Bass numbers over local rings via stable cohomology

TL;DR

This paper investigates the behavior of Ext modules over local rings, revealing a key distinction based on regularity, and employs stable cohomology to compute Bass series in specific cases.

Contribution

It establishes a criterion for the vanishing of Ext maps over local rings using stable cohomology, providing new insights into Bass numbers and series.

Findings

Ext map is non-zero over regular local rings

Ext map is zero over non-regular local rings

Applications include explicit Bass series computations

Abstract

For any non-zero finite module M of finite projective dimension over a noetherian local ring R with maximal ideal m and residue field k, it is proved that the natural map Ext_R(k,M)-->Ext_R(k,M/mM) is non-zero when R is regular and is zero otherwise. A noteworthy aspect of the proof is the use of stable cohomology. Applications include computations of Bass series over certain local rings.