (a) If n ≥ , then H i Z ( M ⊗ R N ) ∼ = H i + Z ( M ⊗ R N ) for all i = , . . . , n − . (b) If H n Z ( M ⊗ R N ) = , then H n − i Z ( M ⊗ R N ) = for all i ≥ and c Tor R − n + j ( M , N ) = for all j ∈ Z . Inparticular, if n is even, then M ⊗ R N is torsion-free.
STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 29
Proof.
Set t : = depth R − depth R ( M ) . As M has a bounded Betti numbers, by [9, Theorem 7.3], theminimal resolution of Ω t M is periodic of period at most two, hence so is the minimal complete resolutionof M . Therefore, we get the following isomorphisms:(7.12.1) c Tor Ri ( M , N ) ∼ = c Tor Ri + ( M , N ) for all i ∈ Z . Note that by Theorem 2.22 and assumption (ii), Supp R ( c Tor Ri ( M , N )) ⊆ Z for all i ∈ Z . Now thefirst assertion follows from Theorem 7.1(i) and (7.12.1). To prove the second assertion, suppose thatH n Z ( M ⊗ R N ) =
0. By Theorem 7.1(ii), c Tor R − n ( M , N ) = c Tor R − n + j ( M , N ) = j ∈ Z . Therefore, by using Theorem 7.1(i) we see that H n − i Z ( M ⊗ R N ) = i ≥ (cid:3) Corollary 7.13.
Assume R is a local hypersurface ring, and let M , N ∈ mod ( R ) . Assume M and N areboth maximal Cohen-Macaulay modules such that NF ( M ) ∩ NF ( N ) ⊆ { m } . If p (respectively, q) is anodd (respectively, even) integer such that max { p , q } < dim R and H q m ( M ⊗ R N ) = H p m ( M ⊗ R N ) = , theneither M or N is free.Proof. Note, in view of Corollary 7.12(b), we have c Tor Ri ( M , N ) = i ∈ Z . Therefore,Tor Ri ( M , N ) = i ≫
0. Hence, by [36, 1.9], it follows that either pd ( M ) < ∞ or pd ( N ) < ∞ ,i.e., either M or N is free. (cid:3) Example 7.14. (i) Let R = k [[ x , y , z , w ]] / ( xy ) and M = N = R / xR . Let p = q =
1. Then max { p , q } < dim R . Notethat M is maximal Cohen-Macaulay and H q m ( M ⊗ R N ) = H p m ( M ⊗ R N ) =
0. Neither M nor N isfree. Moreover, M is not locally free on the punctured spectrum of R . This example shows that theassumption on non-free locus in Corollary 7.13 is necessary.(ii) Let R be a local Cohen-Macaulay ring of dimension d ≥
4, and let M = N = m . Then M and N are not free, but are locally free over the punctured spectrum of R . So, in view of [2, 5.5(i)], wesee that H m ( M ⊗ R N ) = H m ( M ⊗ R N ) =
0. This example shows that the maximal Cohen-Macaulayassumption in Corollary 7.13 is necessary.We end this section by the following result which is an extension of [22, 1.3].
Corollary 7.15.
