  For a given class of modules $\A$, we denote by $\widetilde{\A}$ the class of
exact complexes $X$ having all cycles in $\A$, and by $dw(\A)$ the class of
complexes $Y$ with all components $Y_j$ in $\A$. We consider a two sided
noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF},
\mathcal{GP})$ for the class of Gorenstein injective (flat, projective
respectively) $R$-modules. We prove (Theorem 1) that the following are
equivalent: 1. Every exact complex of injective modules is totally acyclic. 2.
Every exact complex of Gorenstein injective modules is in
$\widetilde{\mathcal{GI}}$. 3. Every complex in $dw(\mathcal{GI})$ is
dg-Gorenstein injective. Theorem 2 shows that the analogue result for complexes
of flat and Gorenstein flat modules also holds. We prove (Corollary 1) that,
over a commutative noetherian ring $R$, the equivalent statements in Theorem 1
(as well as their counterparts from Theorem 2) hold if and only if the ring is
Gorenstein. Thus we improve on a result of Iyengar's and Krause's; in [18] they
proved that for a commutative noetherian ring $R$ with a dualizing complex, the
class of exact complexes of injectives coincides with that of totally acyclic
complexes of injectives if and only if $R$ is Gorenstein. We are able to remove
the dualizing complex hypothesis. In the second part of the paper we focus on
two sided noetherian rings that satisfy the Auslander condition. We prove
(Theorem 6) that for such a ring $R$ that also has finite finitistic flat
dimension, every complex of injective (left and respectively right) $R$-modules
is totally acyclic if and only if $R$ is a Gorenstein ring.
