Exterior powers and Tor-persistence
aa r X i v : . [ m a t h . A C ] A ug EXTERIOR POWERS AND TOR-PERSISTENCE
JUSTIN LYLE, JONATHAN MONTA ˜NO ∗ , AND SEAN K. SATHER-WAGSTAFFA BSTRACT . A commutative Noetherian ring R is said to be Tor-persistent if, for anyfinitely generated R -module M , the vanishing of Tor Ri ( M , M ) for i ≫ M hasfinite projective dimension. An open question of Avramov, et. al. asks whether any such R is Tor-persistent. In this work, we exploit properties of exterior powers of modules andcomplexes to provide several partial answers to this question; in particular, we show thatevery local ring ( R , m ) with m =
1. I
NTRODUCTION
Several conjectures and open questions on the rigidity of Ext and Tor have recentlygained much attention. Among the most well-known of these is the Auslander-Reitenconjecture which poses that the vanishing of Ext iR ( M , M ⊕ R ) for all i > M tobe projective [1]. The Auslander-Reiten conjecture traces its roots to the representationtheory of Artin algebras where it is intimately connected to Nakayama’s conjecture and itsgeneralized version. A significant special case of the Auslander-Reiten conjecture is theTachikawa conjecture which posits that the Auslander-Reiten conjecture holds when R isCohen-Macaulay and M = ω R is a canonical module of R [4]. Inspired by work of S¸ ega[16], Avramov et. al. introduce the following which can be thought of as a version of theAuslander-Reiten conjecture for Tor. Question 1.1 ([5]) . Let R be a commutative Noetherian ring. If, for a finitely generated R -module M , we have Tor Ri ( M , M ) = i ≫
0, must M have finite projective dimension?Rings for which Question 1.1 has an affirmative answer are called Tor -persistent . ThusQuestion 1.1 can be rephrased to ask whether every commutative Noetherian ring is Tor-persistent. Several classes of rings are known to be Tor-persistent, for example, completeintersection rings, Golod rings, and rings of small embedding codimension or multiplicity[2, 5, 13, 14]. The complete intersection case depends on support theory, which is onlyavailable in this setting, while the other known results depend on conditions for the van-ishing of Tor Ri ( M , N ) for all i ≫ M and N , an approach that does not extend tothe general case (see [5]).The main purpose of this work is to provide evidence, and new insights, that this ques-tion may have an affirmative answer. Our first main result provides a new class of ringswhich are Tor-persistent (see Theorem 2.1). Theorem A. If ( R , m ) is a local ring with m = , then R is Tor-persistent. Mathematics Subject Classification.
Key words and phrases.
Exterior squares, Tachikawa’s Conjecture, Tor-persistence. ∗ The second author is supported by NSF Grant DMS ur second main result provides some restrictions in the graded case; here e R ( M ) de-notes the Hilbert-Samuel mutiplicity of the R -module M . Theorem B.
Let R be a Noetherian positively graded k-algebra over the field k with char k = . Let M be a finitely generated graded R-module satisfying the following: ( ) e R ( M ) = e R ( R ) , ( ) Tor Ri ( M , M ) = for i > , and ( ) M ⊗ R M has no embedded primes.Then M ∼ = R. As a consequence of Theorem B, we provide a commutative algebra proof of the Tachi-kawa Conjecture for positively graded rings over a field of characteristic different from2 (see Corollary 3.6). This result also follows from work of Zeng using techniques inrepresentation theory of Artin algebras [17] .Our approach to both of our main results provides an explanation as to why the van-ishing of Tor Ri ( M , N ) is special when M = N ; namely, the vanishing of Tor Ri ( M , M ) hasconsequences for the exterior and symmetric powers of M . For the cases we consider,these consequences come in the form of numerical constraints on the exterior and symmet-ric squares, and are enough to conclude that the module in question is free.We conclude this section with some notation that we use in the subsequent ones. Throughthe remainder of the paper, let ( R , m , k ) be a commutative Noetherian ring which is eitherlocal or positively graded over the field k with maximal homogeneous ideal m . If R has acanonical module, it is denoted by ω R . We let codim R : = µ R ( m ) − dim R be the embed-ding codimension of R . We let M be a finitely generated R -module; in the graded casewe assume M is homogeneous. We write Ω Ri ( M ) for the i th syzygy of an R -module M ,and β Ri ( M ) for the i th Betti number. We use µ R ( M ) = β R ( M ) for its minimal number ofgenerators and l R ( M ) for the length of M . We let ι M : V R ( M ) → M ⊗ R M be the antisym-metrization map defined on elementary wedges by x V x x ⊗ x − x ⊗ x .2. T OR -P ERSISTENCE FOR R INGS WITH R ADICAL C UBE Z ERO
The following main result contains Theorem A from the introduction.