Assume R is a d-dimensional hypersurface with an isolated singularity, and let M , N ∈ mod ( R ) be non-free and maximal Cohen-Macaulay. Then at least one of the following conditions holds: (i) H i m ( M ⊗ R N ) = for all ≤ i ≤ d. (ii) H i m ( M ⊗ R N ) = = H j m ( M ⊗ R N ) for all even integers i and odd integers j with ≤ i , j < d. (iii) H i m ( M ⊗ R N ) = = H j m ( M ⊗ R N ) for all odd integers i and even integers j with ≤ i , j < d.Proof. This claims are consequences of Corollaries 7.12 and 7.13. (cid:3)
8. S
PLITTING CRITERIA FOR VECTOR BUNDLES
In this section, we are concerned with some splitting criteria for vector bundles on schemes. Recallthat a vector bundle is called trivial if it is isomorphic to a direct sum of line bundles. Let E be a vectorbundle on the projective scheme X equipped with a very ample line bundle O ( ) . The dual of E isdenoted by E ∗ : = H om ( E , O X ) . We say that E has the cohomological rigidity property provided thatthe the following condition holds:If H i ∗ ( X , E ⊗ E ∗ ) : = M n ∈ Z H i ( X , E ⊗ E ∗ ( n )) = i ≥ = ⇒ E is trivial.The cohomological rigidity property, considered as above, was initially studied by Kempf [45] andLuk and Yau [43] by using different approaches. A commutative algebra interpretation of the coho-mological rigidity property has been examined by Huneke and Wiegand [36]. More precisely, given areflexive module M over a hypersurface ring R , Huneke and Wiegand determined new criteria for the freeness of the module M in terms of the vanishing of H i m ( M ⊗ R M ∗ ) for some i . Furthermore, Hunekeand Wiegand generalized the aforementioned result of Luk and Yau via the machinery of commutativealgebra.Our work in this section is motivated by the work of Huneke and Wiegand [36]. The main purpose ofthis section is to obtain some new criteria for the freeness of modules in terms of the vanishing of localcohomology. Along the way, as an application, we obtain a new splitting criteria - for an arithmeticallyCohen-Macaulay vector bundle on a smooth complete intersection of an odd dimension – in terms of thecohomological rigidity property; see Corollary 8.15.We start by the following result which is an extension of a result of Huneke and Wiegand; see [36,4.1(1)], and also 2.11 for the terminology. Theorem 8.1.
Let R be a ring and let Z ⊂ Spec
R be a specialization-closed subset where grade R ( Z , R ) ≥ . Assume M ∈ mod ( R ) is such that NF ( M ) ⊆ Z . If H Z ( M ⊗ R M ∗ ) = , thenM = F ⊕ T , where F is free and T is Z -torsion. In particular, if M is torsion-free with respect to Z , then M is free.Proof. In view of Lemma 2.17 and our assumption, grade R ( Z , M ∗ ) ≥ min { , grade R ( Z , R ) } ≥ R ( Tr M , M ∗ ) = . It follows from (8.1.1) and (2.2.5) that Tor R ( Ω − M , M ∗ ) =
0. Note that Ω − M ≈ Tr Ω Tr M ≈ Tr M ∗ (see 2.2). Therefore, Tor R ( Tr M ∗ , M ∗ ) =
0. In other words, by (2.2.4), the natural map M ∗ ⊗ M ∗∗ → Hom ( M ∗ , M ∗ ) is surjective. In view of [6, A.1], we conclude that M ∗ is free. It follows from (8.1.1)that Ext R ( Tr M , R ) =
0. Therefore, by (2.2.3), the natural map e M : M → M ∗∗ is surjective and we get thefollowing exact sequence 0 → Ext R ( Tr M , R ) → M → M ∗∗ →
0. As M ∗∗ is free, the above exact sequencesplits. Hence, M ∼ = F ⊕ T , where T = Ext R ( Tr M , R ) which is Z -torsion and F : = M ∗∗ . (cid:3) Corollary 8.2.
