Comparisons between Annihilators of Tor and Ext
aa r X i v : . [ m a t h . A C ] D ec COMPARISONS BETWEEN ANNIHILATORS OF TOR AND EXT
SOUVIK DEY AND RYO TAKAHASHI
Abstract.
In this paper, we compare annihilators of Tor and Ext modules of finitely generated modulesover a commutative noetherian ring. One of our results refines a theorem of Dao and Takahashi. Introduction
Let R be a d -dimensional Cohen–Macaulay local ring with a canonical module. Let 0 ≤ c ≤ d be aninteger. In this paper, we are mainly interested in the two ideals t cn ( R ) = \ ann R Tor Ri ( M, N ) , e cn ( R ) = \ ann R Ext iR ( M, N )of R , where n ≥ i > n , and the maximalCohen–Macaulay modules M, N that are locally free in codimension less than c . Our first main result isthe theorem below, which is included in Theorems 3.11(3), 4.6(2) and Corollaries 3.6, 4.4(2). Theorem 1.1.
The following assertions hold true. (1) If R is either artinian or Gorenstein, then t cn ( R ) = e cn ( R ) . (2) If R is either locally Gorenstein in codimension less than c or a complete equicharacteristic local ringwith perfect residue field, then p t cn ( R ) = p e cn ( R ) . Denote by mod R the category of finitely generated R -modules, and by CM c ( R ) the full subcategoryof mod R consisting of maximal Cohen–Macaulay modules that are locally free in codimension less than c . As an application of Theorem 1.1 we obtain the following result, which is the same as Corollary 4.10. Theorem 1.2.
Consider the following conditions. (i) CM c ( R ) has finite dimension. (ii) ht e cn ( R ) ≥ c . (iii) ht t cn ( R ) ≥ c . (iv) R is locally regular in codimension less than c . (1) The implications (i) ⇒ (ii) ⇔ (iii) ⇒ (iv) hold. (2) The implications (i) ⇔ (ii) ⇔ (iii) ⇒ (iv) hold when c = d . (3) The implications (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) hold when R is excellent and equicharacteristic. When n = 0, the main result of [6] (i.e. [6, Theorem 1.1]) asserts Theorem 1.2(2) minus the implication(iii) ⇒ (ii), and Theorem 1.2(3) for c = d under the assumption that R is complete and has perfectcoefficient field. Thus, Theorem 1.2 highly refines the main result of [6].The organization of this paper is as follows. In Section 2, we state definitions and properties offundamental notions used in this paper. In Section 3, we explore annihilators of Tor and Ext modulesover a general commutative noetherian ring. In Section 4, we focus on annihilators of Tor and Extmodules over a Cohen–Macaulay local ring.2. Preliminaries
What we state in this section is used in the next sections. We begin with our convention.
Mathematics Subject Classification.
Key words and phrases. annihilator, canonical module, Cohen–Macaulay ring, Ext, maximal Cohen–Macaulay module,cosyzygy, non-Gorenstein locus, punctured spectrum, syzygy, Tor, trace ideal.Takahashi was partly supported by JSPS Grant-in-Aid for Scientific Research 19K03443. Strictly speaking, Theorem 1.2 does not completely include [6, Theorem 1.1]; the implication (ii) ⇒ (i) holds for n = 0and c = d without the assumption that R has a canonical module. Convention 2.1.
Throughout the present paper, let R be a commutative noetherian ring. We assumethat all modules are finitely generated and all subcategories are strictly full. Denote by mod R the categoryof (finitely generated) R -modules, and by CM( R ) the subcategory of mod R consisting of maximal Cohen–Macaulay modules (recall that an R -module M is called maximal Cohen–Macaulay if depth R p M p =dim R p for all p ∈ Supp R M ). We confuse an R -module M with the subcategory of mod R consisting of M . We denote by ( − ) ∗ the algebraic dual Hom R ( − , R ). Whenever R is a local ring, ( − ) ∨ stands for theMatlis dual. Whenever R is a Cohen–Macaulay ring with a canonical module ω , we denote by ( − ) † thecanonical dual Hom R ( − , ω ).From now on, we state the definitions of notions used in the next sections together with a couple oftheir basic properties. Definition 2.2.
Let X be a subcategory of mod R .(1) We denote by add X the subcategory of mod R consisting of direct summands of finite direct sumsof modules in X . Note that add R equals the subcategory of mod R consisting of projective modules.(2) We denote by X the subcategory of mod R consisting of modules M such that there is an isomorphism M ⊕ P ∼ = X ⊕ Q with P, Q ∈ add R and X ∈ X . We say that X is stable if X = X . Note that X = X ,so that X is stable. Also, add R and mod R are stable.(3) Suppose that R is a Cohen–Macaulay ring with a canonical module ω and that X is contained inCM( R ). We denote by X the subcategory of CM( R ) consisting of maximal Cohen–Macaulay modules M such that there is an isomorphism M ⊕ I ∼ = X ⊕ J with I, J ∈ add ω and X ∈ X . We say that X is costable if X = X . Note that add ω and CM( R ) are costable. Definition 2.3.
For an R -module M , we denote by Ω M the (first) syzygy of M , that is, the kernel of anepimorphism from a projective R -module to M . For n ≥ n th syzygy of M byΩ n M = Ω(Ω n − M ), and set Ω M = M . The n th syzygy of M is uniquely determined up to projectivesummands. For a subcategory X of mod R and an integer n ≥
0, we set Ω n X = { Ω n X | X ∈ X } . Wesay that a subcategory X of mod R is closed under syzygies if Ω X ⊆ X . Note that Ω X = Ω X = Ω X . Definition 2.4.