Theorem 2.1.
Assume ( R , m , k ) is a local ring with m = . If M is an R-module such that Tor Ri ( M , M ) = for i , then M is free.Proof. As a notational convenience, we set γ R ( M ) = l R ( M ) µ R ( M ) −
1. We note that γ R ( M ) > M ∼ = k n for some n .Suppose M is not free. Set N = Ω R ( M ) , L = Ω R ( M ) , and b = µ R ( N ) . Let ϕ be themap fitting in the natural exact sequence 0 → L ϕ −→ R b → N →
0. Since N ֒ → m R µ R ( M ) andsince m =
0, we have m N =
0. Similarly, we have m L =
0. By dimension shifting,Tor Ri ( N , L ) = i = ,
2, and so we have m ( L ⊗ R L ) = Ri ( N , N ) = i = , , µ R ( L ) = γ R ( N ) b ,(2) µ R ( m ) = γ R ( N ) , and(3) r ( R ) = γ R ( N ) , where r ( R ) : = dim k Soc ( R ) is the type of R .The map ι L ⊗ R k : V R ( L ⊗ R k ) → ( L ⊗ R k ) ⊗ R ( L ⊗ R k ) is injective because L ⊗ R k is a k -vectorspace, and this map is naturally identified with ι L ⊗ id k : V R ( L ) ⊗ R k → ( L ⊗ R L ) ⊗ R k . As L ⊗ R L is a k -vector space, so is its quotient V R ( L ) , hence ι L ⊗ id k is naturally identifiedwith ι L . In particular, ι L is injective. ow, we have ϕ ⊗ ϕ = ( ϕ ⊗ id R b ) ◦ ( id L ⊗ ϕ ) . The map ϕ ⊗ id R b is injective sinceTor R ( N , R b ) =
0, and id L ⊗ R ϕ is injective since Tor R ( L , N ) = Tor R ( M , M ) =
0. Thus ϕ ⊗ ϕ is also injective.Next, we have the following commutative diagram V R ( L ) L ⊗ R L V R ( R b ) R b ⊗ R R b V R ( ϕ ) ι L ϕ ⊗ ϕι Rb Since ϕ ⊗ ϕ and ι L are both injective, the commutivity of the diagram forces V R ( ϕ ) to beinjective. Since V R ( L ) is a k -vector space, it must thus embed in the socle of V R ( R b ) .The vector space dimension of V R ( L ) is (cid:18) µ R ( L ) (cid:19) = (cid:18) γ R ( N ) b (cid:19) = γ R ( N ) b ( γ R ( N ) b − ) ( V R ( R b )) is r ( R ) (cid:18) b (cid:19) = γ R ( N ) (cid:18) b ( b − ) (cid:19) . It follows that we must have γ R ( N ) b ( γ R ( N ) b − ) γ R ( N ) (cid:18) b ( b − ) (cid:19) . If γ R ( N ) =
0, then N is a k -vector space which cannot be, since N has infinite projectivedimension and Tor R ( N , N ) =
0. Therefore, as b =
0, we have γ R ( N ) b − γ R ( N ) b − γ R ( N ) which forces γ R ( N ) =
1. Thus R is Gorenstein with µ R ( m ) =
2, by items (1)–(2) above,and so R is also a complete intersection.We recall that the complexity of M which is the numbercx R ( M ) : = inf { c ∈ N | ∃ γ ∈ R such that β Rn ( M ) γ n c − for c ≫ } . Since R is a complete intersection, we have cx R ( M ) codim R =
2; see e.g. [12, Theorem1.1 and subsequent paragraph]. Thus, [12, Proposition 2.3] forces Tor Ri ( M , M ) = i >
