The eventual shape of Betti tables of powers of ideals
aa r X i v : . [ m a t h . A C ] J u l THE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS
AMIR BAGHERI, MARC CHARDIN, AND HUY T `AI H `A
Abstract.
Let G be an abelian group and S be a G -graded a Noetherian algebra over acommutative ring A ⊆ S . Let I , . . . , I s be G -homogeneous ideals in S , and let M be afinitely generated G -graded S -module. We show that the shape of nonzero G -graded Bettinumbers of M I t . . . I t s s exhibit an eventual linear behavior as the t i s get large. Introduction
It is a celebrated result (cf. [5, 12, 15]) that if I ⊆ S is a homogeneous ideal in aNoetherian standard N -graded algebra and M is a finitely generated Z -graded S -modulethen the regularity reg( I t M ) is asymptotically a linear function for t ≫
0. This asymptoticlinear function and the stabilization index have also been studied in [1, 4, 7, 8, 9].In the case S is a polynomial ring over a field, a more precise result is proved in [5]: themaximal degree of an i -th syzygy of I t M is eventually a linear function of t . Our interesthere is to understand the eventual behavior of the degrees of all the minimal generators ofthe i -th syzygy module. This is of particular interest when the grading is given by somefinitely generated abelian group G that is not Z , as in this case the result for the regularityof powers do not have an evident analogue. One can in particular consider the Cox ring ofa toric variety, graded by the divisor class group.We shall actually show, in the G -graded setting, that the collection of nonzero G -gradedBetti numbers of M I t . . . I t s s exhibits an asymptotic linear behavior as the t i s get large.Let us also point out that, even when an explicit minimal free resolution of these powers isknown, for instance when I is a complete intersection ideal, the degrees of i -th syzygies of I t do not exhibit trivially a linear behavior.Throughout the paper, let G be a finitely generated abelian group and let S = A [ x , . . . , x n ]be a G -graded algebra over a commutative ring A ⊆ S . Hence A = S/ ( x , . . . , x n ) is a G -graded S -module concentrated in degree 0.Our work hinges on the relationship between multigraded Betti numbers and the gradedpieces of Tor Si ( M I t . . . I t s s , A ); we, thus, examine the support of Tor Si ( M I t · · · I t s s , A ) as the t i s get large. Our main result, in the case of a single ideal, gives the following: Mathematics Subject Classification.
Key words and phrases.
Betti numbers, asymptotic linearity, multigraded.
Theorem 1.1 (see Theorem 4.6) . Let I = ( f , . . . , f r ) be a G -homogeneous S -ideal, and let M be a finitely generated G -graded S -module. Set Γ := { deg G ( f i ) } ri =1 .Let ℓ ≥ , and assume that ℓ = 0 or A is Noetherian.There exist a finite collection of elements δ i ∈ G , a finite collection of integers t i , anda finite collection of non-empty tuples E i = ( γ i, , . . . , γ i,s i ) of elements in Γ , such that theelements ( γ i,j +1 − γ i,j , ≤ j ≤ s i − are linearly independent, satisfying Supp G (Tor Sℓ ( M I t , A )) = m [ i =1 (cid:0) [ c ··· + csi = t − ti,c ,...,csi ∈ N c γ i, + · · · + c s i γ i,s i + δ i (cid:1) , ∀ t ≥ max i { t i } . The condition of linear independence stated for E i implies that c γ i, + · · · + c s i γ i,s i = c ′ γ i, + · · · + c ′ s i γ i,s i if c + · · · + c s i = c ′ + · · · + c ′ s i and ( c , . . . , c s i ) = ( c ′ , . . . , c ′ s i ). Noticethe important fact that the elements in E i are all in Γ, the set of degrees of generators of I .Theorem 4.6 is proved in the last section of the paper. Our proof is based on the twofollowing observations. Firstly, the multi-Rees module (sometimes also referred to as theRees modification) M R = L t i ≥ M I t . . . I t s s is a G × Z s -graded module over the multi-Rees algebra R = L t i ≥ I t . . . I t s s . Secondly, if G ′ is a finitely generated abelian group and R = S [ T , . . . , T r ] is a G × G ′ -graded polynomial extension of S , such that deg G × G ′ ( a ) =(deg G ( a ) ,
0) for any a ∈ S , then for a graded complex G • of free R -modules, H i (cid:0) ( G • ) G ×{ δ } ⊗ S A (cid:1) = H i (cid:0) G • ⊗ S A (cid:1) G ×{ δ } , where ( • ) G ×{ δ } denotes the degree G × { δ } -strand of the corresponding complex. In partic-ular, if G • is a G × G ′ -graded free resolution of an R -module N , as ( G • ) G ×{ δ } is a G -gradedfree resolution of the S -module N G ×{ δ } , it follows thatTor Si ( N G ×{ δ } , A ) = H i ( G • ⊗ S A ) G ×{ δ } , in which G • ⊗ S A is viewed as a G × G ′ -graded complex of free A [ T , . . . , T r ]-modules. Theseobservations allow us to bring the problem to studying the support of graded A [ T , . . . , T r ]-modules. We proceed by making use of the notion of initial submodules to reduce to the casewhen the module is a quotient ring obtained by a monomial ideal. The result then followsby considering the Stanley decomposition of such a quotient ring.In its full generality, our proof of Theorem 4.6 is quite technical, so we start in Section3 by considering first the case when each I i is equi-generated. In fact, in this case, withthe additional assumption that A is a Noetherian ring, our results are much stronger; theasymptotic linearity appears clearer and the proofs are simpler. We can also examine thesupport of local cohomology modules of M I t . . . I t s s . Our results, in the case when each I i is finitely generated in a single degree γ i , are stated as follows. Theorem 1.2 (Theorem 3.3) . Assume that i = 0 or A is Noetherian. Then there exists afinite set ∆ i ⊆ G such that HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 3 (1) For all t = ( t , . . . , t s ) ∈ N s , Tor Si ( M I t · · · I t s s , A ) η = 0 if η ∆ i + t γ + · · · + t s γ s .(2) There exists a subset ∆ ′ i ⊆ ∆ i such that Tor Si ( M I t · · · I t s s , A ) η + t γ + ··· + t s γ s = 0 for t ≫ and η ∈ ∆ ′ i , and Tor Si ( M I t · · · I t s s , A ) η + t γ + ··· + t s γ s = 0 for t ≫ and η ∆ ′ i .(3) Let A → k be a ring homomorphism to a field k . Then for any δ and any j , thefunction dim k Tor Aj (Tor Si ( M I t · · · I t s s , A ) δ + t γ + ··· + t s γ s , k ) is polynomial in the t i s for t ≫ , and the function dim k Tor Si ( M I t · · · I t s s , k ) δ + t γ + ··· + t s γ s is polynomial in the t i s for t ≫ . Theorem 1.3 (Theorem 3.4) . Let b be a G -homogeneous ideal in S such that for any i ≥ and δ ∈ G , H i b ( S ) δ is a finitely generated A -module. Then, if A is Noetherian, for i ≥ ,there exists a subset Λ i ⊆ G such that(1) H i b ( M I t · · · I t s s ) η + t γ + ··· + t s γ s = 0 for t = ( t , . . . , t s ) ≫ if only if η ∈ Λ i .(2) Let A → k be a ring homomorphism to a field k . Then for any δ and any j , thefunction dim k Tor Aj ( H i b ( M I t · · · I t s s ) δ + t γ + ··· + t s γ s , k ) is a polynomial in the t i s for t ≫ . In the simplest scenario, when S is a standard graded polynomial ring over a field k , m ⊆ S the homogeneous maximal S -ideal, s = 1, and I ⊆ S is a homogeneous ideal generated indegree d , Theorems 1.2 and 1.3 give the following result. Corollary 1.4.
For i ≥ , there exist t i and finite sets ∆ ′ i ⊆ ∆ i ⊆ Z such that(1) for all t ∈ N , Tor Si ( M I t , k ) η + td = 0 if η ∆ i ;(2) for t ≥ t i , Tor Si ( M I t , k ) η + td = 0 if and only if η ∈ ∆ ′ i ;(3) for any η ∈ Z , the function dim k Tor Si ( M I t , k ) η + td is a polynomial in t for t ≥ t i ;(4) for any θ ∈ Z , the function dim k H i m ( M I t ) θ + td is a polynomial in t , for t ≫ . While writing this paper, we were informed by Whieldon that in her recent work [16],a similar result to Corollary 1.4 (1)–(3) is proved. We later learned that Pooja Singla alsoproved these results, in the second chapter of her thesis [14], independently. She also shows in[14] that if I is graded ideal, then for any a, b ∈ Z , dim k Tor Si ( I t , k ) a + bt is a quasi-polynomialin t for t ≫
0, and give results describing the regularity of I t · · · I t s s for t ≫ A is not necessarily a field and S is not necessarily standardgraded, the problem is more subtle, and requires more work and a different approach.We choose not to restrict to a Noetherian base ring in all our results as it seemed tous that it makes the presentation more clear to put this hypothesis only in the statementswhere it is of use in the proof. AMIR BAGHERI, MARC CHARDIN, AND HUY T `AI H `A
Acknowledgement.
Our work was done when the third named author visited the otherauthors at the Institut de Math´ematiques de Jussieu, UPMC. The authors would like tothank the Institut for its support and hospitality. The third named author is partiallysupported by NSA grant H98230-11-1-0165.2.
Preliminaries
In this section, we collect necessary notations and terminology used in the paper, andprove a few auxiliary results. For basic facts in commutative algebra, we refer the reader to[6, 13].From now on, G denotes a finitely generated abelian group, S = A [ x , . . . , x n ] is a G -graded algebra over a commutative ring with identity, A ⊆ S and M is a G -graded S -module. By abusing notation, we shall use 0 to denote the identity of all abelian groupsconsidered in the paper; the particular group will be understood from the context of its use. Definition 2.1.
Let E ⊆ G be a collection of elements in G . We say that E is a linearlyindependent subset of G if E forms a basis for a free submonoid of G . Definition 2.2.
The support of M in G is defined to beSupp G ( M ) = { γ ∈ G (cid:12)(cid:12) M γ = 0 } . Remark 2.3.
When A is a field, let F • be a minimal G -graded free resolution of M over S ,where F i = M η ∈ G S ( − η ) β iη ( M ) . The numbers β iη ( M ) are called the multigraded (or G -graded) Betti numbers of M and β iη ( M ) = dim A Tor Si ( M, A ) η as, by definition, the maps in F • ⊗ S A are zero maps.More generally, we shall prove the following lemma relating the multigraded Betti numbersof M and the support of Tor S ∗ ( M, A ). Lemma 2.4.
Let F • be a G -graded free resolution of a G -graded S -module M . Then(1) F i has a summand S ( − γ ) for any γ ∈ Supp G (Tor Si ( M, A )) .(2) Assume that there exists φ ∈ Hom Z ( G, R ) such that φ (deg( x i )) > for all i and M isfinitely generated. Then there exists a G -graded free resolution F ′• of M such that F ′ i = M ℓ ∈ E ′ i S ( − γ i,ℓ ) with γ i,ℓ ∈ [ j ≤ i Supp G (Tor Sj ( M, A )) , ∀ ℓ. Notice that, without further restrictions on A and/or M one cannot in general choose F ′ i so that γ i,ℓ ∈ Supp G (Tor Si ( M, A )) , ∀ ℓ . HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 5
Proof.