For an R -module M we denote by Tr M the (Auslander) transpose of M . This is definedas follows. Take a projective presentation P f −→ P → M →
0. Dualizing this by R , we get an exactsequence 0 → M ∗ → P ∗ f ∗ −→ P → Tr M →
0, that is, Tr M is the cokernel of the map f ∗ . The transposeof M is uniquely determined up to projective summands; see [1] for basic properties. For a subcategory X of mod R , we set Tr X = { Tr X | X ∈ X } . We say that X is closed under transposes if Tr X ⊆ X . Notethat there are equalities Tr X = Tr X = Tr X . Definition 2.5.
Let Φ be a subset of Spec R . We define the (Krull) dimension of Φ by dim Φ =sup { dim R/ p | p ∈ Φ } . Definition 2.6.
For an R -module M we denote by NF( M ) the nonfree locus of M , that is, the setof prime ideals p of R such that the R p -module M p is nonfree. It is well-known and easy to see thatNF( M ) is a closed subset of Spec R in the Zariski topology. We set Spec ( R ) = Spec R \ Max R andcall it the punctured spectrum of R . Note that an R -module M is locally free on Spec ( R ) if and only ifdim NF( M ) ≤
0. For each n ≥
0, we denote by mod n ( R ) the subcategory of mod R consisting of modules M such that dim NF( M ) ≤ n . It is easy to see that mod n ( R ) is stable and closed under syzygies andtransposes. For an integer n ≥ n ( R ) = CM( R ) ∩ mod n ( R ). Note that CM n ( R ) is stable andclosed under syzygies if R is Cohen–Macaulay. Definition 2.7. An R -module M is said to be totally reflexive if Ext iR ( M, R ) = Ext iR (Tr M, R ) = 0 forall i >
0. This is equivalent to saying that the canonical map M → M ∗∗ is an isomorphism (i.e., M is reflexive) and Ext iR ( M, R ) = Ext iR ( M ∗ , R ) = 0 for all i >
0. Every totally reflexive module is thesyzygy of some totally reflexive module. If R is Cohen–Macaulay, then every totally reflexive R -moduleis maximal Cohen–Macaulay. Also, R is Gorenstein if and only if every maximal Cohen–Macaulay R -module is totally reflexive. For more details of totally reflexive modules, we refer the reader to [1, 3]. Wedenote by G ( R ) the subcategory of mod R consisting of totally reflexive modules. Note that G ( R ) is stable Note that in the case where R is not local, a canonical module is in general not unique up to isomorphism (see [2,Remark 3.3.17]). We thus fix one of the canonical modules in the non-local case. OMPARISONS BETWEEN ANNIHILATORS OF TOR AND EXT 3 and closed under syzygies and transposes. For an integer n ≥ G n ( R ) = G ( R ) ∩ mod n ( R ), whichis stable and closed under syzygies and tranposes as well. If R is Gorenstein, then G n ( R ) = CM n ( R ). Definition 2.8.
Let R be a Cohen–Macaulay ring with a canonical module ω .(1) For a maximal Cohen–Macaulay R -module M , we denote by ✵ M the (first) cosyzygy of M , that is,the maximal Cohen–Macaulay cokernel of a monomorphism to a module in add ω . Cosyzygies alwaysexist: taking an exact sequence 0 → Ω( M † ) → P → M † → P projective and dualizing itby ω , one obtains an exact sequence 0 → M → P † → (Ω( M † )) † → M † )) † is the cosyzygy of M . For n ≥ n th cosyzygy of M by ✵ n M = ✵ ( ✵ n − M ), and set ✵ M = M . Note that the n th cosyzygy of M is uniquelydetermined up to direct summands that belong to add ω . For a subcategory X of CM( R ) and aninteger n ≥
0, we set ✵ n X = { ✵ n X | X ∈ X } . We say that X is closed under cosyzygies if ✵ X ⊆ X .(2) For a subcategory X of CM( R ) we set X † = { X † | X ∈ X } . Then X † ⊆ X ⇐⇒ X ⊆ X † ⇐⇒ X = X † . We say that X is closed under canonical duals if one of these equivalent conditions holds. Remark 2.9. (1) Suppose that a subcategory X of mod R is closed under syzygies. Then for an exactsequence 0 → M → P → X → R -modules with X ∈ X and P ∈ add R , one has M ∈ X . Theconverse holds if X is stable.(2) Suppose that a subcategory X of mod R is closed under transposes. Then for an exact sequence P f −→ P → X → R -modules with X ∈ X and P , P ∈ add R , the cokernel of f ∗ belongs to X .The converse holds if X is stable.(3) Let R be a Cohen–Macaulay ring with a canonical module ω . Let X be a subcategory of CM( R ).Suppose that X is closed under cosyzygies. Then for an exact sequence 0 → X → I → M → R -modules with X ∈ X and I ∈ add ω , one has M ∈ X . The converseholds if X is costable.3. Annihilators over a commutative noetherian ring
In this section we investigate annihilators of Tor and Ext over an arbitrary commutative noetherianring R . First of all, we give their definitions. Definition 3.1.
