Local homology and Gorenstein flat modules

TL;DR

This paper explores the relationship between local homology and Gorenstein flat modules over Noetherian rings, establishing isomorphisms and bounds that deepen understanding of Gorenstein homological properties.

Contribution

It introduces a natural isomorphism linking local homology of complexes to Gorenstein flat modules, providing new bounds and criteria in Gorenstein homological algebra.

Findings

Established a natural isomorphism in derived categories.

Bound the Gorenstein flat dimension of local homology.

Provided a criterion for regularity of Gorenstein local rings.

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $\mathcal{D}(R)$ denote the derived category of $R$-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let $X$ be a homologically bounded to the right complex and $Q$ a bounded to the right complex of Gorenstein flat $R$-modules such that $Q$ and $X$ are isomorphic in $\mathcal{D}(R)$. We establish a natural isomorphism ${\bf L}\Lambda^{\fa}(X)\simeq \Lambda^{\fa}(Q)$ in $\mathcal{D}(R)$ which immediately asserts that $\sup {\bf L}\Lambda^{\fa}(X)\leq \Gfd_RX$. This isomorphism yields several consequences. For instance, in the case $R$ possesses a dualizing complex, we show that $\Gfd_R {\bf L}\Lambda^{\fa}(X)\leq \Gfd_RX$. Also, we establish a criterion for regularity of Gorenstein local rings.