Assume R is local and, M ∈ mod ( R ) , and a is an ideal of R of positive grade. Then thefollowing statements hold: (i) If NF ( M ) ⊆ V ( a ) and H a ( M ⊗ R M ∗ ) = , then M = F ⊕ T , where F is free and T is a -torsion.Therefore, if M is torsion-free with respect to a , then M is free. (ii) If NF ( M ) ⊆ V ( m ) and H m ( M ⊗ R M ∗ ) = , then M = F ⊕ T , where F is free and T is a finite lengthmodule. Therefore, if depth R ( M ) ≥ , then M is free.Proof. This follows immediately from Theorem 8.1. (cid:3)
To facilitate things, we bring the following:
Discussion 8.3. (i) By isolated Gorenstein singularity we mean a non Gorenstein ring ( R , m ) which isGorenstein over the punctured spectrum.(ii) Let ( R , m ) be a Cohen-Macaulay local ring of dimension d > i m ( ω R ⊗ R ω ∗ R ) = i ≤ i = d .(iii) Let us reprove the claim H i m ( ω R ⊗ R ω ∗ R ) = ≤ i < dim ( R ) , by the methods developed in thispaper. Indeed, ω R is locally free on the punctured spectrum of R . Let i ≥
1. Then, in view of Lemma3.2, we have that H i m ( ω R ⊗ R ω ∗ R ) = Ext i − R ( ω R , ω R ) = i where 2 ≤ i < depth R ( ω R ) = dim ( R ) .(iv) Any one-dimensional integral domain which is not Gorenstein is a Cohen-Macaulay ring with iso-lated Gorenstein singularity. For example, let R : = k [[ x , x , x ]] .(v) Any two-dimensional normal domain which is not Gorenstein is a Cohen-Macaulay ring with iso-lated Gorenstein singularity. For example, R : = k [[ x , x y , xy , y ]] .(vi) To see a 3-dimensional situation, let S : = k [[ x , y , z , u , v ]] and put R : = S / ( yv − zu , yu − xv , xz − y ) ,where k is an algebraically closed field of characteristic different from 2. By [42, Example 16.2], R is a 3-dimensional Cohen-Macaulay ring with isolated Gorenstein singularity. In fact, its canonicalmodule is isomorphic to the ideal ( u , v ) . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 31
Example 8.4.
The first item shows the importance of the locally free assumption. The second (and thethird) item shows that the spot 1 in the vanishing of local cohomology module is crucial.(i) Let R = k [[ x , y , z ]] / ( x ) and let M = R / xR . Note that M ∼ = M ∗ and H m ( M ⊗ R M ∗ ) =
0. It is clear that M is not of the form F ⊕ T , where F is free and T has finite length.(ii) Let R be a Cohen–Macaulay local ring with isolated Gorenstein singularity with a canonical module ω R , and of dimension d >
2. Such a thing exists, e.g., look at R : = C [[ x , y , z , u , v ]]( yv − zu , yu − xv , xz − y ) . By the abovediscussion, ω R is locally free on the punctured spectrum of R and H m ( ω R ⊗ R ω ∗ R ) =
0. It is clearthat ω R is not of the form F ⊕ T , where F is free and T has finite length.(iii) Let ( R , m , k ) be a d -dimensional ( d > ) Cohen–Macaulay local ring and set M : = m . Thendepth R ( M ) = M is locally free. We know that m ∗ ≃ R . It is easy to see H i m ( M ⊗ R M ∗ ) ≃ H i m ( m ) = ≤ i ≤ d −
1. However, M is not free. In fact pd R ( M ) = ∞ provided R issingular. Also, pd R ( M ) = d − > R is nonsingular. Corollary 8.5.
Assume R is local complete intersection ring of dimension d ≥ and let Z ⊂ Spec
Rbe specialization-closed. Let M ∈ mod ( R ) be such that NF ( M ) ⊆ Z . If H n Z ( M ⊗ R M ∗ ) = for an oddinteger n with ≤ n ≤ grade R ( Z , M ) , then pd R ( M ) < ∞ .Proof. If n =
1, then the assertion is clear by Corollary 8.2. Now let n >
1. In view of Proposition2.16 and [14, 1.4.1(b)], M is reflexive and so M ≈ M ∗∗ ≈ Ω Tr M ∗ . It follows from Lemma 3.5 thatExt n − R ( M , M ) ∼ = Ext n + R ( Tr M ∗ , M ) =
0. Hence, by [8, 4.2], pd R ( M ) < ∞ . (cid:3) Example 8.6. (i) Here, we show that the complete-intersection assumption in Corollary 8.5 is important. To thisend, let R be of isolated Gorenstein singularity and of dimension at least four. Let n : = M : = ω R . In view of Discussion 8.3(ii) we know H m ( M ⊗ R M ∗ ) =
0. In order to complete this item,we need to recall that pd ( M ) = ∞ .(ii) Let R = k [[ x , y , z , u , v ]] / ( xy ) , M = R / xR and let n =
3. Then 4 = depth R ( M ) ≥ n , M ≃ M ∗ and that M ⊗ R M ≃ M . Therefore, H m ( M ⊗ R M ∗ ) ≃ H m ( M ) =
0. However, pd ( M ) = ∞ . This example showsthe importance of the assumption on the non-free locus in Corollary 8.5.(iii) Let R = k [[ x , y , u , v ]] / ( xy − uv ) and let M = ( x , u ) ⊆ R . Recall from Example 3.9 that M is locally freeon the punctured spectrum of R and is maximal Cohen-Macaulay. Set n =
2. Then n < depth R ( M ) and H m ( M ⊗ R M ∗ ) ≃ H m ( m ) =
0. However, we have pd R ( M ) = ∞ . This example shows that theoddness of the integer n is necessary in Corollary 8.5. Corollary 8.7.
