Tate cohomology with respect to semidualizing modules

TL;DR

This paper explores Tate cohomology relative to semidualizing modules over commutative noetherian rings, establishing new classes of modules with Tate resolutions and proving a general balance theorem.

Contribution

It introduces a framework for Tate cohomology with respect to semidualizing modules and analyzes the relationship between relative and Tate cohomology, including a balance result.

Findings

Identified classes of modules admitting Tate resolutions

Analyzed interaction between relative and Tate cohomology modules

Proved a general balance theorem for Tate cohomology

Abstract

We investigate Tate cohomology of modules over a commutative noetherian ring with respect to semidualizing modules. We identify classes of modules admitting Tate resolutions and analyze the interaction between the corresponding relative and Tate cohomology modules. As an application of our approach, we prove a general balance result for Tate cohomology. Our results are based on an analysis of Tate cohomology in abelian categories.