Let X , Y be subcategories of mod R , and let n ≥ T n ( X , Y ) = M i>n M X ∈X M Y ∈Y Tor Ri ( X, Y ) , E n ( X , Y ) = M i>n M X ∈X M Y ∈Y Ext iR ( X, Y ) . We define the ideals T n ( X , Y ) and E n ( X , Y ) of R by T n ( X , Y ) = ann R T n ( X , Y ) = \ i>n \ X ∈X \ Y ∈Y ann R Tor Ri ( X, Y ) , E n ( X , Y ) = ann R E n ( X , Y ) = \ i>n \ X ∈X \ Y ∈Y ann R Ext iR ( X, Y ) . Note that if X is closed under syzygies, T n ( X , Y ) = T X ∈X T Y ∈Y ann R Tor Rn +1 ( X, Y ) and E n ( X , Y ) = T X ∈X T Y ∈Y ann R Ext n +1 R ( X, Y ). We put T n ( X ) = T n ( X , X ) and E n ( X ) = E n ( X , X ). Remark 3.2.
Let X , Y be subcategories of mod R and n ≥ R ( T n ( X , Y )) ⊆ V( T n ( X , Y )) , Supp R ( E n ( X , Y )) ⊆ V( E n ( X , Y )) . As we will see in Remark 4.11(3), these inclusions are not necessarily equalities. Thus, it is not sufficientto investigate the supports of the modules T n ( X , Y ) and E n ( X , Y ) to get the structure of (the radicalsof) the ideals T n ( X , Y ) and E n ( X , Y ).Let M, N be R -modules. We denote by Hom R ( M, N ) the quotient of Hom R ( M, N ) by homomorphisms M → N factoring through projective R -modules. We set End R ( M ) = Hom R ( M, M ). The followinglemma yields an isomorphism between Tor and Ext modules.
Lemma 3.3.
For R -modules M, N one has an isomorphism
Ext R (Tr Ω Tr Ω M, N ) ∼ = Tor R (Tr Ω M, N ) . SOUVIK DEY AND RYO TAKAHASHI
Proof.
An exact sequence 0 → Ext R (Tr Ω Tr Ω M, N ) → Tor R (Tr Ω M, N ) → Hom R (Ext R (Tr Ω M, R ) , N )exists by [1, Theorem (2.8)]. As Ω M is 1-torsionfree ([1, Definition (2.15)]), we have Ext R (Tr Ω M, R ) = 0.Thus the isomorphism in the lemma is obtained. (cid:4)
Remark 3.4.
Here is another proof of Lemma 3.3, which may be easier for the reader who is familiarwith Auslander’s approximation theory: There is an exact sequence 0 → Ω M f −→ P → Tr Ω Tr Ω M → f is a left (add R )-approximation. An exact sequence Hom R ( P, N ) f ′ −→ Hom R (Ω M, N ) → Ext R (Tr Ω Tr Ω M, N ) → f ′ coincides with the setof homomorphisms Ω M → N factoring through projective modules. We obtain an isomorphismExt R (Tr Ω Tr Ω M, N ) ∼ = Hom R (Ω M, N ). Combining this with [10, Lemma (3.9)] deduces the lemma.Using the above lemma, we obtain the following proposition on annihilators.
Proposition 3.5.
Let X be a subcategory of mod R . Suppose that X is contained in Ω(mod R ) , andclosed under syzygies and transposes. Then for all integers n ≥ and subcategories Y of mod R one has T n ( X , Y ) = E n ( X , Y ) .Proof. Note that E n ( X , Y ) = E n ( X , Y ) and T n ( X , Y ) = T n ( X , Y ). Replacing X by X , we may assume X is stable. Any R -module M satisfies Tr Tr M ∼ = M up to projective summands by [1, Proposition(2.6)], which implies X = Tr X . Let X ∈ X . Then X is a syzygy, and hence it is 1-torsionfree by [1,Theorem (2.17)]. Therefore X ∼ = Ω Tr Ω Tr X up to projective summands by [1, Theorem (2.17)] again.As X is closed under syzygies and transposes, we get Tr Ω Tr X ∈ X , and X ∈ Ω X . Thus X = Ω X , and X = Tr X = Tr Ω X . There are equalities T n ( X , Y ) = T (Ω n X , Y ) = T ( X , Y ) = T X ∈X ,Y ∈Y ann R Tor R ( X, Y ) , (3.5.1) E n ( X , Y ) = E (Ω n X , Y ) = E ( X , Y ) = T X ∈X ,Y ∈Y ann R Ext R ( X, Y ) . (3.5.2)Using the equalities X = Tr Ω X = Tr Ω Tr Ω X and Lemma 3.3, we observe that the last terms in (3.5.1)and (3.5.2) coincide. (cid:4) The following corollary is a direct consequence of Proposition 3.5, which gives part of Theorem 1.1(1).
Corollary 3.6.
Let n, t ≥ be any integers and let C be any subcategory of mod R . Then the equality T n ( G t ( R ) , C ) = E n ( G t ( R ) , C ) holds. In particular, T n (CM t ( R )) = E n (CM t ( R )) if R is Gorenstein. To prove our next proposition, we establish a lemma.
Lemma 3.7.
Let M be an R -module. Then T ( M, mod R ) = ann R End R ( M ) = ann R Ext R ( M, Ω M ) = E ( M, mod R ) . Proof.
We call the four ideals (1), (2), (3) and (4) in order. Clearly, (3) contains (4). Let a be an elementof (3). Then the multiplication map M a −→ M factors through a projective module by the proof of [8,Lemma 2.14]. Hence M a −→ M is zero in End R ( M ), and so is the composition of any endomorphism of M with M a −→ M . Thus (2) contains (3). Let b be an element of (2). Then b · id M is zero in End R ( M ),which means that the multiplication map M b −→ M factors through a projective module. There is adiagram M f −→ P g −→ M of homomorphisms of R -modules with P projective such that gf = ( M b −→ M ).Applying Tor Ri ( − , N ) and Ext iR ( − , N ) with i > N ∈ mod R , we see that the multiplication mapsTor Ri ( M, N ) b −→ Tor Ri ( M, N ) and Ext iR ( M, N ) b −→ Ext iR ( M, N ) are zero as Tor Ri ( P, N ) = Ext iR ( P, N ) = 0.Thus (1) and (4) contain (2). There is an isomorphism End R ( M ) ∼ = Tor R ( M, Tr M ) by [10, Lemma(3.9)], from which we see that (2) contains (1). (cid:4) Now we obtain inclusions among annhilators of Tor and Ext.