For (1), let K be defined by the exact sequence0 → K → F → M → F ⊗ S A → M ⊗ S A is onto. As ( F ⊗ S A ) γ = 0 if and only if S ( − γ ) is a directsummand of F , the result holds for i = 0. Furthermore, · · · → F → K → K , Tor S ( M, A ) is a graded submodule of K ⊗ S A and Tor Si ( M, A ) ≃ Tor Si − ( K, A ) for i ≥
2, which implies the result by induction on i .To prove the second statement, we relax the finite generation of M by the followingcondition, which will enable us to make induction : ∃ q ∈ R , φ (deg a ) > q, ∀ a ∈ M . Noticethat if a module satisfies this condition, any of its submodules satisfies the same condition.Set T j := Supp G (Tor Sj ( M, A )). Let ψ : F = L ℓ ∈ E S ( − γ ,ℓ ) → M be the augmentationand E ′ := { ℓ ∈ E | γ ,ℓ ∈ T } . Denote by ψ ′ the restriction of ψ to F ′ := L ℓ ∈ E ′ S ( − γ ,ℓ ).We now prove that ψ ′ is onto. First notice that ψ ′ ⊗ S A is surjective. Assume that ψ ′ isnot surjective, let M ′ be the image of ψ ′ and h := inf { φ ( γ ) , M γ = M ′ γ } ≥ inf { φ (deg( a )) , a ∈ M } ∈ R . Set ǫ := min { φ (deg( x i )) , i = 1 , . . . , n } > m ∈ M ν \ M ′ ν for some ν with h ≤ φ ( ν ) < h + ǫ . As ψ ′ ⊗ S A is onto, there exists m ′ ∈ M ′ such that m is of theform m ′ + P ni =1 m i x i . Now, for some value i we have m i M ′ . It then follows that φ (deg( m i )) = φ ( ν ) − φ (deg( x i )) < h , contradicting the definition of h .We will now prove (2) by induction on i . To end this, assume that there exists a gradedcomplex 0 → F ′ i → · · · → F ′ → M → i ≥
0, and such that F ′ i = L ℓ ∈ E ′ i S ( − γ i,ℓ )with γ i,ℓ ∈ S j ≤ i T j .If the complex is exact, our claim is proved; otherwise, by setting K i ⊂ F ′ i for the i -thhomology module of the complex and Q i := ker( F ′ i ⊗ S A → F ′ i − ⊗ S A ), one hasSupp G ( K i ⊗ S A ) = Supp G (Tor Si +1 ( M, A )) ∪ Supp G (ker( Q i → Tor Si ( M, A ))) . Applying the argument above to a graded onto map F → K i , and using that by inductionSupp G ( K i ⊗ S A ) ⊆ T i +1 ∪ Supp G ( F ′ i ⊗ S A ) ⊆ ∪ j ≤ i +1 T j one obtains a graded free S -module F ′ i +1 as claimed mapping onto K i . (cid:3) Let t = ( t , . . . , t s ) ∈ Z s . We shall write t ≥ t > t ≤ t = 0)if t i ≥ t i > , t i ≤ t i = 0) for all i = 1 , . . . , s . For a property thatdepends on a parameter t ∈ Z s , one says that the property holds for t ≫ t ∈ Z s such that it holds for t ∈ t + N s . The following semi-classical lemma will be of use. AMIR BAGHERI, MARC CHARDIN, AND HUY T `AI H `A
Lemma 2.5.
Let R be a finitely generated N s -graded algebra over a commutative ring A .Let M be a finitely generated Z s -graded R -module. Then either M t = 0 for t ≫ or M t = 0 for t ≫ .Proof. Let b = L t i ≥ R ( t ,...,t s ) be the ideal generated by elements of strictly positive degrees.If M = H b ( M ) one has M t = 0 for any t ∈ t + N s , where t is the degree of a non zeroelement in M/H b ( M ).If M = H b ( M ) then any generator a of M spans a submodule of M that is zero in degreesdeg( a ) + b a (1 , . . . ,
1) + N s , where b a is such that for any product p of b a elements among thefinitely many generators of the R -ideal b , one has pa = 0. As M is finitely generated, theresult follows. (cid:3) We shall make use of the notion of initial modules with respect to a monomial order. Thisis a natural extension of the familiar notion of initial ideals in a polynomial ring. Let F bea free S -module. We can write F = L i ∈ I Se i . A monomial in F is of the form x α e i , where x α is a monomial in S and i ∈ I . A monomial order in F is a total order, say ≺ , on themonomials of F satisfying the following condition:if u ≺ v and w = 1 is a monomial in S then u ≺ uw ≺ vw .It can be seen that ≺ is a well ordering, i.e., every non-empty subset of the monomialsin F has a minimal element. We refer the reader to [6, 15.2] for more details on monomialorders on free modules. Definition 2.6.
Let F be a free S -module, and let K be an S -submodule of F . Let ≺ bea monomial order in F . The initial module of K , denoted by in ≺ ( K ), is defined to be the S -submodule of F generated by { x α e i (cid:12)(cid:12) ∃ f = x α e i + smaller terms ∈ K } . Proposition 2.7.