Let R be a complete intersection local ring of dimension d ≥ , and let M ∈ mod ( R ) . IfM is maximal Cohen–Macaulay and H i m ( M ⊗ M ∗ ) = for an odd integer i with < i < d, then M is free.Proof. This is an immediate consequence of Proposition 8.5. (cid:3)
Example 8.8.
The first item shows that oddness of i in Corollary 8.7 is crucial. The second item showsthat the maximal Cohen–Macaulay assumption is crucial.(i) R = k [[ x , y , u , v ]] / ( xy − uv ) and M = ( x , u ) . Then, by Example 3.9, it follows that M is maximalCohen-Macaulay and H m ( M ⊗ R M ∗ ) =
0. However M is not free.(ii) Let R = k [[ x , y , z , v , w ]] and let M = Ω d − k . Then, by [2, 5.5(ii)], we see H m ( M ⊗ R M ∗ ) =
0. However, M is not free. As M is not maximal Cohen–Macaulay, this example shows that the depth assumptionon M is necessary in Corollary 8.7. Proposition 8.9.
Assume R is local and a complete intersection of dimension d, where d ≥ , and let Z ⊆ Spec
R be a specialization-closed subset. Assume further M ∈ mod ( R ) such that NF ( M ) ⊆ Z and grade R ( Z , M ) ≥ . Then the following hold: (i) If H Z ( M ⊗ R M ∗ ) = , then it follows that pd R ( M ) ≤ . (ii) If H Z ( M ⊗ R M ∗ ) = = H Z ( M ⊗ R M ∗ ) , then it follows that M is free. Proof.
First note, by Proposition 2.16, we have that M satisfies ( S ) . Therefore, M is reflexive andso M ≈ M ∗∗ ≈ Ω Tr M ∗ . Assume that H i Z ( M ⊗ R M ∗ ) = i with 2 ≤ i ≤
3. Then, in view ofLemma 3.5 we conclude that Ext i − R ( M , M ) ∼ = Ext i + R ( Tr M ∗ , M ) = . Now the assertions follow from [37,2.5]. (cid:3)
Next we prove a generalization of [36, 4.2]. In the following, for simplicity, we denote the Matlis dualfunctor by ( − ) v : = Hom R ( − , E R ( k )) , where E R ( k ) is the injective hull of the residue field k . Proposition 8.10.
Assume R is Gorenstein and local of dimension d, and let Z ⊂ Spec
R be aspecialization-closed subset. Let M , N ∈ mod ( R ) be such that NF ( N ) ∪ NF ( M ) ⊆ Z . Then, if < i < grade ( Z , R ) , it follows that H i Z ( M ∗ ⊗ R N ∗ ) ∼ = H d − i + m ( M ⊗ R N ) v . Proof.
There is a natural map Φ : M ∗ ⊗ R N ∗ → ( M ⊗ R N ) ∗ taking f ⊗ g to the map x ⊗ y f ( x ) · g ( y ) . Itis easy to see that Φ is an isomorphism if either M or N is free. Hence, by our assumption, ker ( Φ ) andcoker ( Φ ) are Z -torsion. It follows from parts (ii) and (iii) of Theorem 2.11 that(8.10.1) H i Z ( M ∗ ⊗ R N ∗ ) ∼ = H i Z (( M ⊗ R N ) ∗ ) for all i > . Set L = M ⊗ R N . Consider the natural map Ψ : L → L ∗∗ . As NF ( L ) ⊆ Z , we deduce that Ψ p is an isomorphism for all p ∈ Spec R \ Z . Hence, by using Proposition 2.16 and (6.12.1) we ob-serve that grade R ( Z , R ) ≤ min { grade R ( ker ( Ψ )) , grade R ( coker ( Ψ )) } . In other words, we have thatExt iR ( ker ( Ψ ) , R ) = = Ext iR ( coker ( Ψ ) , R ) for i < grade R ( Z , R ) . Hence, the natural map Ψ inducesthe following isomorphism:(8.10.2) Ext iR ( L , R ) ∼ = Ext iR ( L ∗∗ , R ) for all i < grade R ( Z , R ) − . Note that L ∗∗ ≈ Ω Tr L ∗ . Therefore, by Lemma 3.5(i), we obtain the following isomorphisms:(8.10.3) H i Z ( L ∗ ) ∼ = Ext i + R ( Tr L ∗ , R ) ∼ = Ext i − R ( L ∗∗ , R ) for all 1 < i < grade R ( Z , R ) . Now the assertion is clear, by (8.10.1), (8.10.2), (8.10.3) and the Theorem 2.14. (cid:3)
Theorem 8.11.
Assume R is local and a complete intersection of dimension d, where d ≥ , and letM ∈ mod ( R ) such that depth R ( M ) ≥ . Then the following hold: (i) If H i m ( M ⊗ R M ∗ ) = for some i ∈ { , d − } , then it follows pd R ( M ) ≤ . (ii) If H i m ( M ⊗ R M ∗ ) = = H j m ( M ⊗ R M ∗ ) for some i ∈ { , d − } and j ∈ { , d − } , then M is free.Proof. This follows easily from Proposition 8.9 and Proposition 8.10. (cid:3)
Example 8.12.
Let R = k [[ x , y , z , u , v ]] and let M = Ω k . Then depth R ( M ) =
4. In view of [2, 5.5(ii)], weknow that H m ( M ⊗ R M ∗ ) =
0. Recall that pd R ( M ) =
1. Thus the example shows:(i) In Theorem 8.11(i), the bound on the projective dimension is sharp.(ii) In Theorem 8.11(ii), the vanishing in two spots is necessary.At this point we do not know whether or not the depth assumption on M in Theorem 8.11 is necessary.In some cases one can deduce a similar result for reflexive modules. Theorem 8.13. ( ˇCesnaviˇcius [24]) Let R be a graded normal complete intersection over a field of di-mension d ≥ . Assume M ∈ mod ( R ) , where depth R ( M ) ≥ . Then M is free provided that at least oneof the following conditions holds: (i) H m ( M ⊗ R M ∗ ) = H m ( M ⊗ R M ∗ ) = . (ii) H d − m ( M ⊗ R M ∗ ) = H d − m ( M ⊗ R M ∗ ) = . STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 33
Proof.
Note that there is a homogeneous ideal I of k [ X , . . . , X n ] such that R = k [ X ,..., X n ] I . Let X : = Proj ( R ) and E : = e M . Then X ⊂ P n is globally complete intersection, E is a vector bundle and that L n ∈ Z H ( X , O X ( n )) ≃ R . Also, M i ∈ Z H ( X , E nd O X ( E )( i )) = M i ∈ Z H ( X , E ⊗ E ∗ ( i )) = H m ( M ⊗ R M ∗ ) = . Similarly, one can show that H ∗ ( X , E nd O X ( E )) =
0. In view of [24, 1.2], E is direct sum of powers of O ( ) . It follows that M = H ∗ ( X , E ) is free. The second part follows from the first part and Proposition8.10. (cid:3) If R is a Gorenstein local ring of even dimension, and M ∈ mod ( R ) is maximal Cohen–Macaulaysuch that CI-dim R ( M ) < ∞ (e.g., R is complete intersection) and depth R ( M ⊗ R M ∗ ) ≥
1, then M is free;see [18, 3.9]. Note that one needs a ring of even dimension for this result to hold; see [18, 3.12]. Inthe following we show that one can replace the condition depth R ( M ⊗ R M ∗ ) ≥ n m ( M ⊗ R M ∗ ) for some integer n with 0 ≤ n < d , and generalize [18, 3.9] to obtain a stronger result. Theorem 8.14.
