An extension of a depth inequality of Auslander

TL;DR

This paper extends a known depth inequality for Tor-rigid modules over local rings to 2-Tor-rigid modules and identifies new classes of Tor-rigid modules over hypersurfaces using Dao's eta function.

Contribution

It generalizes Auslander's depth inequality to 2-Tor-rigid modules and introduces new classes of Tor-rigid modules over hypersurfaces.

Findings

Extended depth inequality to 2-Tor-rigid modules.

Identified new classes of Tor-rigid modules over hypersurfaces.

Confirmed the inequality for generically free 2-Tor-rigid modules.

Abstract

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.