Vanishing of stable homology with respect to a semidualizing module

TL;DR

This paper explores the conditions under which stable homology vanishes relative to a semidualizing module, revealing that such vanishing implies the module is trivial and the ring is Gorenstein.

Contribution

It extends existing results on stable homology vanishing by establishing new conditions that link the triviality of the semidualizing module to the Gorenstein property of the ring.

Findings

Vanishing of stable homology implies the semidualizing module is trivial.

The ring must be Gorenstein if stable homology vanishes with respect to the module.

Results generalize and improve previous vanishing theorems.

Abstract

We investigate stable homology of modules over a commutative noetherian ring $R$ with respect to a semidualzing module $C$, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces $C$ to be trivial and $R$ to be Gorenstein.