Vanishing of cohomology over complete intersection rings

TL;DR

This paper investigates how the vanishing of Ext modules over complete intersection rings constrains the complete intersection dimension of modules, providing new bounds and estimates based on cohomological vanishing patterns.

Contribution

It establishes a link between the vanishing of Ext modules and bounds on the complete intersection dimension of modules over complete intersection rings, including estimates for dual modules.

Findings

Vanishing of Ext^i_R(M,N) for consecutive i bounds the CI-dimension of M.

Provides estimates for the CI-dimension of the dual module M^*.

Shows that cohomological vanishing imposes structural constraints on modules over complete intersection rings.

Abstract

Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n-1. We also estimate the complete intersection dimension of the dual of M, in terms of vanishing of the cohomology modules, Ext^i_R(M,N).