0. By [2, Theorem 4.2], this contradicts the fact that M is not free. The result follows. (cid:3)
3. S
OME R ESTRICTIONS FOR T OR - PERSISTENCE
Unless otherwise stated, throughout this section we let R = ⊕ i > R i be a positivelygraded algebra over a field R = k and m = ⊕ i > R i its homogeneous maximal ideal. Let M = ⊕ i ∈ Z M i be a finitely generated graded R -module. The Hilbert series of M is H M ( t ) = ∑ i ∈ Z ( dim k M i ) t i . We recall that if M =
0, then there exist positive integers a , . . . , a dim M and a Laurentpolynomial ε M ( t ) ∈ Z [ t , t − ] with ε M ( ) > H M ( t ) can be written as H M ( t ) = ε M ( t ) ∏ dim Mi = ( − t a i ) . oreover, the set { a , . . . , a dim M } is the same for all M of maximal dimension, i.e., dim M = dim R [6, Proposition 4.4.1, Remark 4.4.2]. We refer to ε M ( t ) as the multiplicity polynomial of M .We denote by e R ( M ) the Hilbert-Samuel multiplicity of Me R ( M ) = lim n → ∞ ( dim R ) ! l R ( M / m n M ) n dim R ;we also write e ( R ) for e R ( R ) . We note that e R ( M ) > M = dim R , and inthis case we have e R ( M ) = ε M ( ) .For a graded complex of finite rank graded free R -modules X = · · · X i + → X i → X i − → · · · with X i = ⊕ j ∈ Z R ( − j ) b i , j , we denote by P X ( t , z ) = ∑ i , j ∈ Z b i , j t j z i the (graded) Poincar´e series of X . If F is a graded free resolution of M , then we set P M ( t , z ) : = P F ( t , z ) . The additivity of length gives the following comparison of Hilbert andPoincar´e series. Fact 3.1.
For R and M as above, we have H M ( t ) = H R ( t ) P M ( t , − ) . We now describe a construction of Buchsbaum-Eisenbud [7], following the presentationof Frankild-Sather-Wagstaff-Taylor[9]. Assume for the remainder of this paragraph thatchar k =
2. Let X be as above, and let α X : X ⊗ R X → X ⊗ X be the map defined onhomogeneous generators by α X ( x ⊗ x ′ ) = x ⊗ x ′ − ( − ) | x || x ′ | x ′ ⊗ x . Let S R ( X ) be the complex Coker ( α X ) and call it the second symmetric power of X .In the following statement we summarize some important properties of S R ( X ) . Weremark that although the statements in [9] are in the local case, the arguments thereinreadily extend to account for the grading in R . Fact 3.2 ([9, 3.8, 4.1, 3.12]) . Assume char k =
2. Let X be a graded complex of finite rankgraded free R -modules.(1) The following exact sequences are split exact.0 → Ker ( α X ) → X ⊗ R X → Im ( α X ) → → Im ( α X ) → X ⊗ R X → S R ( X ) → H ( S R ( X )) ∼ = S R ( H ( X )) .(3) P S R ( X ) ( t , z ) = h P X ( t , z ) + P X ( t , − z ) i The following lemma is essential in the proof of our main result.
Lemma 3.3.
