A Rigidity Theorem for Ext

TL;DR

This paper proves a rigidity theorem for Ext modules over unramified hypersurfaces, showing vanishing of Ext groups under certain conditions and confirming non-vanishing in specific cases, advancing understanding in commutative algebra.

Contribution

It establishes a new rigidity result for Ext modules over unramified hypersurfaces, extending previous work and answering a question posed by Jorgensen.

Findings

Ext vanishing extends from n to all i ≤ n under given conditions

Non-vanishing of Ext for M with itself up to grade(M)

Results relate to and build upon Dao's work

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 $\operatorname{Ext}_{R}^{n}(M,N)=0$ for some $n\leq\operatorname{grade}{M}$, then $\operatorname{Ext}_{R}^{i}(M,N)=0$ for $i\leq n$. A corollary of this says that $\operatorname{Ext}_{R}^{i}(M,M)\neq 0$ for $i\leq\operatorname{grade}{M}$ and $M\neq 0$. These results are related to a question of Jorgensen and results of Dao.