Assume R is local and Gorenstein of even dimension d. If M ∈ mod ( R ) is maximalCohen–Macaulay such that CI-dim R ( M ) < ∞ and H n m ( M ⊗ R M ∗ ) = for some integer n with ≤ n < d,then M is free.Proof. Note that, since M ∈ mod ( R ) , we may assume d ≥
2. Note also that, the case where n = n = n ≥ n is odd, then the assertion follows from Corollary 8.5. Next assume n is even. Then Proposition8.10 implies that H d − n + m ( M ⊗ R M ∗ ) =
0. As M is reflexive, M ≈ M ∗∗ ≈ Ω Tr M ∗ . Therefore, by Lemma3.5, we have that Ext d − nR ( M , M ) ∼ = Ext d − n + R ( Tr M ∗ , M ) =
0. Consequently, as d − n is even, we concludethat pd R ( M ) < ∞ , i.e., M is free; see [8, 4.2]. (cid:3) Let X be a projective variety over an algebraically closed field k . For each vector bundle E , we denote Γ ∗ ( E ) : = L i ∈ Z Γ ( X , E ( i )) . Recall that E is called arithmetically Cohen–Macaulay if H i ( X , E ( j )) = i = , . . . , dim X − j ∈ Z . The following result should be compared with [28, 1.5] and [24,1.2]. In fact, for an arithmetically Cohen-Macaulay vector bundle on a smooth complete intersection ofodd dimension, we have a stronger result as we state next: Corollary 8.15.
Let k be an algebraically closed and let X ⊂ P mk be a globally complete intersection ofdimension d ≥ . Assume that E is a vector bundle and that H n ( X , E ⊗ E ∗ ( j )) = for all j ∈ Z and forsome < n < d. The following statements hold: (i) If n is even and depth ( Γ ∗ ( E )) ≥ n + , then pd ( Γ ∗ ( E )) < ∞ over its affine cone. In particular, if E is arithmetically Cohen–Macaulay, then it is a direct sum of powers of O ( ) . (ii) If d is odd and E is arithmetically Cohen–Macaulay, then E is a direct sum of powers of O ( ) .Proof. Note that there is a homogeneous ideal I of k [ X , . . . , X m ] generated by regular sequence such that X = Proj ( R ) where R = k [ X ,..., X m ] I . Recall that L n ∈ Z H ( X , O X ( n )) ≃ R and that dim ( R ) = dim ( X ) + R -module M such that E = e M . In fact M = Γ ∗ ( E ) which is graded and reflexive. Recallthat arithmetically Cohen-Macaulay bundles correspond to maximal Cohen–Macaulay modules over theassociated graded ring. From this, M m is maximal Cohen–Macaulay if and only if E is arithmeticallyCohen-Macaulay. Also, we have the following isomorphism:(8.15.1) H n + m ( M ⊗ R M ∗ ) ∼ = M i ∈ Z H n ( X , E ⊗ E ∗ ( i )) = . (i) In view of Proposition 8.5 and (8.15.1), we see pd ( M m ) is finite over R m . Also, due to [14,1.5.15(e)], we know that pd R ( M ) < ∞ . (ii) Note that M = Γ ∗ ( E ) is maximal Cohen–Macaulay and that E = e M . In view of the Theorem 8.14and (8.15.1), pd ( M m ) is zero over R m . Also, due to [14, 1.5.15(e)], we know that pd R ( M ) = M is graded, there is a finite set L ⊂ Z such that M = L ℓ ∈ L R ( ℓ ) . Therefore the following observationcompletes the proof: E = e M = L ℓ ∈ L g R ( ℓ ) = L ℓ ∈ L O ( ) ℓ . (cid:3) Example 8.16. (i) Let R = k [ x , y , u , v ] / ( xy − uv ) and M = ( x , u ) . Let X = Proj ( R ) and E : = e M . Then, by Example8.8(i), we have H ∗ ( X , E nd O X ( E )) =
0. However, E is not trivial.(ii) Let X ⊂ P k be a smooth hypersurface of degree d ≥
2, and let E be an indecomposable arith-metically Cohen–Macaulay vector bundle of rank two. Then, by [46, 1.1.2], it follows thatH ∗ ( X , E nd O X ( E )) =
0. Since E is indecomposable, this example shows that n needs to be aneven integer in Corollary 8.15(i).Next is an application of Lemma 6.13; we use it to prove Theorem 8.19. Proposition 8.17.