Proposition 3.8.
Let X , Y be subcategories of mod R . Assume that X is closed under syzygies and that Y contains X . Then for any integer n ≥ one has E n ( X ) = T X ∈X ann R Ext n +1 R ( X, Ω n +1 X )= T X ∈X ann R End R (Ω n X ) = E n ( X , Y ) ⊆ T n ( X , Y ) ⊆ T n ( X ) . If moreover Y contains Tr X , then one has the equality T n ( X , Y ) = E n ( X , Y ) . OMPARISONS BETWEEN ANNIHILATORS OF TOR AND EXT 5
Proof. As Y contains X , the ideals E n ( X ) and T n ( X ) contain E n ( X , Y ) and T n ( X , Y ), respectively. Since X is closed under syzygies, we have an equality E n ( X ) = T X,X ′ ∈X ann R Ext n +1 R ( X, X ′ ), the right-handside of which is contained in I := T X ∈X ann R Ext n +1 R ( X, Ω n +1 X ) as Ω n +1 X ∈ X . Lemma 3.7 implies I = T X ∈X ann R End R (Ω n X ) = T X ∈X E (Ω n X, mod R ) = E n ( X , mod R ) ⊆ E n ( X , Y ) ,I = T X ∈X ann R End R (Ω n X ) ⊆ T X ∈X T (Ω n X, mod R ) = T n ( X , mod R ) ⊆ T n ( X , Y ) . Now the proof of the first assertion of the proposition is completed.Next we show the last assertion of the proposition. We already know that T n ( X , Y ) contains E n ( X , Y ).Let a ∈ T n ( X , Y ) and X ∈ X . The assumption implies Ω n X ∈ X and Tr Ω n X ∈ Y . The isomorphismsEnd R (Ω n X ) ∼ = Tor R (Tr Ω n X, Ω n X ) ∼ = Tor R (Ω n X, Tr Ω n X ) ∼ = Tor Rn +1 ( X, Tr Ω n X )hold, where the first isomorphism follows from [10, Lemma (3.9)]. The last term Tor Rn +1 ( X, Tr Ω n X ) isannihilated by the element a , and so is the first term End R (Ω n X ). It follows that T n ( X , Y ) is containedin T X ∈X ann R End R (Ω n X ) = E n ( X , Y ). (cid:4) Remark 3.9.
Proposition 3.8 also deduces Corollary 3.6 for those C which contain G t ( R ).To derive our next annihilator relations, we need a lemma, generalizing [6, Propositions 4.5 and 4.6(1)].Assertions (1) and (2) are shown similarly as in the proof of [6, Proposition 4.5], while (3) is deducedanalogously as in the proof of [6, Proposition 4.6(1)] by using (1) and (2) instead of [6, Proposition 4.5]. Lemma 3.10.
Let R be a d -dimensional Cohen–Macaulay local ring. Let n ≥ be an integer. (1) Let a ∈ T n (CM ( R )) . Then a d Tor Ri ( M, N ) = 0 for all i > n + 4 d and M, N ∈ mod R . (2) Let a ∈ E n (CM ( R )) . Then a d ( d +1) Ext iR ( M, N ) = 0 for all i > n + d and M, N ∈ mod R . (3) It holds that
Sing R ⊆ V( T n +4 d (mod R )) ∩ V( E n + d (mod R )) ⊆ V( T n (CM ( R ))) ∩ V( E n (CM ( R ))) . Now we can prove the following theorem, which contains part of Theorem 1.1(2).
Theorem 3.11.
Let ( R, m , k ) be a equicharacteristic, local Cohen–Macaulay ring of dimension d . Let n ≥ be an integer. Let CM ( R ) ⊆ X ⊆ Y be subcategories of mod R , where X is closed under syzygies.Then Sing R = V( T n ( X , Y )) = V( E n ( X , Y )) , if one of the following three conditions is satisfied. (1) R is excellent and n ≥ d . (2) R is complete, k is perfect and n ≥ d . (3) R is complete, k is perfect and X ⊆
CM( R ) .Proof. Applying Lemma 3.10(3) and Proposition 3.8 gives rise to inclusions Sing R ⊆ V( T n (CM ( R ))) ⊆ V( T n ( X , Y )) ⊆ V( E n ( X , Y )). It remains to show that V( E n ( X , Y )) is contained in Sing R .(1) V( E n ( X , Y )) is contained in V( E d (mod R )), which is equal to Sing R by [8, Theorem 5.3].(2) V( E n ( X , Y )) is contained in V( E d (mod R )), which is equal to Sing R by [9, Corollary 5.15].(3) V( E n ( X , Y )) is contained in V( E ( X , Y )), which is equal to Sing R by [6, Proposition 4.8]. (cid:4) Annihilators over a Cohen–Macaulay local ring
In this section, we consider annihilators of Tor and Ext modules over a Cohen–Macaulay local ring.Let M be an R -module. The trace ideal of M , denoted tr M , is defined by the image of the canonicalmap Hom R ( M, R ) ⊗ R M → R given by f ⊗ x f ( x ). To prove the proposition below, we establish alemma. Lemma 4.1.
