Elsevier

Journal of Algebra

Volume 573, 1 May 2021, Pages 620-640
Journal of Algebra

Bass and Betti numbers of A/In

https://doi.org/10.1016/j.jalgebra.2020.12.041Get rights and content

Abstract

Let (A,m,k) be a Gorenstein local ring of dimension d1. Let I be an ideal of A with ht(I)d1. We prove that the numerical functionn(ExtAi(k,A/In+1)) is given by a polynomial of degree d1 in the case when id+1 and curv(In)>1 for all n1. We prove a similar result for the numerical functionn(ToriA(k,A/In+1)) under the assumption that A is a Cohen-Macaulay local ring. We note that there are many examples of ideals satisfying the condition curv(In)>1, for all n1. We also consider more general functions n(ToriA(M,A/In) for a filtration {In} of ideals in A. We prove similar results in the case when M is a maximal Cohen-Macaulay A-module and {In=In} is the integral closure filtration, I an m-primary ideal in A.

Introduction

Let (A,m,k) be a Noetherian local ring of dimension d1 and let I be an arbitrary ideal of A. It is well known that for a finitely generated A-module M the numerical functionn(ExtAi(k,M/In+1M)) is given by a polynomial of degree at most d1 for i1, see [15, Theorem 2]. We denote this polynomial by εM,Ii(x). When M=A we simply denote this by εIi(x). Note that (ExtAi(k,M/In+1M)) is the ith Bass number of M/In+1M. Dually we have the numerical functionn(ToriA(k,M/In+1M)) giving the ith Betti numbers of M/In+1M. Again, by a theorem of Kodiyalam [15, Theorem 2], this is given by a polynomial of degree at most d1. We denote this polynomial by tI,iA(M,x). These polynomials are collectively called the Hilbert polynomials associated to derived functors because they generalize the usual Hilbert polynomial. It is of some interest to find the degrees of the polynomials εM,Ii(x) and tI,iA(M,x). For instance, in [17, Theorem 18] it was proved that if M is a maximal Cohen-Macaulay A-module and I=m thendegtm,1A(M,x)<d1if and only ifMis free. In [11, Theorem I] this result was generalized to arbitrary finitely generated modules with projective dimension at least 1 over Cohen-Macaulay local rings. In [12] it was proved that for an ideal I of analytic deviation one,ifdegtI,1A(M,x)<d1thenFI(M)is freeF(I)-module, where F(I)=n0In/mIn is the fiber cone of I and FI(M)=n0InM/mInM is the fiber module of M with respect to ideal I. Katz and Theodorescu in [13] prove that if I integrally closed then the degree of tI,iA(k,x) is equal to the analytic spread of I minus one under some mild conditions on the ring A. In the case of Hilbert polynomials associated to extension functor see [7], [14], [21] giving estimates on the degree of the polynomials in some cases of interest.

In this paper, we provide a new class of ideals, namely ideals I satisfying the condition curv(In)>1 for all n1 (see 2.2 for definition of curvature) for which the numerical functions giving the ith Bass numbers and the ith Betti numbers of A/In are polynomials of degree equal to d1 for n>>0. It follows from [1, Corollary 5] that there many examples of ideals with curv(In)>1 for all n1, we note this in 2.4 and 2.5 of the section on preliminaries. Our results in the case of Bass numbers require the Gorenstein hypothesis on the ring A while in the case of Betti numbers we require A to be a Cohen-Macaulay ring. More precisely our results state:

Theorem 1.1

Let (A,m,k) be a Gorenstein local ring of dimension d1. Let I be an ideal of A with ht(I)d1 and curv(In)>1 for all n1. Let x1,,xd1 be a superficial sequence w.r.t. I and Al(x_)=A/(x1,,xdl) for 1ld. Then for id+1, we have degεAl,Ii(x)=l1. In particular when l=d, we get degεIi(x)=d1.

Theorem 1.2

Let (A,m) be a Cohen-Macaulay ring of dimension d1. Let I be an ideal with ht(I)d1 and curv(In)>1 for all n1. Then for id1,n(ToriA(k,A/In)) is given by a polynomial tI,iA(k,z) of degree d1 for n>>0.

We then consider more general functions for i1n(ToriA(M,A/In)) where J={In} is an I-admissible filtration of ideals in A. Here we assume that A is Cohen-Macaulay of dimension d1 and that M is a non-free maximal Cohen-Macaulay A-module. Our main results show that these functions are given by polynomials of degree equal to d1 for the following cases of filtrations of ideals {In}:
  • (i)

    J={In=In} is an I-adic filtration where I is m-primary with r(I)=1 i.e. ideals having reduction number one and ToriA(M,A/I)0.

  • (ii)

    J={In=In} is the integral closure filtration of ideals in A where A is analytically unramified ring.

