Abstract:
Let
$R$
be a left Noetherian ring satisfying
the Auslander condition.
It is proven that a left
$R$-module
$M$
is Gorenstein injective
if and only if
$M$
is strongly cotorsion and $\mathrm{Ext}_R^{i\geq1}(I,M)=0$
for any injective
left
$R$-module
$I$;
a right
$R$-module
$M$
is Gorenstein flat if and only if
$M$
is
strongly torsion-free and Tor$^R_{i\geq1}(M,J)=0$
for any injective left
$R$-module
$J$.
We
also prove that if
$R$
is a commutative Noetherian ring with splf $R$
finite, then the
local ring
$R_{\mathfrak{p}}$
is Gorenstein for every prime ideal
$\mathfrak{p}$
of
$R$
if
and only if the cycles of every acyclic complex of PGF-modules are PGF-modules if and only
if every complex of PGF-modules is a dg-PGF complex.