Let R be a d -dimensional Cohen–Macaulay local ring with a canonical module ω . Let M, N be R -modules. Let i ≥ be an integer. Suppose that N is maximal Cohen–Macaulay. (1) Let t be an integer with dim NF( M ) ≤ t ≤ d . Then one has Q tj =0 ann R Ext d − jR (Tor Ri + j ( M, N ) , ω ) ⊆ ann R Ext d + iR ( M, N † ) , Q tj =0 ann R Ext d − jR (Ext d + i − jR ( M, N ) , ω ) ⊆ ann R Tor Ri ( M, N † ) . If t = 0 , then ann R Tor Ri ( M, N ) = ann R Ext d + iR ( M, N † ) . (2) For any integer r ≥ , one has (tr ω ) r · ann R Ext r + iR ( N, Ω r M ) ⊆ ann R Ext iR ( N, M ) ⊇ (tr ω ) r · ann R Ext r + iR ( ✵ r N, M ) . SOUVIK DEY AND RYO TAKAHASHI
Proof. (1) There is a spectral sequenceE pq = Ext pR (Tor Rq ( M, N ) , ω ) = ⇒ H p + q = Ext p + qR ( M, N † ) . Clearly, E pq = 0 if p < q <
0. As ω has injective dimension d , we have that E pq = 0 if p > d . Thesupport of Tor Rq ( M, N ) is contained in NF( M ) if q >
0. Local duality ([2, Corollary 3.5.11(a)]) showsthat E pq = 0 if q > p < d − t . The filtration induced from the spectral sequence isH d + i = · · · = H d + id − t ⊇ H d + id − t +1 ⊇ · · · ⊇ H d + id ⊇ H d + id +1 = · · · = 0with H d + id − j / H d + id − j +1 = E d − j,i + j ∞ for each 0 ≤ j ≤ t . In general, E pqr is a subquotient of E pqr − for all p, q, r , and hence ann R E pq ∞ contains ann R E pq for all p, q . The filtration shows that Q tj =0 ann R E d − j,i + j ⊆ Q tj =0 ann R E d − j,i + j ∞ ⊆ ann R H d + i . The first inclusion in the assertion follows from this. The secondinclusion is deduced by a dual argument. Namely, there is a spectral sequenceE pq = Ext pR (Ext − qR ( M, N ) , ω ) = ⇒ H p + q = Tor R − p − q ( M, N † ) , and E pq = 0 if (i) p <
0, or (ii) q >
0, or (iii) p > d , or (iv) q < p < d − t . A filtration H − i = · · · =H − id − t ⊇ H − id − t +1 ⊇ · · · ⊇ H − id ⊇ H − id +1 = · · · = 0 is induced, where H − id − j / H − id − j +1 = E d − j, − d − i + j ∞ for each0 ≤ j ≤ t . This shows Q tj =0 ann R E d − j, − d − i + j ⊆ Q tj =0 ann R E d − j, − d − i + j ∞ ⊆ ann R H − i .Now let t = 0. The first spectral sequence yields Ext dR (Tor Ri ( M, N ) , ω ) ∼ = Ext d + iR ( M, N † ). As M islocally free on Spec ( R ), the R -modules Tor Ri ( M, N ) , Ext d + iR ( M, N † ) have finite length. By [2, Corollary3.5.9], we getExt d + iR ( M, N † ) ∨ ∼ = Ext dR (Tor Ri ( M, N ) , ω ) ∨ ∼ = H m (Tor Ri ( M, N )) = Tor Ri ( M, N ) . Hence ann R Tor Ri ( M, N ) is equal to ann R Ext d + iR ( M, N † ) ∨ , which coincides with ann R Ext d + iR ( M, N † ) by[2, Proposition 3.2.12(c)].(2) For each j ≥ → Ω j +1 M → R ⊕ m j → Ω j M →
0, which in-duces an exact sequence Ext i + jR ( N, R ) ⊕ m j → Ext i + jR ( N, Ω j M ) → Ext i + j +1 R ( N, Ω j +1 M ). By [4, The-orem 2.3], the ideal tr ω annihilates Ext lR ( X, R ) for all l > X ∈ CM( R ). It is observed that(tr ω ) · ann R Ext i + j +1 R ( N, Ω j +1 M ) ⊆ ann R Ext i + jR ( N, Ω j M ), and hence(tr ω ) r · ann R Ext r + iR ( N, Ω r M ) ⊆ (tr ω ) r − · ann R Ext r + i − R ( N, Ω r − M ) ⊆ · · · ⊆ (tr ω ) · ann R Ext i +1 R ( N, Ω M ) ⊆ ann R Ext iR ( N, M ) . It is also seen from [4, Theorem 2.3] that tr ω annihilates Ext lR ( ω, X ) for all l > X ∈ mod R . Adual argument using this and exact sequences 0 → ✵ j N → ω ⊕ n j → ✵ j +1 N → ω ) r · ann R Ext r + iR ( ✵ r N, M ) ⊆ ann R Ext iR ( N, M ). (cid:4) We can now prove the following proposition, which is an essential part of the theorem stated below.
Proposition 4.2.
