Bass and Betti numbers of A/In
Introduction
Let be a Noetherian local ring of dimension and let I be an arbitrary ideal of A. It is well known that for a finitely generated A-module M the numerical function is given by a polynomial of degree at most for , see [15, Theorem 2]. We denote this polynomial by . When we simply denote this by . Note that is the ith Bass number of . Dually we have the numerical function giving the ith Betti numbers of . Again, by a theorem of Kodiyalam [15, Theorem 2], this is given by a polynomial of degree at most . We denote this polynomial by . 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 and . For instance, in [17, Theorem 18] it was proved that if M is a maximal Cohen-Macaulay A-module and then 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, where is the fiber cone of I and 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 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 for all (see 2.2 for definition of curvature) for which the numerical functions giving the ith Bass numbers and the ith Betti numbers of are polynomials of degree equal to for . It follows from [1, Corollary 5] that there many examples of ideals with for all , 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 be a Gorenstein local ring of dimension . Let I be an ideal of A with and for all . Let be a superficial sequence w.r.t. I and for . Then for , we have . In particular when , we get .
Theorem 1.2 Let be a Cohen-Macaulay ring of dimension . Let I be an ideal with and for all . Then for , is given by a polynomial of degree for .
- (i)
is an I-adic filtration where I is -primary with i.e. ideals having reduction number one and .
- (ii)
is the integral closure filtration of ideals in A where A is analytically unramified ring.
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 in the case when A is Gorenstein local ring of dimension one. In section 4, we proceed to estimate the degree of 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 giving the Betti numbers of is 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 . 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 .
Section snippets
Preliminaries
Throughout this section, we assume that is a Noetherian local ring with residue field and M a finitely generated A-module. The nth Betti number of A-module M is denoted by while the nth Bass number of M is denoted by . 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 in the case when is a Gorenstein local ring of dimension and I is an ideal of analytic spread . 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 be Cohen-Macaulay local ring of dimension 1. Let I be an -primary ideal. If for some then I is a principal ideal. Proof We may assume that the residue field of A is infinite. Suppose
General case
We now do the general case where is a Gorenstein local ring of dimension . Recall from 2.7 that is a graded -module.
Lemma 4.1 Let be a Gorenstein local ring of dimension d. Let M be a finitely generated A-module with . For we have Proof Since A is Gorenstein, gives . Consider the following short exact sequence of graded -modules Applying we get
Betti numbers of
Let be a Cohen-Macaulay local ring of dimension with infinite residue field k. We consider the following numerical function It is well known that this numerical function is given by a polynomial for of degree at most for , see [15, Theorem 2]. We denote this polynomial by . Note that gives the ith Betti number of . We show that if for all then for , .
Lemma 5.1 Let be a Cohen-Macaulay local
Hilbert polynomials associated to derived functors
Let be a Noetherian Cohen-Macaulay local ring of dimension with infinite residue field k and I be an -primary ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. We now consider the following numerical function for , It is known from [15, Theorem 2] that this function coincides with a polynomial denoted by for of degree at most . We now recall the notion of reduction of an ideal. We say that is a reduction of I if
Integral closure filtration
Suppose is an admissible I-filtration of -primary ideals in A. Then as in the I-adic case, the numerical function for any is given by a polynomial for , denoted by . When A is analytically unramified and I is an ideal of A then by a theorem of D. Rees [19], it is known that is an admissible I-filtration. Proposition 7.1 Let be analytically unramified Cohen-Macaulay local ring of dimension 1 and I an -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.