Assume char k = . If M is a graded R-module such that Tor Ri ( M , M ) = forevery i > , then we haveH S R ( M ) ( t ) = H M ( t ) H R ( t ) + H M ( t ) H R ( t ) H R ( t ) and H V R ( M ) ( t ) = H M ( t ) H R ( t ) − H R ( t ) H M ( t ) H R ( t ) . roof. Let F be a minimal graded free resolution of M . By the vanishing of Tor assump-tion, the complex F ⊗ R F is acyclic and therefore a minimal free resolution of M ⊗ R M .Therefore, Fact 3.2(1)–(2) imply that S R ( F ) is acyclic and a minimal free resolution of S R ( M ) . From Facts 3.1 and 3.2(3) we obtain H S R ( M ) ( t ) = H R ( t ) P S R ( F ) ( t , − ) = H R ( t ) h P F ( t , − ) + P F ( t , − ) i = H M ( t ) H R ( t ) + H R ( t ) H R ( t ) P F ( t , − ) H R ( t )= H M ( t ) H R ( t ) + H R ( t ) H M ( t ) H R ( t ) . We note that H M ⊗ R M ( t ) = H R ( t ) P F ⊗ R F ( t , − ) = H R ( t ) P F ( t , − ) = H M ( t ) H R ( t ) . Thus, it suffices to show H M ⊗ R M ( t ) = H S R ( M ) ( t ) + H V R ( M ) ( t ) . For this, we consider theantisymmetrization map ι M : V R ( M ) → M ⊗ R M defined in the introduction. This map isa split injection where the splitting map is given by x ⊗ y x V y . Since Coker ( ι M ) = S R ( M ) , we have M ⊗ R M ∼ = S R ( M ) ⊕ V R ( M ) , and the result follows. (cid:3) We are now ready to prove Theorem B.
Theorem 3.4.
Let R be a Noetherian positively graded k-algebra with char k = . Let Mbe a finitely generated graded R-module satisfying the following: ( ) e R ( M ) = e ( R ) , ( ) Tor Ri ( M , M ) = for i > , and ( ) M ⊗ R M has no embedded primes.Then M ∼ = R.Proof.
We proceed by contradiction. Suppose that V R ( M ) = M ⊗ R M ∼ = V R ( M ) ⊕ S R ( M ) . Since M ⊗ R M has no embedded primes,it follows that V R ( M ) has maximal dimension. By Lemma 3.3, we have H R ( t ) H R ( t ) H V R ( M ) ( t ) = H M ( t ) H R ( t ) − H M ( t ) H R ( t ) . As each module in question has maximal dimension, we may clear denominators to obtaina formula for multiplicity polynomials ε R ( t ) ε R ( t ) ε V R ( M ) ( t ) = ε M ( t ) ε R ( t ) − ε M ( t ) ε R ( t ) . Evaluating these at t = ε V R ( M ) ( ) =
0, a contradiction. Therefore, V R ( M ) = M is cyclic. Thus M ∼ = R / I for some homogeneous ideal I . As I / I ∼ = Tor R ( R / I , R / I ) ∼ = Tor R ( M , M ) =
0, it follows that I =
0, concluding the proof. (cid:3)
In what follows, we set ( − ) ∨ = Hom R ( − , ω R ) Corollary 3.5.
Let R be a positively graded Cohen-Macaulay k-algebra with char k = .If M is a finitely generated graded maximal Cohen-Macaulay R-module such that: ( ) e R ( M ) = e ( R ) , and ( ) Ext iR ( M , M ∨ ) = for i > .Then M ∼ = R. roof. From [14, Lemma 3.4 (1)], we have that M ⊗ R M is maximal Cohen-Macaulay andthat Tor Ri ( M , M ) = i >
0. The result then follows from Theorem 3.4. (cid:3)
As an immediate consequence of Corollary 3.5, we prove the (commutative) graded caseof the Tachikawa conjecture in charactersitic different from 2, which also follows fromwork of Zeng. Notably, Zeng’s approach requires passing to noncommutative algebras,whereas our proof uses only techniques in commutative algebra.
Corollary 3.6 ([17, Theorem 1.3]) . Let R be a positively graded Cohen-Macaulay k-algebra with char k = . If Ext iR ( ω R , R ) = for every i > , then R is Gorenstein. For Artinian rings, condition (3) of Theorem 3.4 is automatically satisfied, so we obtainthe following.
Corollary 3.7.