Let R be a d -dimensional Cohen–Macaulay local ring with a canonical module ω . Let X , Y be subcategories of mod R with Y ⊆
CM( R ) . Let n ≥ be an integer. (1) Let ≤ t ≤ d be an integer. If X is contained in mod t ( R ) , then ( T n ( X , Y )) t +1 ⊆ E d + n ( X , Y † ) , ( E d + n − t ( X , Y )) t +1 ⊆ T n ( X , Y † ) , and the equalities hold when t = 0 . (2) For each integer r ≥ one has an inclusion (tr ω ) r · E r + n ( Y , Ω r X ) ⊆ E n ( Y , X ) ⊇ (tr ω ) r · E r + n ( ✵ r Y , X ) . Proof. (1) Fix i > n , 0 ≤ j ≤ t , M ∈ X and N ∈ Y . Note then that dim NF( M ) ≤ t .Take any element a j ∈ T n ( X , Y ). Since i + j ≥ i > n , the element a j belongs to ann R Tor Ri + j ( M, N ),which is contained in ann R Ext d − jR (Tor Ri + j ( M, N ) , ω ). The first inclusion in Lemma 4.1(1) yields a · · · a t ∈ Q tj =0 ann R Ext d − jR (Tor Ri + j ( M, N ) , ω ) ⊆ ann R Ext d + iR ( M, N † ) . As we fix i > n , M ∈ X and N ∈ Y , we get a · · · a t ∈ E d + n ( X , Y † ). Thus ( T n ( X , Y )) t +1 ⊆ E d + n ( X , Y † ). OMPARISONS BETWEEN ANNIHILATORS OF TOR AND EXT 7
The other inclusion is similarly deduced. Pick a j ∈ E d + n − t ( X , Y ). As d + i − j ≥ d + i − t > d + n − t ,we have a j ∈ ann R Ext d + i − jR ( M, N ) ⊆ ann R Ext d − jR (Ext d + i − jR ( M, N ) , ω ). The second inclusion in Lemma4.1(1) implies a · · · a t ∈ Q tj =0 ann R Ext d − jR (Ext d + i − jR ( M, N ) , ω ) ⊆ ann R Tor Ri ( M, N † ) , and hence a · · · a t ∈ T n ( X , Y † ). Therefore, the inclusion ( E d + n − t ( X , Y )) t +1 ⊆ T n ( X , Y † ) follows.When t = 0, the equality T n ( X , Y ) = E d + n ( X , Y † ) follows from the last assertion of Lemma 4.1(1).Replacing Y with Y † , we see that the equality T n ( X , Y † ) = E d + n ( X , Y ) also holds .(2) The assertion immediately follows from Lemma 4.1(2). (cid:4) Here are two immediate consequences of the above proposition.
Corollary 4.3.
Let R be a d -dimensional Cohen–Macaulay local ring with a canonical module. Let n ≥ be an integer. Then T n (CM ( R )) = E d + n (CM ( R )) if R is locally Gorenstein on Spec ( R ) .Proof. Since R is locally Gorenstein on the punctured spectrum, CM ( R ) is closed under canonical duals,that is to say, (CM ( R )) † = CM ( R ). Let t = 0 and X = Y = CM ( R ) in Proposition 4.2(1). (cid:4) Corollary 4.4.
Let R be an artinian local ring. Let n ≥ be an integer. (1) Let X , Y be subcategories of mod R . If Y is closed under Matlis duals, then T n ( X , Y ) = E n ( X , Y ) . (2) There is an equality T n (mod R ) = E n (mod R ) .Proof. Letting d = t = 0 in Proposition 4.2(1) yields the assertion. (cid:4) To state our theorem below, we recall the definition of the non-Gorenstein locus of R . Definition 4.5.
We denote by NonGor( R ) the non-Gorenstein locus of R , that is, the set of prime ideals p of R such that the local ring R p is non-Gorenstein. If R is a Cohen–Macaulay ring with a canonicalmodule ω , it holds that NonGor( R ) = NF( ω ) = V(tr ω ) (see [7, Lemma 2.1]).Now we can state and prove the theorem below. The second assertion is nothing but Theorem 1.1(2). Theorem 4.6.
Let R be a d -dimensional Cohen–Macaulay local ring with a canonical module ω . Let n ≥ and ≤ t ≤ d be integers. (1) Let X be a subcategory of CM t ( R ) . Let Y be a subcategory of CM( R ) closed under canonical duals.Assume either that Y is closed under syzygies or that X is closed under cosyzygies. Then (tr ω ) d · ( T n ( X , Y )) t +1 ⊆ E n ( X , Y ) , ( E n ( X , Y )) t +1 ⊆ T n ( X , Y ) . (2) If dim NonGor( R ) ≤ t and CM ( R ) ⊆ X ⊆ CM t ( R ) , then p T n ( X , CM t ( R )) = p E n ( X , CM t ( R )) .Proof. (1) First of all, since Y is closed under canonical duals, we have Y = Y † .The following argument deduces the first inclusion in the assertion.(tr ω ) d · ( T n ( X , Y )) t +1 (a) ⊆ (tr ω ) d · E d + n ( X , Y ) (b) ⊆ ( (tr ω ) d · E d + n ( X , Ω d Y ) if Y is closed under syzygies , (tr ω ) d · E d + n ( ✵ d X , Y ) if X is closed under cosyzygies (c) ⊆ E n ( X , Y ) . Here are the reasons why (a)–(c) hold. (a): Applying Proposition 4.2(1), we have an inclusion( T n ( X , Y )) t +1 ⊆ E d + n ( X , Y ). (b): If Y (resp. X ) is closed under syzygies (resp. cosyzygies), thenΩ d Y (resp. ✵ d X ) is contained in Y (resp. X ), and hence E d + n ( X , Y ) is contained in E d + n ( X , Ω d Y )(resp. E d + n ( ✵ d X , Y )). (c): Use Proposition 4.2(2).Next we show the second inclusion in the assertion. Since d − t ≥
0, the ideal E n ( X , Y ) is containedin the ideal E d + n − t ( X , Y ), whose ( t + 1)st power is contained in T n ( X , Y ) by Proposition 4.2(1).(2) When d = 0, we have t = 0 and X = CM t ( R ) = mod R . By Corollary 4.4(2) the assertion holds.Hence we assume d >
