A Rigidity Theorem for Ext

Abstract

The goal of this paper is to show that if $R$ is an unramified hypersurface, if $M$ and $N$ are finitely generated $R$ modules, and if ${\mathrm{Ext}}_R^n(M,N)=0$ for some $n\le \mathrm{grade}M$, then ${\mathrm{Ext}}_R^i(M,N)=0$ for $i\le n$. A corollary of this says that ${\mathrm{Ext}}_R^i(M,M)\ne 0$ for $i\le \mathrm{grade}M$ and $M\ne 0$. These results are related to a question of Jorgensen and results of Dao.