Homological properties and finiteness of reducing invariants

TL;DR

This paper investigates homological properties and finiteness conditions of reducing invariants in modules, establishing key inequalities and conditions under which modules satisfy important homological conjectures.

Contribution

It introduces new results linking finite reducing invariants to classical homological conditions like the Auslander and Reiten conjectures.

Findings

Established grade inequalities for modules with finite reducing projective dimension.

Proved modules satisfy key homological conditions if they have finite reducing invariants.

Connected finiteness of reducing invariants to the validity of major homological conjectures.

Abstract

We study reducing invariants of modules related to certain homological properties. For modules of finite reducing projective dimension, we establish grade inequalities. We prove that if $\mathbb{P}$ is the (uniform) Auslander condition, or the generalized Auslander--Reiten conjecture, or dependence of the total reflexivity conditions, then a module satisfies $\mathbb{P}$ provided that it has finite reducing invariant with respect to $\mathbb{P}$.