0, so that we get equalities V((tr ω ) d ) = V(tr ω ) = NonGor( R ). These two equalities can also be deduced from the inclusions given in the first part of Proposition 4.2(1). In fact, letting t = 0 yields T n ( X , Y ) ⊆ E d + n ( X , Y † ) and E d + n ( X , Y ) ⊆ T n ( X , Y † ). Then replace Y with Y † . SOUVIK DEY AND RYO TAKAHASHI
We claim that for each maximal Cohen–Macaulay R -module M there is an inclusionNF( M † ) ⊆ NF( M ) ∪ NonGor( R ) . Indeed, if p is a prime ideal of R which does not belong to NF( M ) ∪ NonGor( R ), then both M p and ω p are R p -free, and so is ( M † ) p .Combining this claim with the assumption dim NonGor( R ) ≤ t , we observe that CM t ( R ) is closedunder canonical duals. The subcategory CM t ( R ) is always closed under syzygies. Applying (1) to X and Y := CM t ( R ), we get the first line below, which yields the second.(tr ω ) d · ( T n ( X , CM t ( R ))) t +1 ⊆ E n ( X , CM t ( R )) , ( E n ( X , CM t ( R ))) t +1 ⊆ T n ( X , CM t ( R )) . V( E n ( X , CM t ( R ))) ⊆ NonGor( R ) ∪ V( T n ( X , CM t ( R ))) , V( T n ( X , CM t ( R ))) ⊆ V( E n ( X , CM t ( R ))) . There are inclusions NonGor( R ) ⊆ Sing R ⊆ V( T n (CM ( R ))) ⊆ V( T n ( X , CM t ( R ))), where the secondone follows from Lemma 3.10(3). Hence V( E n ( X , CM t ( R ))) is contained in V( T n ( X , CM t ( R ))). Now weobtain the equality V( E n ( X , CM t ( R ))) = V( T n ( X , CM t ( R ))), which completes the proof. (cid:4) A natural question arises.
Question 4.7.
Let R be a Cohen–Macaulay local ring. Does the following equality always hold?(4.7.1) T (CM ( R )) = E (CM ( R )) . Theorem 4.6(2) and Corollaries 3.6, 4.4 guarantee that (4.7.1) holds if R is either Gorenstein orartinian, and holds up to radicals when R is locally Gorenstein on the punctured spectrum. Corollary 4.3says that, when R is locally Gorenstein on the punctured spectrum, (4.7.1) is equivalent to the equality E (CM ( R )) = E d (CM ( R )), where d = dim R .We denote by Sing R the singular locus of R , that is, the set of prime ideals p of R such that R p issingular. Note that R has an isolated singularity if and only if dim Sing R ≤ ⇒ (a) in [6, Theorem 1.1] remains valid for a local Cohen–Macaulay ringadmitting a canonical module even if we replace E (CM ( R )) with E n (CM ( R )). For the undefinedterminology in the following proposition, we refer the reader to [5, 6]. Proposition 4.8. (1)
Let R be a d -dimensional excellent equicharacteristic local ring. (a) The subcategory Ω d (mod R ) of mod R has finite size and radius. (b) Suppose that R is Cohen–Macaulay and admits a canonical module. Then the subcategory CM( R ) of mod R has finite rank, size, dimension and radius. (2) Let ( R, m , k ) be a d -dimensional local Cohen–Macaulay ring admitting a canonical module. Let n ≥ be an integer. Suppose that the ideal E n (CM ( R )) of R contains some power of m . Then R has anisolated singularity, and CM ( R ) = CM( R ) has finite rank, size, dimension and radius.Proof. (1a) We modify the proof of [5, Theorem 5.7]. Replace “ d ”, “ d − R M ” with “ 2 d ”,“ 2( d −
1) ” and “ Ω R M ” respectively. Then it proves the assertion. Indeed, the assumption of [5, Theorem5.7] that R is complete and has perfect coefficient field is used only to apply [9, Corollary 5.15] to findan ideal J of R which satisfies Sing R = V( J ) and annihilates Ext d +1 R ( M, N ) for all R -modules M, N (inthe case where R is a singular domain with d > J ′ of R which satisfies Sing R = V( J ′ ) and annhilates Ext d +1 R ( M, N ) for all R -modules M, N .Here is the flow of the proof. We use induction on d , and the case d = 0 follows by the original argument.Let d >
0. As in the original argument, we may assume that R is a singular domain. Take J ′ as above,and find an element 0 = x ∈ J ′ . Put N = Ω dR M . Then x is N -regular. As in the original argument, N is a direct summand of Ω R ( N/xN ), and
N/xN ∼ = Ω d − R/xR (Ω R M/x Ω R M ) ∈ Ω d − R/xR (mod
R/xR ).(1b) We modify the proof of [5, Corollary 5.9]. Replace “ d ” with “ 2 d ” in it, and apply (1a) instead of[5, Theorem 5.7]. Then we observe that [5, Corollary 5.9] remains valid for any excellent equicharacteristiclocal ring. Combining this with [5, Proposition 5.10] deduces the assertion.(2) Using Lemma 3.10(3), we have Sing R ⊆ V( E n (CM ( R ))) ⊆ { m } . This particularly says that R has an isolated singularity, and hence CM ( R ) = CM( R ). Similarly as in the proof of [6, Proposition6.1(2a)], we observe that there exists an integer r > n (CM( R )) ⊆ | Ω d k | r . An analogous OMPARISONS BETWEEN ANNIHILATORS OF TOR AND EXT 9 argument as in the proof of [5, Corollary 5.9] yields that CM( R ) ⊆ | Ω d k ⊕ W | r ( n +1) for some W ∈ CM( R ).We obtain CM( R ) = | Ω d k ⊕ W | r ( n +1) , which shows that CM( R ) has rank less than r ( n + 1). (cid:4) Remark 4.9.
