Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

TL;DR

This paper investigates the asymptotic behavior of Ext functors for modules with finite complete intersection dimension over local rings, providing improved bounds on vanishing conditions especially in low-dimensional cases.

Contribution

It introduces a generalized pairing extending Buchweitz's Herbrand difference and uses it to refine bounds on the vanishing of Ext modules for modules of finite complete intersection dimension.

Findings

Provides new bounds on the number of consecutive Ext vanishings needed for eventual total vanishing.

Generalizes Buchweitz's Herbrand difference to a broader setting.

Improves known results particularly for rings of dimension at most two.

Abstract

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules $\Ext_R^i(M,N)$ which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of $\Ext_R^i(M,N)$ needed to ensure that $\Ext_R^i(M,N)=0$ for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two.