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Let 
$\mathfrak{a}$ 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 
$\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and 
${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$. Let 
$M$ be an $R$-module such that 
$\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all 
$i\,\ne \,n$. It is proved that if 
$T\left( M \right)\,<\,\infty$, then 
$\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$, 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.