Assume R is Gorenstein and local, and let M ∈ mod ( R ) . If M is locally free on X n − ( R ) for some integer n ≥ and satisfies Serre’s condition ( S n + ) , then the following hold: (i) Ext iR ( M , M ) ∼ = Ext iR ( M ⊗ R M ∗ , R ) for all i with ≤ i ≤ n − . (ii) There is an injection
Ext nR ( M , M ) ֒ → Ext nR ( M ⊗ R M ∗ , R ) .Proof. First note that, by [5, 4.25], M is ( n + ) -torsionfree, i.e., Ext iR ( Tr M , R ) = i = , . . . , n + M is reflexive and Ext iR ( M ∗ , R ) = i = , . . . , n −
1. By our assumption,grade R ( Tor Ri ( M , M ∗ )) ≥ n for all i ≥
1. Now the assertion is clear by Lemma 6.13. (cid:3)
From now on we use the following result of Jothilingam without further reference; see [39, Theorem]and [30, 3.1.2].
Lemma 8.18. (Jothilingam) Assume R is local ring and let M ∈ mod ( R ) be Tor-rigid. If Ext nR ( M , M ) = n ≥
1, then pd R ( M ) < n .In the following G ( R ) Q , i.e., G ( R ) Q denotes the reduced Grothendicek group with rational coeffi-cients; we refer the reader to [28] for the definition and basic properties of this group. Theorem 8.19.
Assume R is d-dimensional, Gorenstein and local, and let M ∈ mod ( R ) . Assume M islocally free on X n − ( R ) for some integer n, where < n < d. Assume further that M satisfies Serre’scondition ( S n + ) and H d − n m ( M ⊗ R M ∗ ) = . (i) If n is even and
CI-dim R ( M ) < ∞ , then it follows that pd R ( M ) < d − n. (ii) If M is Tor-rigid, then it follows that pd R ( M ) < n. (iii) If R is a hypersurface which is quotient of an unramified regular ring, and the class of M is zero inG ( R ) Q , then it follows that pd R ( M ) < n.Proof. We have, by Theorem 2.14, that Ext nR ( M ⊗ R M ∗ , R ) =
0. Then, in view of Proposition 8.17, itfollows that:(8.19.1) Ext nR ( M , M ) = . (i) This follows from (8.19.1) and [8, 4.2].(ii) This follows from (8.19.1) and Lemma 8.18.(iii) This follows from (8.19.1) and [28, 5.4]. (cid:3) The following is an immediate consequence of Theorem 8.19(ii).
Corollary 8.20.
If R is regular and local of dimension d ≥ , and M ∈ mod ( R ) is reflexive such that H d − m ( M ⊗ R M ∗ ) = , then M is free. The following result is to be compared with [36, 4.1(2)].
STUDY OF THE COHOMOLOGICAL RIGIDITY PROPERTY 35
Proposition 8.21.