Let F be a free G -graded S -module, and let K be a G -graded S -submoduleof F . Let ≺ be a monomial order in F . Then in ≺ ( K ) is a G -graded S -module of F , and Supp G ( F/K ) = Supp G ( F/ in ≺ ( K )) . Proof.
It is clear from the definition that in ≺ ( K ) is a G -graded S -module. To prove theproposition, we need to show that for any µ ∈ G , K µ = F µ if and only if in ≺ ( K ) µ = F µ .Clearly, if K µ = F µ then all monomials of degree µ in F are elements of K , and thus, areelements of in ≺ ( K ). Therefore, in this case, in ≺ ( K ) µ = F µ . Suppose now that in ≺ ( K ) µ = F µ . Let x α e i be the smallest monomial of degree µ in F but not in K , if it exists. Then x α e i ∈ in ≺ ( K ) µ . Thus, there exists an element f ∈ K of theform f = x α e i + g, where g consists of monomials that are smaller than x α e i with respect to ≺ . Since K isa G -graded S -module, we can choose f to be G -homogeneous of degree µ . That is, all its HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 7 monomials are of degree µ . This implies that all monomials, and thus, all terms in g areelements in K . In particular, g ∈ K . Therefore, x α e i = f − g ∈ K , a contradiction. Hence, F µ = K µ . The proposition is proved. (cid:3) One of our techniques is to take the collection of elements of certain degree from a complex.This construction gives what we shall call strands of the complex.
Definition 2.8.
Let F • be a G -graded complex of S -modules and let Γ ⊆ G . The Γ -strand of F • , often denoted by ( F • ) Γ , is obtained by taking elements of degrees belonging to Γin F • and the boundary maps between these elements (since the complex is graded, theboundary maps are of degree 0). In particular, if F = L γ ∈ G F γ is a G -graded S -module,then F Γ := L γ ∈ Γ F γ . Note that the degree Γ-strand of a complex/module is not necessarilya complex/module over S . 3. Forms of the same degree
In this section, we consider the case when every ideal I i is generated in a single degree.That is, when deg G ( f i,j ) = γ i for all j . We keep the notations of Section 2.Let G ′ denote a finitely generated abelian group. The following result, with G ′ = Z s , willbe a key ingredient of our proof. Theorem 3.1.
Let R = S [ T , . . . , T r ] be a G × G ′ -graded polynomial extension of S with deg G × G ′ ( a ) ∈ G × for all a ∈ S and deg G × G ′ ( T j ) ∈ × G ′ for all j . Let M be a finitelygenerated G × G ′ -graded R -module and let i be an integer. Assume that i = 0 or A is aNoetherian ring. Then(1) There exists a finite subset ∆ i ⊆ G such that, for any t , Tor Si ( M ( ∗ ,t ) , A ) δ = 0 for all δ ∆ i .(2) Assume that G ′ = Z s . For δ ∈ ∆ i , Tor Si ( M ( ∗ ,t ) , A ) δ = 0 for t ≫ or Tor Si ( M ( ∗ ,t ) , A ) δ =0 for t ≫ . If, furthermore, A → k is a ring homomorphism to a field k , then forany j , the function dim k Tor Aj (Tor Si ( M ( ∗ ,t ) , A ) δ , k ) is polynomial in the t i s for t ≫ , and the function dim k Tor Si ( M ( ∗ ,t ) , k ) δ is polynomial in the t i s for t ≫ .Proof. Let F • be a graded free resolution of M over R , where F i = L η,j R ( − η, − j ) β iη,j is offinite rank for i = 0, and for any i when A is Noetherian. For t ∈ G ′ , the ( ∗ , t )-strand of F • ,denoted by F t • , is a G -graded free resolution of M ( ∗ ,t ) over S = R ( ∗ , , that is not necessarilyminimal. Its i -th term is F ti = M η,j S ( − η ) β iη,j ⊗ A B t − j , AMIR BAGHERI, MARC CHARDIN, AND HUY T `AI H `A where B = A [ T , . . . , T r ].Let ∆ i = { η (cid:12)(cid:12) ∃ j : β iη,j = 0 } . The module Tor Si ( M ( ∗ ,t ) , A ) = H i ( F t • ⊗ S A ) is a subquotientof the module L η,j A ( − η ) β iη,j ⊗ A B t − j , and (1) is proved.To prove (2), observe first that Tor Si ( M ( ∗ ,t ) , A ) δ = H i ( F t • ⊗ S A ) δ and (cid:0) A ( − η ) ⊗ A B t − j (cid:1) δ = A δ − η ⊗ A B t − j is zero if η = δ . Thus, H i ( F t • ⊗ S A ) δ is equal to H i ( F [ δ ] • ⊗ S A ) t , where F [ δ ] • isthe subcomplex of F • given by F [ δ ] i = M j R ( − δ, − j ) β iδ,j = M j [ S ( − δ ) ⊗ A B ( − j )] β iδ,j . As F [ δ ] • ⊗ S A is a graded complex of finitely generated B -modules, H i ( F [ δ ] • ⊗ S A ) is afinitely generated B -module for any i when A is Noetherian. Similarly, Tor Si ( M ( ∗ ,t ) , k ) δ = H i ( F t • ⊗ S k ) δ = H i ( F [ δ ] • ⊗ S k ) is a finitely generated k [ T , . . . , T r ]-module for any i when A isNoetherian.This proves (2) in view of Lemma 2.5 and [11, Theorem 1]. (cid:3) Remark 3.2.
