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$\Ext$-finite modules were introduced and studied by
Enochs and Jenda. We prove under some conditions that the depth of a
local ring is equal to the sum of the Gorenstein injective dimension
and $\Tor$-$\depth$ of an $\Ext$-finite module of finite Gorenstein
injective dimension. Let $(R,\fm)$ be a local ring. We say that an
$R$-module $M$ with $\dim_R M=n$ is a \emph{Grothendieck module} if
the $n$-th local cohomology module of $M$ with respect to $\fm$,
$\H_{\fm} ^n (M)$, is non-zero. We prove the Bass formula for this
kind of modules of finite Gorenstein injective dimension and of
maximal Krull dimension. These results are dual versions of the
Auslander-Bridger formula for the Gorenstein dimension. We also
introduce $GF$-perfect modules as an extension of quasi-perfect
modules introduced by Foxby.
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