On modules with finite reducing Gorenstein dimension

TL;DR

This paper explores whether modules with finite injective and reducing Gorenstein dimensions over Noetherian local rings imply the ring is Gorenstein, extending known results for classical Gorenstein dimension.

Contribution

It investigates the relationship between finite reducing Gorenstein dimension and the Gorenstein property of the ring, generalizing previous results.

Findings

Finite Gorenstein dimension implies finite projective dimension.

Reducing Gorenstein dimension is a finer invariant than classical Gorenstein dimension.

The paper provides conditions under which the ring is Gorenstein.

Abstract

If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective dimension, and hence a result of Foxby implies that $R$ is Gorenstein. We investigate whether the same conclusion holds for nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general.