On modules with finite reducing Gorenstein dimension

TL;DR

This work extends Holm's result by replacing classical Gorenstein and projective dimensions with their reducing counterparts. It proves that if a nonzero finitely generated module $M$ over a local ring $R$ has finite injective dimension, then the reducing Gorenstein-dimension and reducing-projective-dimension coincide, i.e., $ ext{red-G-dim}_R(M)= ext{red-pd}_R(M)$, and derives that $R$ is Gorenstein under additional hypotheses such as $ ext{red-G-dim}_R(M) le 1$ or finite reducing dimension with $ ext{depth}_R(M) ge d-1$. The approach develops a framework of reducing dimensions, establishes key propositions via Ext-vanishing and reduction mod a non-zero-divisor, and uses dimension-based induction (with Omega, pushouts, and Taksyz-type arguments) to extend Foxby-Holm phenomena to this finer invariant setting. The results yield concrete consequences for low-dimensional rings (e.g., 1- and 2-dimensional cases) and deepen the understanding of when finiteness conditions on modules enforce Gorenstein properties of the ambient ring.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Example 2.5
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9
• proof : Proof of Theorem \ref{['mainthm']}
• Lemma 3.1
• proof