On Auslander`s depth formula

TL;DR

This paper extends the validity of Auslander`s depth formula from Cohen-Macaulay local rings of dimension 1 to rings of higher dimension and depth, under certain conditions.

Contribution

It generalizes the conditions under which Auslander`s depth formula holds, from dimension 1 to higher dimensions and depths, for Tor-independent modules.

Findings

Depth formula holds for Cohen-Macaulay rings of dimension 1

Extension of depth formula validity to higher dimensions and depths

Applicable to modules with finite Cohen-Macaulay dimension

Abstract

We show that if Auslander`s depth formula holds for non-zero Tor-independent modules over Cohen-Macaulay local rings of dimension 1, then it holds for such modules over any Cohen-Macaulay local ring. More generally, we show that the depth formula for non-zero Tor-independent modules which have finite Cohen-Macaulay dimension over depth 1 local rings implies the depth formula for such modules over any positive depth local ring.