Assume R is Cohen-Macaulay and local, and let Z ⊂ Spec
R be a specialization-closed subset. Let M ∈ mod ( R ) be Tor-rigid and NF ( M ) ⊆ Z . If H n Z ( M ⊗ R M ∗ ) = for some n, where ≤ n ≤ grade R ( Z , M ) , then pd R ( M ) < n − .Proof. First note, by Proposition 2.16, that grade R ( Z , M ) = inf { depth R p ( M p ) | p ∈ Z } ≥
2. Therefore, M p is free for all p ∈ X ( R ) . It follows from [14, 1.4.1(b)] that M is reflexive and so M ≈ M ∗∗ ≈ Ω Tr M ∗ .Hence, by Lemma 3.5 and our assumption, we have Ext n − R ( M , M ) ∼ = Ext n + R ( Tr M ∗ , M ) =
0. Now theassertion is clear by Lemma 8.18. (cid:3)
The Tor-rigidity condition is necessary in Proposition 8.21; see Example 3.9. We now give an exampleto consider the other hypotheses.
Example 8.22. (i) Let R = k [[ x , y , z , u , v ]] and let M = Ω k . Then depth R ( M ) =
4. In view of [2, 5.5(ii)] we knowthat H m ( M ⊗ R M ∗ ) =
0. We apply Proposition 8.21 for the case where n = < depth R ( M ) to seepd R ( M ) < n − =
2. Note that pd R ( M ) =
1. This example shows that the bound on the projectivedimension of M in Proposition 8.21 is sharp.(ii) Let R = k [[ x , y ]] , M = Ω k = m and let a = xR for some 0 = x ∈ m . Then H a ( M ⊗ R M ∗ ) = a is principal. However, pd R ( M ) = = n −
1. This example shows, setting n =
2, the bound on n is sharp in Proposition 8.21.In the following we establish a non-vanishing result, which is new even for regular local rings. Corollary 8.23.
Assume R is Gorenstein and local ring, and let M ∈ mod ( R ) be Tor-rigid (e.g. Ris regular.) Assume further R has odd dimension d ≥ and depth R ( M ) = d + . Then it follows that H i m ( M ⊗ R M ∗ ) = for all i with < i < d.Proof. Set t : = d + . Assume contrarily that H i m ( M ⊗ R M ∗ ) = i < d . First we deal withcase 0 < i ≤ t . By Corollary 8.2 we may assume that i =
1. Hence, 2 ≤ i ≤ t . In view of Proposition8.21 we have pd R ( M ) < i −
1. It follows from the Auslander–Buchsbaum formula that d − t < i − ≤ t −
1. Hence d < t − = d + − = d which is a contradiction. Now let i > t . By Proposition 8.10,H d − i + m ( M ⊗ R M ∗ ) =
0. Note that d − i + < d − t + = t + (cid:3) The following example shows that the locally free assumption in Corollary 8.23 is crucial.
Example 8.24.
Assume R is 3-dimensional, Gorenstein, and local (e.g., R = k [[ x , y , z ]] ). Let M = R / xR for a non zero-divisor x ∈ m . Then depth R ( M ) = = + and pd R ( M ) =
1. Hence, M is Tor-rigid. Notethat, as M is torsion, we have M ∗ =
0. Thus H i m ( M ⊗ R M ∗ ) = i .An application of Proposition 8.17 is a slight generalization of Theorem 8.11. Theorem 8.25.
Assume R is local, a complete intersection, and has dimension d ≥ . Let M ∈ mod ( R ) and assume M is locally free on X ( R ) and it satisfies ( S ) . Then the following hold: (i) If H d − m ( M ⊗ R M ∗ ) = , then pd R ( M ) ≤ . (ii) If H d − m ( M ⊗ R M ∗ ) = = H d − m ( M ⊗ R M ∗ ) , then M is free.Proof. This follows from the Theorem 2.14, Proposition 8.17 and [37, 2.5]. (cid:3) A CKNOWLEDGEMENTS
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OHSEN A SGHARZADEH , H
AKIMIEH , 16599-19556, T
EHRAN , I
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