In the context of point (2) above, there is a graded spectral sequence ofgraded B -modules with second term E j,i ( t ) = Tor Aj (Tor Si ( M ( ∗ ,t ) , A ) δ , k ) that converges toTor Si + j ( M ( ∗ ,t ) , k ) δ . It follows that all terms E pj,i ( t ) for p ≥ k [ T , . . . , T r ]-modules. In particular, one can write :dim k Tor Sℓ ( M ( ∗ ,t ) , k ) δ = X i + j = ℓ dim k E ∞ j,i ( t ) ≤ X i + j = ℓ dim k E pj,i ( t ) , ∀ p ≥ , which provides a control on the Hilbert function (and polynomial) of Tor Sℓ ( M , k ) δ in termsof the ones of Tor Aj (Tor Si ( M , A ) δ , k ) for i + j = ℓ .We are now ready to examine the asymptotic linear behavior of nonzero G -graded Bettinumbers and non-vanishing degrees of local cohomology modules of M I t · · · I t s s .In the next two theorems, let R = S [ T i,j | ≤ i ≤ s, ≤ j ≤ r i ]. Then R is equipped witha G × Z s -graded structure in which deg G × Z s ( x i ) = (deg G ( x i ) ,
0) and deg G × Z s ( T i,j ) = (0 , e i ),where e i is the i -th element in the canonical basis of Z s . Recall that γ i = deg G ( f i,j ) and let γ := ( γ , . . . , γ s ). For t = ( t , . . . , t s ) ∈ Z s , let I t := I t . . . I t s s , T t := T t . . . T t s s , I t T t ( γ.t ) := I t ( t γ ) T t . . . I t s s ( t s γ s ) T t s s and M I t T t ( γ.t ) := M I t ( t γ ) T t . . . I t s s ( t s γ s ) T t s s . Theorem 3.3.
Assume that i = 0 or A is Noetherian. Then there exists a finite set ∆ i ⊆ G such that(1) For all t ∈ N s , Tor Si ( M I t · · · I t s s , A ) η = 0 if η ∆ i + t γ + · · · + t s γ s .(2) There exists a subset ∆ ′ i ⊆ ∆ i such that Tor Si ( M I t · · · I t s s , A ) η + t γ + ··· + t s γ s = 0 for t ≫ and η ∈ ∆ ′ i and Tor Si ( M I t · · · I t s s , A ) η + t γ + ··· + t s γ s = 0 for t ≫ and η ∆ ′ i . HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 9 (3) If, furthermore, A → k is a ring homomorphism to a field k , then for any δ and any j , the function dim k Tor Aj (Tor Si ( M I t · · · I t s s , A ) δ + t γ + ··· + t s γ s , k ) is polynomial in the t i s for t ≫ , and the function dim k Tor Si ( M I t · · · I t s s , k ) δ + t γ + ··· + t s γ s is polynomial in the t i s for t ≫ .Proof. Let R := L t ≥ I t T t ( γ.t ) and M R := L t ≥ M I t T t ( γ.t ) denote the (shifted) multi-Rees algebra and the multi-Rees module with respect to I , . . . , I s , and M . The naturalsurjective map φ : R ։ R that sends x i to x i and T i,j to f i,j T i makes M R a finitelygenerated G × Z s -graded module over R .Observe that, for any δ ∈ G and t ∈ Z s , M R ( δ,t ) ≃ [ M I t ( γ.t )] δ = [ M I t ] δ + γ.t , where in thelast term δ + γ.t := δ + t γ + · · · + t s γ s . Thus, the assertion follows by applying Theorem3.1 to the R -module M := M R . (cid:3) Theorem 3.4.
Let b be a G -homogeneous ideal in S such that for any i ≥ and δ ∈ G , H i b ( S ) δ is a finitely generated A -module. Then, if A is Noetherian, for i ≥ , there exists asubset Λ i ⊆ G such that(1) H i b ( M I t · · · I t s s ) η + t γ + ··· + t s γ s = 0 for t = ( t , . . . , t s ) ≫ if only if η ∈ Λ i .(2) If, furthermore, A → k is a ring homomorphism to a field k , then for any δ and any j , the function dim k Tor Aj ( H i b ( M I t · · · I t s s ) δ + t γ + ··· + t s γ s , k ) is a polynomial in the t i s for t ≫ .Proof. Since taking local cohomology respects the G -homogeneous degree, we have H i b ( M I t ) δ + γ.t = H i b R ( M R ) ( δ,t ) = (cid:0) H i b R ( M R ) ( δ, ∗ ) (cid:1) t . Let B = A [ T i,j , ≤ i ≤ s, ≤ j ≤ r i ]. Since B is a flat extension of A , H i b R ( R ) ( δ, ∗ ) = H i b R ( B ⊗ A S ) ( δ, ∗ ) is a finitely generated B -module. Let F • be the minimal G × Z s -graded freeresolution of M R over R . Since A is Noetherian, each term F j of F • is of finite rank. Thisimplies that for all δ ∈ G , i ≥ j ≥ H i b R ( F j ) ( δ, ∗ ) is a finitely generated B -module. Thespectral sequence H i b R ( F j ) ⇒ H i − j b R ( M R ) implies that H i b R ( M R ) ( δ, ∗ ) is a finitely generatedmultigraded B -module. This proves (2) in view of [11, Theorem 1].Notice that H i b R ( M R ) ( δ,t ) = 0 for all t ≫ K δ = H i b R ( M R ) ( δ, ∗ ) is annihilatedby a power of the ideal a := T si =1 ( T i, , . . . , T i,r i ). Hence (1) holds withΛ i := { δ ∈ G (cid:12)(cid:12) K δ = H a ( K δ ) } . (cid:3) Forms of arbitrary degrees
This section is devoted to proving our main result in its full generality, when the ideals I i s are generated in arbitrary degrees. We start by recalling the notion of a Stanley decom-position of multigraded modules. Definition 4.1.
