A generalized Dade's Lemma for local rings

TL;DR

This paper extends Dade's Lemma to quotients of local rings by regular sequences, providing a necessary and sufficient condition for the vanishing of higher Ext or Tor between modules, based on their behavior over simpler quotient rings.

Contribution

It introduces a generalized version of Dade's Lemma applicable to a broader class of local rings with regular sequence ideals, linking module vanishing conditions across related quotient rings.

Findings

Established a necessary and sufficient condition for vanishing Ext and Tor.

Connected module properties over the original ring to those over quotient rings.

Extended classical results to a more general algebraic setting.

Abstract

We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence.