Polynomial growth of Betti sequences over local rings

TL;DR

This paper investigates the polynomial growth patterns of Betti number sequences over complete intersection local rings, establishing conditions under which even and odd subsequences share polynomial formulas and providing new proofs and bounds.

Contribution

It introduces new criteria for the coincidence of Betti polynomial subsequences and offers a novel proof of existing results, enhancing understanding of Betti sequence behavior.

Findings

Betti subsequences are polynomial with equal leading terms under certain ideal height conditions.

The degree bounds for the equality of Betti polynomial terms are optimal.

A new proof for the degree bound result is provided, based on intrinsic ring characterizations.

Abstract

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, $R$. The subsequences $\{\beta^R_{i}(M)\}_{i\geq 0}$ with even, respectively, odd $i$ are known to be eventually given by polynomials in i with equal leading terms. We show that these polynomials coincide if $I^\square$, the ideal generated by the quadratic relations of the associated graded ring of $R$, satisfies ${\rm height}\ I^\square \ge {\rm codim}\ R -1$, and that the converse holds if $R$ is homogeneous or ${\rm codim}\ R \le 4$. Subsequently Avramov, Packauskas, and Walker proved that the terms of degree $j > {\rm codim}\ R - {\rm height}\ I^\square$ of the even and odd Betti polynomials are equal. We give a new proof of that result, based on an intrinsic characterization of residue rings of c.i. local rings of minimal multiplicity obtained in this paper. We also show that that bound is optimal.