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Totally acyclic complexes

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 tot