On flat and Gorenstein flat dimensions of local cohomology modules

TL;DR

This paper investigates the flat and Gorenstein flat dimensions of local cohomology modules over Noetherian local rings, providing bounds and characterizations related to Cohen-Macaulay modules, dualizing modules, and Gorenstein rings.

Contribution

It establishes bounds on the flat and Gorenstein flat dimensions of local cohomology modules and characterizes key classes of modules and rings using these dimensions.

Findings

Bound $d_R ext{ or } ext{Gfd}_R$ of local cohomology modules by the original module's dimension plus the cohomology degree.

Equality of dimensions holds for finitely generated modules.

Characterizations of Cohen-Macaulay modules, dualizing modules, and Gorenstein rings based on these dimensions.

Abstract

Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that $\H_{\fa}^i(M)=0$ for all $i\neq n$. It is proved that if $\T(X)<\infty$, then $\T(\H_{\fa}^n(M))\leq\T(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.