Moreover we also identify the normalized leading coefficient in the case (i) above. In case (ii), we first prove the result in dimension one by using the fact that for integrally closed m-primary ideal quotient I, ring A/I acts as a test module for finite projective dimension, see [6, Corollary 3.3]. We then prove the result by induction on dimension d by using Lemma 7.2.

Here is an overview of the contents of the paper. In section 2 on preliminaries, we give all the basic definitions, notations and also discuss some preliminary facts that we need. In section 3, we estimate the degree of the polynomial εIi(x) in the case when A is Gorenstein local ring of dimension one. In section 4, we proceed to estimate the degree of εIi(x) in the general case of Gorenstein local ring of arbitrary dimension. This proves one of our main theorems (Theorem 1.1) stated above. In section 5, we prove Theorem 1.2 stated above, showing that the degree of the polynomial tI,iA(k,z) giving the Betti numbers of A/In is d1 in the case of interest. In section 6, we consider more general Hilbert polynomials associated to derived functors of torsion functor. In this section, we provide some conditions on ideal I and module M under which the associated Hilbert polynomials have degree exactly d1. Finally, in section 7, we consider the case of integral closure filtration when the ring A is analytically unramified Cohen-Macaulay. We prove in Theorem 7.5 that in this case again, the degree of the associated Hilbert polynomial attains the upper bound of d1.

Section snippets

Preliminaries

Throughout this section, we assume that (A,m) is a Noetherian local ring with residue field k=A/m and M a finitely generated A-module. The nth Betti number of A-module M is denoted by βnA(M) while the nth Bass number of M is denoted by μAn(M). We first define the notions of complexity and curvature of modules. We also mention some of their basic properties needed in this paper. For detailed proofs and other additional information, see [2, Section 4].

One dimensional case

We now study the growth of εIi(x) in the case when (A,m) is a Gorenstein local ring of dimension d=1 and I is an ideal of analytic spread l(I)=1. We first need a lemma. The proof of the following Lemma 3.1 was suggested by the referee and it is considerably simpler than our previous proof.

Lemma 3.1

Let (A,m) be Cohen-Macaulay local ring of dimension 1. Let I be an m-primary ideal. If μ(In)=1 for some n1 then I is a principal ideal.

Proof

We may assume that the residue field of A is infinite. Suppose In=(y)

General case

We now do the general case where (A,m) is a Gorenstein local ring of dimension d1. Recall from 2.7 that LI(M) is a graded R(I)-module.

Lemma 4.1

Let (A,m) be a Gorenstein local ring of dimension d. Let M be a finitely generated A-module with projdimM<. For id+1 we haveExtAi(k,LI(M)(1))ExtAi+1(k,R(I,M))

Proof

Since A is Gorenstein, projdimM< gives injdimM<. Consider the following short exact sequence of graded R(I)-modules0R(I,M)M[t]LI(M)(1)0 Applying ExtAi(k,) we getExtAi(k,M[t])ExtAi(k,LI(M)(1))

Betti numbers of A/In

Let (A,m) be a Cohen-Macaulay local ring of dimension d1 with infinite residue field k. We consider the following numerical functionn(ToriA(k,A/In)) It is well known that this numerical function is given by a polynomial for n>>0 of degree at most d1 for i1, see [15, Theorem 2]. We denote this polynomial by tI,iA(k,z). Note that (ToriA(k,A/In)) gives the ith Betti number of A/In. We show that if curv(In)>1 for all n1 then for id1, degtI,iA(k,z)=d1.

Lemma 5.1

Let (A,m) be a Cohen-Macaulay local

Hilbert polynomials associated to derived functors

Let (A,m) be a Noetherian Cohen-Macaulay local ring of dimension d1 with infinite residue field k and I be an m-primary ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. We now consider the following numerical function for i1,n(ToriA(M,A/In+1A)) It is known from [15, Theorem 2] that this function coincides with a polynomial denoted by tI,iA(M,n) for n>>0 of degree at most d1. We now recall the notion of reduction of an ideal. We say that JI is a reduction of I if

Integral closure filtration

Suppose I={In} is an admissible I-filtration of m-primary ideals in A. Then as in the I-adic case, the numerical function n(ToriA(M,A/In)) for any i1 is given by a polynomial for n>>0, denoted by tI,iA(M,z). When A is analytically unramified and I is an ideal of A then by a theorem of D. Rees [19], it is known that I={In} is an admissible I-filtration.

Proposition 7.1

Let (A,m) be analytically unramified Cohen-Macaulay local ring of dimension 1 and I an m-primary ideal in A. Let M be a non-free maximal

Acknowledgement

We thank the referee for many pertinent comments. The first author would like to thank DST-SERB for the financial assistance under the project ECR/2017/00790.

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The first author thanks DST-SERB for financial support under the project ECR/2017/00790.

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