Let G be a finitely generated abelian group and let B = A [ T , . . . , T r ] bea G -graded polynomial ring over a commutative ring A . Let M be a finitely generated G -graded B -module. A Stanley decomposition of M is a finite decomposition of the form M = m M i =1 u i A [ Z i ] , where the direct sum is as A -modules, u i s are G -homogeneous elements in M , Z i s are subsets(could be empty) of the variables { T , . . . , T r } , and u i A [ Z i ] denotes the A -submodule of M generated by elements of the form u i m where m is a monomial in the polynomial ring A [ Z i ].The following lemma is well known in N -graded or standard N n -graded situations (cf.[2, 3, 10]). Lemma 4.2.
Let G be a finitely generated abelian group and let B = A [ T , . . . , T r ] be a G -graded polynomial ring. Let I be a monomial ideal in B . Then a Stanley decompositionof B/I exists.Proof.
The proof follows along the same lines as in the proof of [10, Corollary 6.4] or [2,Theorem 2.1], as one notices that any monomial is a homogeneous element. (cid:3)
Theorem 4.3.
Let G be a finitely generated abelian group, B = A [ T , . . . , T r ] be a G -gradedpolynomial ring over a commutative ring A and M be a finitely generated G -graded B -module.Let Γ denote the set of subsets of { deg G ( T i ) } ri =1 whose elements are linearly independent over Z . Then there exist a collection of pairs ( δ p , E p ) ∈ G × Γ , for p = 1 , . . . , m , such that Supp G ( M ) = m [ p =1 (cid:0) δ p + h E p i (cid:1) , where h E p i represents the free submonoid of G generated by elements in E p .Proof. Since M is a finitely generated G -graded B -module, there exists a homogeneous sur-jective map φ : F ։ M from a free B -module F to M . We can write F = L mi =1 Be i , wheredeg G ( e i ) represents the degree of the i -th generator of M . Let K = ker φ . Then M ≃ F/K .In particular, Supp G ( M ) = Supp G ( F/K ).Extend any monomial order on B to a monomial order on F = L mi =1 Be i , by orderingthe e i ’s. Since M is G -graded, so is K . Thus, by Proposition 2.7, we have Supp G ( F/K ) =Supp G ( F/ in ≺ ( K )). Therefore,Supp G ( M ) = Supp G ( F/ in ≺ ( K )) . HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 11
Observe that in ≺ ( K ) is generated by monomials of the form T α e i (where T α = T α · · · T α r r for α = ( α , . . . , α r ) ∈ N r ). Let I i be the monomial ideal in B generated by all monomials T α for which T α e i ∈ in ≺ ( K ). Clearly, F/ in ≺ ( K ) ≃ L mi =1 BI i e i . By Lemma 4.2, for each i = 1 , . . . , m , there exists a Stanley decomposition of BI i e i BI i e i ≃ m i M j =1 u ij A [ T ij ] , where u ij are homogeneous elements of BI i e i and T ij are subsets (could be empty) of thevariables { T , . . . , T r } in B . This gives F/ in ≺ ( K ) ≃ m M i =1 m i M j =1 u ij A [ T ij ] . Thus, Supp G ( F/ in ≺ ( K )) can be written as a finite union of the formSupp G ( F/ in ≺ ( K )) = [ i,j Supp G (cid:0) u ij A [ T ij ] (cid:1) . Let δ ij = deg G ( u ij ). ThenSupp G ( F/ in ≺ ( K )) = [ i,j (cid:0) δ ij + Supp G ( A [ T ij ]) (cid:1) . To prove the theorem, it now suffices to show that Supp G ( A [ T ij ]) can be decomposed into aunion of free submonoids of G of the form h E i , where E is a linearly independent subset inΓ. Since A [ T ij ] is a polynomial ring whose variables are variables of B , without loss ofgenerality, we may assume that T ij = { T , . . . , T r } , i.e., A [ T ij ] = B . Let H be the binomialideal in B generated by { T α − T β (cid:12)(cid:12) deg G ( T α ) = deg G ( T β ) } . Then taking the quotient B/H is the same as identifying monomials of the same degree in B . Thus, we haveSupp G ( B ) = Supp G ( B/H ) = Supp G ( B/ in ≺ ( H ))and (cid:16) BH (cid:17) γ = (cid:16) B in ≺ ( H ) (cid:17) γ = (cid:26) A if γ ∈ Supp G ( B/H )0 otherwise.(4.1)By Lemma 4.2, a Stanley decomposition of B/ in ≺ ( H ) exists. That is, we can write B/ in ≺ ( H ) = s M j =1 u j A [ Z j ] where u j are G -homogeneous elements of B/ in ≺ ( H ) and Z j are subsets (could be empty) ofthe variables { T , . . . , T r } . Let E j be the set of degrees of variables in Z j . It further followsfrom (4.1) that B/ in ≺ ( H ) has at most one monomial in each degree. This implies that thesupport of u j A [ Z j ] are all disjoint and each set E j is linearly independent. Hence, by letting σ j = deg G ( u j ), we have Supp G ( B ) = s a j =1 (cid:0) σ j + h E j i (cid:1) . The theorem is proved. (cid:3)
Remark 4.4.
It would be nice if the union in Theorem 4.3 is a disjoint union. However,this is not true. Let B = A [ x, y ] be a Z -graded polynomial ring with deg( x ) = 4 anddeg( y ) = 7 (hence, Γ = { , } ). Let M = B/ ( x ) ⊕ B/ ( y ) ≃ A [ y ] ⊕ A [ x ]. Then Supp Z ( M ) = { a + 7 b (cid:12)(cid:12) a, b ∈ Z ≥ } . Moreover, linearly independent subsets of Γ are { } and { } . It canbe easily seen that Supp Z ( M ) cannot be written as disjoint union of shifted free submonoidsof Z generated by 4 and/or by 7.For a vector c = ( c , . . . , c s ) ∈ Z s and a tuple E = ( ν , . . . , ν s ) of elements in G , we shalldenote ∆ E the empty tuple if s ≤ s − ν − ν , . . . , ν s − ν s − ) else, and by c .E the G -degree P sj =1 c j ν j . If E and E ′ are tuples, we denote by E | E ′ the concatenationof E and E ′ . Remark 4.5.
With some simple linear algebra arguments, it can be seen that for tuples E , . . . , E s of elements of G , the tuple of elements of G × Z s , E × { e }| · · · | E s × { e s } , where e i is the i -th basis element of Z s , is linearly independent if and only if ∆ E | · · · | ∆ E s islinearly independent. These equivalent conditions imply that for ( c , . . . , c s ) = ( c ′ , . . . , c ′ s ),with c i , c ′ i ∈ Z | E i |≥ and | c i | = | c ′ i | for all i , one has c .E + · · · + c s .E s = c ′ .E + · · · + c ′ s .E s .This last fact is a direct corollary of the linear independence of E × { e }| · · · | E s × { e s } , as c . ( E × { e } ) + · · · + c s . ( E s × { e s } ) = ( c .E + · · · + c s .E s ) × ( | c | e + · · · + | c s | e s ).We shall now prove our main result. Recall that I i = ( f i, , . . . , f i,r i ) and let γ i,j =deg G ( f i,j ). Theorem 4.6.
Let G be a finitely generated abelian group and let S = A [ x , . . . , x n ] be a G -graded algebra over a commutative ring A ⊆ S . Let I i = ( f i, , . . . , f i,r i ) for i = 1 , . . . s be G -homogeneous ideals in S , and let M be a finitely generated G -graded S -module. Set Γ i = { deg G ( f i,j ) } r i j =1 . Let ℓ ≥ and assume that ℓ = 0 or A is Noetherian.There exist a finite collection of elements δ ℓp ∈ G , a finite collection of integers t ℓp,i ,and a finite collection of non-empty tuples E ℓp,i ⊆ Γ i , such that ∆ E ℓp, | · · · | ∆ E ℓp,s is linearlyindependent for all p , satisfying : HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 13
Supp G (Tor Sℓ ( M I t · · · I t s s , A )) = m [ p =1 (cid:0) δ ℓp + [ c i ∈ Z | Eℓp,i |≥ , | c i | = t i − t ℓp,i c .E ℓp, + · · · + c s .E ℓp,s (cid:1) , if t i ≥ max p { t ℓp,i } for all i .Proof. As before, we use t to denote ( t , . . . , t s ) ∈ Z s , and let R := L t ≥ I t T t and M R := L t ≥ M I t T t . Consider R = A [ x , . . . , x n ][ T i,j , ≤ i ≤ s, ≤ j ≤ r i ], the G × Z s -gradedpolynomial ring over A [ x , . . . , x n ] with deg G × Z s ( x i ) = (deg G ( x i ) ,
0) and deg G × Z s ( T i,j ) =(deg G ( f i,j ) , e i ), where e i denotes the i -th canonical generator of Z s . The natural surjectivemap φ : R ։ R that sends x i to x i and T i,j to f i,j T i makes M R a finitely generated G × Z s -graded module over R .Let F • be a G × Z s -graded free resolution of M R over R . If A is Noetherian, then each F i can be chosen of finite rank, and we make such a choice. For t ∈ Z s , the degree ( ∗ , t )-strand F t • of F • provides a G -graded free resolution of M I t over S = R ( ∗ , . Thus,Tor Si ( M I t , A ) = H i ( F t • ⊗ S A ) . Moreover, taking homology respects the graded structure, and therefore, H i ( F t • ⊗ S A ) = H i ( F • ⊗ R R/ m R ) ( ∗ ,t ) , where m = ( x , . . . , x n ) is the homogeneous irrelevant ideal in S .Let Γ ′ = { ( γ i,j , e i ) ∈ G × Z s } = ` i Γ i × { e i } be the set of degrees of the variables T i,j .Observe that H i ( F • ⊗ R R/ m R ) is a finitely generated G × Z s -graded module over R/ m R ≃ B for any i if A is Noetherian, and for i = 0 in any case. Applying Theorem 4.3 to the G × Z s -graded module H i ( F • ⊗ R R/ m R ) we obtain a finite collection of elements θ ℓp ∈ G × Z s and afinite collection of linearly independent subsets E ℓp ⊆ Γ ′ (which we view in a fixed order astuples), for p = 1 , . . . , m , such thatSupp G × Z s (cid:0) H i ( F • ⊗ R R/ m R ) (cid:1) = m [ p =1 (cid:0) θ ℓp + h E ℓp i (cid:1) . Let θ ℓp = ( δ ℓp , t ℓ ,p , . . . , t ℓs,p ), where δ ℓp ∈ G and t ℓi,p ∈ Z . One has E ℓp = ` si =1 E ℓp,i × { e i } .The linear independence of the elements in E ℓp is equivalent to the fact that the elements of∆ E ℓp, | · · · | ∆ E ℓp,s are linearly independent.Taking the degree ( ∗ , t )-strand of H i ( F • ⊗ R R/ m R ), we getSupp G (Tor Sℓ ( M I t · · · I t s s , A )) = m [ p =1 (cid:0) δ ℓp + [ c i ∈ Z | Eℓp,i |≥ , | c i | = t i − t ℓp,i c .E ℓp, + · · · + c s .E ℓp,s (cid:1) . (cid:3) Example 4.7.
