A study of Tate homology via the approximation theory with applications to the depth formula

TL;DR

This paper explores Tate homology using approximation theory, generalizing existing results, and applies these findings to establish new conditions under which the depth formula holds for modules with finite Gorenstein or injective dimensions.

Contribution

It generalizes a key result connecting absolute, relative, and Tate Tor modules using Auslander-Buchweitz approximation theory and extends the depth formula to broader module classes.

Findings

Established a new exact sequence linking Tor modules.

Generalized the depth formula for modules with finite Gorenstein dimension.

Provided new sufficient conditions for the depth formula to hold.

Abstract

In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.