On reducing homological dimensions over noetherian rings

TL;DR

This paper introduces new homological invariants for modules over noetherian rings, providing conditions for torsionfreeness, exploring totally reflexive modules, and relating these invariants to classical properties like Gorenstein dimension.

Contribution

It develops the concepts of upper reducing projective and Gorenstein dimensions, establishing their properties and connections to existing homological invariants in noetherian rings.

Findings

Equivalent conditions for torsionfree modules and their transpose

Inequality relating upper reducing projective dimension and complexity

Connections between upper reducing projective dimension and ring properties

Abstract

Let $\Lambda$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate totally reflexive modules and reducing Gorenstein dimension. Next, we introduce homological invariants for $\Lambda$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $\Lambda$ is commutative and local. Using it, we consider how upper reducing projective dimension relates to reducing projective dimension, and the complete intersection and AB properties of a commutative noetherian local ring.