Let I ⊆ S be a complete intersection ideal of three forms f , f , f of degrees a, b, c . On easily sees, for instance by the Hilbert Burch theorem, that0 / / R ( − a − b − c, − L R ( − a − b − c, − T T T f f f / / R ( − b − c, − L R ( − a − c, − L R ( − a − b, − f T − f T f T − f T f T − f T / / R / / R I / / R -resolution of R I .It follows that Tor S ( I t , A ) µ = B µ,t , Tor S ( I t , A ) µ = B µ − a − b − c,t − , andTor S ( I t , A ) µ = coker B µ − a − b − c,t − (cid:16) T T T (cid:17) / / B µ − b − c,t − L B µ − a − c,t − L B µ − a − b,t − . Now, Tor S ( R I , A ) has the following Stanley decomposition: A [ T , T , T ]( − b − c, − M A [ T , T ]( − a − c, − M A [ T , T , T ]( − a − b, − . The ideal H generated by the binomials T α − T β with deg( T α ) = deg( T β ) is the kernel ofthe map A [ T , T , T ] / / A [ u, v ] T / / uv a T / / uv b T / / uv c and is therefore generated by a single irreducible and homogeneous binomial.If, for example, ( a, b, c ) = (2 , ,
8) then this relation is T − T T , and a Stanley decom-position of, for instance, A [ T , T , T ] / ( T ) is A [ T , T ] ⊕ T A [ T , T ]( − , − S ( R I , A ) : A [ T , T ]( − , − ⊕ T A [ T , T ]( − , − ⊕ A [ T , T ]( − , − ⊕ A [ T , T ]( − , − ⊕ T A [ T , T ]( − , − . Setting E t := { α + 8 β | α, β ∈ Z + , α + β = t } = 2 t + 6 { , · · · , t } , one gets that for t ≥ Z (Tor S ( I t , A )) = E t ∪ (5 + E t − ),– Supp Z (Tor S ( I t , A )) = (13 + E t − ) ∪ (18 + E t − ) ∪ (10 + E t − ) ∪ (7 + E t − ) ∪ (12 + E t − ),– Supp Z (Tor S ( I t , A )) = (15 + E t − ) ∪ (20 + E t − ).Notice that one has the simplified expression :– Supp Z (Tor S ( I t , A )) = (5 + E t ) ∪ (10 + E t − ). HE EVENTUAL SHAPE OF BETTI TABLES OF POWERS OF IDEALS 15
References [1] D. Berlekamp.
Regularity defect stabilization of powers of an ideal . Preprint. arXiv:1105.2260 .[2] K. Baclawski and A.M. Garsia.
Combinatorial decompositions of a class of rings . Adv. in Math. (1981), 155-184.[3] W. Bruns, C. Krattenthaler, and J. Uliczka. Stanley decompositions and Hilbert depth in the Koszulcomplex.
J. Commutative Algebra, (2010), no. 3, 327-357.[4] M. Chardin. Powers of ideals and the cohomology of stalks and fibers of morphisms . Preprint. arXiv:1009.1271 .[5] S.D. Cutkosky, J. Herzog, and N.V. Trung.
Asymptotic behaviour of the Castelnuovo-Mumford regu-larity.
Composito Mathematica, (1999), 243-261.[6] D. Eisenbud. Commutative Algebra: with a View Toward Algebraic Geometry. Springer-Verlag, NewYork, 1995.[7] D. Eisenbud and J. Harris.
Powers of ideals and fibers of morphisms . Math. Res. Lett. (2010), no.2, 267-273.[8] D. Eisenbud and B. Ulrich. Stabilization of the regularity of powers of an ideal . Preprint. arXiv:1012.0951 .[9] H.T. H`a.
Asymptotic linearity of regularity and a ∗ -invariant of powers of ideals . Math. Res. Lett. (2011), no. 1, 1-9.[10] J. Herzog and D. Popescu. Finite filtrations of modules and shellable multicomplexes.
ManuscriptaMath., (2006), no. 3, 385-410.[11] V. Kodiyalam.
Homological invariants of powers of an ideal.
Proceedings of the American MathematicalSociety, , no. 3, (1993), 757-764.[12] V. Kodiyalam.
Asymptotic behaviour of Castelnuovo-Mumford regularity.
Proceedings of the AmericanMathematical Society, , no. 2, (1999), 407-411.[13] H. Matsumura. Commutative Ring Theory. Cambridge, 1986.[14] P. Singla. Convex-geometric, homological and combinatorial properties of graded ideals. genehmitgeDissertation. Universit¨at Duisburg-Essen, December 2007.[15] N.V. Trung and H. Wang.
On the asymptotic behavior of Castelnuovo-Mumford regularity . J. PureAppl. Algebra, (2005), no. 1-3, 42-48.[16] G. Whieldon.
Stabilization of Betti tables . Preprint. arXiv:1106.2355 . Institut de Math´ematiques de Jussieu, UPMC, Boite 247, 4, place Jussieu, F-75252 ParisCedex, France
E-mail address : [email protected] Institut de Math´ematiques de Jussieu, UPMC, Boite 247, 4, place Jussieu, F-75252 ParisCedex, France, http://people.math.jussieu.fr/~chardin
E-mail address : [email protected] Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans,LA 70118, USA,
E-mail address ::