Some characterizations of local rings via reducing dimensions

TL;DR

This paper introduces reducing homological dimensions to characterize Gorenstein and complete intersection local rings, extending classical results and exploring connections with module complexity.

Contribution

It provides new characterizations of Gorenstein and complete intersection rings using reducing homological dimensions, generalizing previous theorems.

Findings

A local ring is Gorenstein iff all finitely generated modules have finite reducing Gorenstein dimension.

Established links between complexity and reducing projective dimension.

Extended classical results of Auslander and Bridger.

Abstract

In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings via reducing homological dimensions. For example, we extend a classical result of Auslander and Bridger, and prove that a local ring is Gorenstein if and only if each finitely generated module over it has finite reducing Gorenstein dimension. Along the way, we prove various connections between complexity and reducing projective dimension of modules.