Let R be an Artinian positively graded k-algebra with char k = . If M is afinitely generated graded R-module such that l R ( M ) = l ( R ) and Tor Ri ( M , M ) = for everyi > , then M ∼ = R. Next we observe that our techniques also apply to a generalization of Tachikawa’sConjecture that is due to Frankild-Sather-Wagstaff-Taylor [8, Question 1.2, cf. Proposi-tion 7.5]. For this, recall that an R -module C is semidualizing provided that Ext iR ( C , C ) = i > R ( C , C ) ∼ = R . For example, R is semidualizing over R . Corollary 3.8.
Let R be a Cohen-Macaulay positively graded k-algebra with char k = and let C be a graded semidualizing R-module. ( ) If Ext iR ( C , C ∨ ) = for i > , then C ∼ = R. ( ) If Ext iR ( C ∨ , C ) = for i > , then C ∼ = ω R .Proof. First we note that C is maximal Cohen-Macaulay [15, Proposition 2.1.16(b), Theo-rem 2.2.6(c)]. Since Ext iR ( C , C ) = i >
0, [3, Theorem 1] implies H R ( t − ) H R ( t ) = H C ( t − ) H C ( t ) . Similar to the proof of Theorem 3.4, we clear denominators to get an equality of multiplic-ity polynomials ε R ( t − ) ε R ( t ) = ε C ( t − ) ε C ( t ) . Evaluating at t = e ( R ) = e R ( C ) from which we obtain e R ( C ) = e ( R ) . Both ( ) and ( ) now follow from Corollary 3.5. (cid:3) We close this section with versions of Theorem 3.4 and Corollary 3.5 for the local case.
Proposition 3.9.
Let ( R , m , k ) be a Noetherian local ring with char k = . Assume M is afinitely generated R-module satisfying the following: ( ) M has rank , and ( ) M ⊗ R M has no embedded primes.Then M is cyclic. If, in addition,
Tor R ( M , M ) = , then M ∼ = R.Proof.
As in the proof of Lemma 3.3, we have M ⊗ R M ∼ = V R ( M ) ⊕ S R ( M ) . Since M ⊗ R M has no embedded primes, it follows that V R ( M ) has maximal dimension if it is not 0. Onthe other hand rank V R ( M ) = (cid:18) rank M (cid:19) =
0. It follows that V R ( M ) = M is cyclic.If, in addition, Tor R ( M , M ) =
0, then, writing M ∼ = R / I , we have I / I ∼ = Tor R ( R / I , R / I ) =
0, which forces I = (cid:3) he following corollary provides an extension of the generically Gorenstein case ofthe Tachikawa conjecture proved independently by Avramov-Buchweitz-S¸ega [4, 2.1] andHanes-Huneke [10, Corollary 2.2]. Corollary 3.10.
Let ( R , m , k ) be a Cohen-Macaulay local ring with canonical module ω R and with char k = . Let M be a finitely generated maximal Cohen-Macaulay R-modulesatisfying the following:(1) M has rank , and(2) Ext iR ( M , M ∨ ) = for i dim R.Then M is cyclic. If, in addition,
Ext dim R + R ( M , M ∨ ) = , then M ∼ = R.Proof.
By [14, Lemma 3.4 (1)], we have we have that M ⊗ R M is maximal Cohen-Macaulay.That M is cyclic then follows immediately from Proposition 3.9.If, in addition, we have Ext dim R + R ( M , M ∨ ) , then [14, Lemma 3.4 (2)] implies thatTor R ( M , M ) =
0, and Proposition 3.9 again gives the result. (cid:3) A CKNOWLEDGMENTS
We would like to thank Luchezar Avramov and Srikanth Iyengar for helpful discussionson Tor-persistence. We thank Ryo Takahashi for helpful feedback, in particular for pointingout [17] to us. This work began during the Thematic Program in Commutative Algebra andits Interaction with Algebraic Geometry at the University of Notre Dame. We thank theorganizers for a lovely conference. R
EFERENCES[1] Maurice Auslander and Idun Reiten. On a generalized version of the nakayama conjecture.
