A Note on Symmetry in the Vanishing of Ext

TL;DR

This paper generalizes a symmetry property of the vanishing of Ext modules over complete intersection rings, extending the conditions under which vanishing of Ext in one direction implies vanishing in the opposite direction.

Contribution

It extends the known symmetry result of Ext vanishing to new cases involving modules with different finiteness and completeness conditions.

Findings

Proves symmetry of Ext vanishing when M is finitely generated and N is arbitrary.

Establishes symmetry when N has finite length and M is arbitrary.

Shows symmetry when M is complete and N is finitely generated.

Abstract

Avramov and Buchweitz proved that for finitely generated modules $M$ and $N$ over a complete intersection local ring $R$, $\Ext^i_R(M,N)=0$ for all $i\gg 0$ implies $\Ext^i_R(N,M)=0$ for all $i\gg 0$. In this note we give some generalizations of this result. Indeed we prove the above mentioned result when (1) $M$ is finitely generated and $N$ is arbitrary, (2) $M$ is arbitrary and $N$ has finite length and (3) $M$ is complete and $N$ is finitely generated.