Proposition 4.8(1a) refines the latter statement of [8, Theorem 5.3], which asserts that thesubcategory Ω d (mod R ) of mod R has finite size.As an application of Theorem 4.6, one can refine the main results of [6]. More precisely, it is assertedin [6, Theorem 1.1 and Corollary 7.2] that among the four conditions (a)–(d) given in the corollary below,(A) the implication (a) ⇒ (d) holds,(B) the implications (a) ⇔ (b) ⇒ (c) ⇒ (d) hold for t = 0, and(C) the equivalences (a) ⇔ (b) ⇔ (c) ⇔ (d) hold for t = 0 provided that R is complete and has perfectcoefficient field.When R admits a canonical module, by using Theorem 4.6(2) we can improve the above statements (A),(B), (C) as follows. Corollary 4.10.
Let R be a d -dimensional Cohen–Macaulay local ring. Suppose that R admits a canon-ical module. Let n ≥ and ≤ t ≤ d be integers. Consider the following conditions. (a) dim CM t ( R ) < ∞ . (b) dim V( E n (CM t ( R ))) ≤ t. (c) dim V( T n (CM t ( R ))) ≤ t. (d) dim Sing R ≤ t. (1) The implications (a) ⇒ (b) ⇒ (d) and (a) ⇒ (c) ⇒ (d) hold. (2) The equivalence (b) ⇔ (c) holds. (3) The implications (a) ⇔ (b) ⇔ (c) ⇒ (d) hold when t = 0 . (4) The equivalences (a) ⇔ (b) ⇔ (c) ⇔ (d) hold when R is excellent and equicharacteristic.Proof. (1) We begin with proving that (a) implies both (b) and (c). Note that the sets V( E n (CM t ( R )))and V( T n (CM t ( R ))) are contained in V( E (CM t ( R ))) and V( T (CM t ( R ))), respectively. It suffices toshow that V( E (CM t ( R ))) and V( T (CM t ( R ))) have dimension at most t . We make a similar argumentas in the proof of [6, Proposition 6.1(1a)]. Write CM t ( R ) = [ G ] r with G ∈ CM t ( R ) and r >
0. We haveV( T (CM t ( R ))) = V( T ( G, CM t ( R ))) ⊆ V( T ( G, mod R )) = NF( G ) , V( E (CM t ( R ))) = V( E ( G, CM t ( R ))) ⊆ V( E ( G, mod R )) = NF( G )by [6, Lemma 5.3(1) and Proposition 5.1(1)]. It follows that the dimensions of V( T (CM t ( R ))) andV( E (CM t ( R ))) are at most the dimension of NF( G ), which is at most t since G ∈ CM t ( R ).The assertion that each of the conditions (b) and (c) implies (d) follows from the inclusions Sing R ⊆ V( T n (CM ( R ))) ∩ V( E n (CM ( R ))) ⊆ V( T n (CM t ( R ))) ∩ V( E n (CM t ( R ))) by Lemma 3.10(3).(2) Suppose that (b) (resp. (c)) holds. Then it follows from (1) that (d) holds, i.e., dim Sing R ≤ t .Since NonGor( R ) ⊆ Sing R , we get dim NonGor( R ) ≤ t . Hence (c) (resp. (b)) holds by Theorem 4.6(2).(3) For t = 0, Proposition 4.8(2) shows (b) implies (a). The rest implications follow from (1) and (2).(4) Suppose that R is excellent and equicharacteristic. Then Proposition 4.8(1b) shows dim CM( R ) < ∞ . If (d) holds, then CM t ( R ) = CM( R ) and (a) follows. Combining this with (1) completes the proof. (cid:4) Remark 4.11. (1) In view of (3) of Corollary 4.10, it is natural to ask whether (b) implies (a) for t > t = d >
0. Indeed, in this case, (b) automaticallyholds and CM t ( R ) = CM( R ). We do not know in general whether CM( R ) has finite dimension when R is not equicharacteristic, even if we assume that R is complete and has an isolated singularity.(2) Making an analogous argument as in its proof, one actually obtains a more general statement thanCorollary 4.10(1):Let X , Y be subcategories of mod R . Suppose that X has finite dimension and is containedin mod t ( R ). Then V( T ( X , Y )) and V( E ( X , Y )) have dimension at most t .This is a generalization of [6, Proposition 6.1(1a)]; letting t = 0 recovers it.(3) Let ( R, m ) be a local ring not having an isolated singularity, and let n = 0 and X = Y = CM ( R ).Then no nonmaximal prime ideal of R belongs to the supports of the modules T n ( X , Y ) and E n ( X , Y ),while neither T n ( X , Y ) nor E n ( X , Y ) is m -primary by Corollary 4.10(1). Hence we get strict inclusionsSupp R ( T n ( X , Y )) ( V( T n ( X , Y )) , Supp R ( E n ( X , Y )) ( V( E n ( X , Y )) . References [1]
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Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
Email address : [email protected] Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
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