Totally acyclic complexes

Abstract

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.