Proceedings ofthe American Mathematical Society , 52(1):69–74, 1975.[2] Luchezar L. Avramov and Ragnar-Olaf Buchweitz. Support varieties and cohomology over complete inter-sections.
Invent. Math. , 142(2):285–318, 2000.[3] Luchezar L Avramov, Ragnar-Olaf Buchweitz, and Judith D Sally. Laurent coefficients and ext of finitegraded modules.
Mathematische Annalen , 307(3):401–415, 1997.[4] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, and Liana M. S¸ega. Extensions of a dualizing complex byits ring: commutative versions of a conjecture of Tachikawa.
J. Pure Appl. Algebra , 201(1-3):218–239,2005.[5] Luchezar L. Avramov, Srikanth B. Iyengar, Saeed Nasseh, and Sean K. Sather-Wagstaff. Persistence ofhomology over commutative noetherian rings. preprint (2020), arxiv:2005.10808 .[6] Winfried Bruns and J¨urgen Herzog.
Cohen-Macaulay rings , volume 39 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 1993.[7] David A. Buchsbaum and David Eisenbud. Algebra structures for finite free resolutions, and some structuretheorems for ideals of codimension 3.
Amer. J. Math. , 99(3):447–485, 1977.[8] Anders J. Frankild, Sean Sather-Wagstaff, and Amelia Taylor. Relations between semidualizing complexes.
J. Commut. Algebra , 1(3):393–436, 2009.[9] Anders J. Frankild, Sean Sather-Wagstaff, and Amelia Taylor. Second symmetric powers of chain com-plexes.
B. Iran. Math. Soc. , 37:39–75, 2011.[10] Douglas Hanes and Craig Huneke. Some criteria for the Gorenstein property.
J. Pure Appl. Algebra , 201(1-3):4–16, 2005.[11] Craig Huneke, Liana M. S¸ega, and Adela N. Vraciu. Vanishing of Ext and Tor over some Cohen-Macaulaylocal rings.
Illinois J. Math. , 48(1):295–317, 2004.[12] David A. Jorgensen. Complexity and Tor on a complete intersection.
J. Algebra , 211(2):578–598, 1999.[13] David A. Jorgensen. A generalization of the Auslander-Buchsbaum formula.
J. Pure Appl. Algebra ,144(2):145–155, 1999.[14] Justin Lyle and Jonathan Monta˜no. Extremal growth of betti numbers and trivial vanishing of (co)homology.
Transactions of the American Mathematical Society , 2020. to appear.[15] S. Sather-Wagstaff. Semidualizing modules. in preparation https://ssather.people.clemson.edu/DOCS/sdm.pdf .
16] Liana M. S¸ega. Self-tests for freeness over commutative Artinian rings.
J. Pure Appl. Algebra , 215(6):1263–1269, 2011.[17] Qiang Zeng. Vanishing of Hochschild’s cohomologies H i ( A ⊗ A ) and gradability of a local commutativealgebra A . Tsukuba J. Math. , 14(2):263–273, 1990.J
USTIN L YLE , D
EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF K ANSAS , 405 S
NOW H ALL , 1460J
AYHAWK B LVD ., L
AWRENCE , KS 66045
E-mail address : [email protected] URL : http://people.ku.edu/~j830l811/ J ONATHAN M ONTA ˜ NO , D EPARTMENT OF M ATHEMATICAL S CIENCES , N EW M EXICO S TATE U NIVER - SITY , PO B OX AS C RUCES , NM 88003-8001
E-mail address : [email protected] URL : https://web.nmsu.edu/~jmon/ S EAN
K. S
ATHER -W AGSTAFF , S
CHOOL OF M ATHEMATICAL AND S TATISTICAL S CIENCES , C
LEMSON U NIVERSITY , O-110 M
ARTIN H ALL , B OX LEMSON , S.C. 29634 USA
E-mail address : [email protected] URL : https://ssather.people.clemson.edu/https://ssather.people.clemson.edu/