Totally Acyclic Complexes

Abstract

We prove first (Proposition 3) that, over any ring $R$, an acyclic complex of projective modules is totally acyclic if and only if the cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. The dual result for injective and Gorenstein injective modules also holds over any ring $R$ (Proposition 4). And, when $R$ is a GF-closed ring, the analogue result for flat/Gorenstein flat modules is also true (Proposition 5). Then we show (Theorem 2) that over a left noetherian ring $R$, a third equivalent condition can be added to those in Proposition 4, more precisely, we prove that the following are equivalent: 1. Every acyclic complex of injective modules is totally acyclic. 2. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 3. Every complex of Gorenstein injective modules is dg-Gorenstein injective. Theorem 3 shows that the analogue result for complexes of flat and Gorenstein flat modules holds over any left coherent ring $R$. We prove (Corollary 1) that, over a commutative noetherian ring $R$, the equivalent statements in Theorem 3 hold if and only if the ring is Gorenstein. We also prove (Theorem 4) that when moreover $R$ is left coherent and right $n$-perfect (that is, every flat right $R$-module has finite projective dimension $\leq n$) then statements 1, 2, 3 in Theorem 2 are also equivalent to the following: 4. Every acyclic complex of projective right $R$-modules is totally acyclic. 5. Every acyclic complex of Gorenstein projective right $R$-modules is in $\widetilde{\mathcal{GP}}$. 6. Every complex of Gorenstein projective right $R$-modules is dg-Gorenstein projective. Corollary 2 shows that when $R$ is commutative noetherian of finite Krull dimension, the equivalent conditions (1)-(6) from Theorem 4 are also equivalent to those in Theorem 3 and hold if and only if $R$ is an Iwanaga-Gorenstein ring. Thus we improve slightly on a result of Iyengar's and Krause's; in [22] they proved that for a commutative noetherian ring $R$ with a dualizing complex, the class of acyclic 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 and add more equivalent conditions. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 7) 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 an Iwanaga-Gorenstein ring.