On flat and Gorenstein flat dimensions of local cohomology modules

Let $\text{fa}$ 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 ${\text{fd}}_{R}X$, ${\text{Gfd}}_{R}X$ and ${\text{GCfd}}_{R}X$ by $\text{T}(X)$. Let $M$ be an $R$-module such that \H_{\fa}^i(M)=0 for all $i\mathrm{\ne }n$. It is proved that if $\text{T}(X)<\mathrm{\infty }$, 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.