Ext functors, support varieties and Hilbert polynomials over complete intersection rings

TL;DR

This paper investigates the asymptotic behavior of Ext functors over complete intersection rings, establishing invariants related to support varieties and Hilbert polynomials, with implications for regularity and syzygy properties.

Contribution

It introduces new invariants for modules over complete intersection rings based on Ext functor degrees and links these to support varieties and regularity bounds.

Findings

The degrees of Ext polynomial functions stabilize for large indices.

The invariant r^I(M) depends only on the ideal, ring, and support variety.

Boundedness of regularity for certain syzygies when r^I(M) ≤ 0.

Abstract

Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d \geq 1$ and codimension $c \geq 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i \geq 1$ let $\psi_i^I(M)$ be the degree of the polynomial type function $n \rightarrow \ell(Ext^i_A(M, A/I^n))$. We show that for $j = 0, 1$ and for all $i \gg 0$ we have $\psi_{2i +j}^I(M)$ is a constant and let $r_0^I(M)$ and $r_1^I(M)$ denote these constant values. Set $r^I(M) = \max\{ r_0^I(M), r_1^I(M) \}$. We show that $r^I(M)$ is an invariant of $I, A$ and the support variety of $M$. We set the degree of the zero polynomial to be $-\infty$. If $r^I(M) \leq 0$ then we show that $reg \ G_I(\Omega^i(M))$ for $i \geq 0$ is bounded. We give an application of this result to syzgetic Artin-Rees property of $M$. We also give several examples which illustrate our results.