# Reducing invariants and total reflexivity

## TL;DR

This paper introduces reducing invariants for modules over local rings, providing criteria for total reflexivity and characterizing Gorenstein rings via the reducing Gorenstein dimension of the canonical module.

## Contribution

It develops reducing versions of invariants for modules, linking finite reducing Gorenstein dimension to total reflexivity and Gorenstein properties of rings.

## Key findings

- Modules with finite reducing Gorenstein dimension are totally reflexive if Ext vanishes.
- A Cohen-Macaulay local ring with canonical module is Gorenstein iff the canonical module has finite reducing Gorenstein dimension.
- Provides examples and applications of reducing invariants in commutative algebra.

## Abstract

Motivated by a recent result of Yoshino, and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over commutative Noetherian local rings. Our main result considers modules which have finite reducing Gorenstein dimension, and determines a criterion for such modules to be totally reflexive in terms of the vanishing of Ext. Along the way we give examples and applications, and in particular, prove that a Cohen-Macaulay local ring with canonical module is Gorenstein if and only if the canonical module has finite reducing Gorenstein dimension.