Acyclic complexes and Gorenstein rings

Abstract

For a given class of modules $\mathcal{A}$, we denote by $\widetilde{\mathcal{A}}$ the class of exact complexes $X$ having all cycles in $\mathcal{A}$, and by $dw(\mathcal{A})$ the class of complexes $Y$ with all components $Y_j$ in $\mathcal{A}$. We use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (Gorenstein flat, Gorenstein projective respectively) $R$-modules, $\mathcal{DI}$ for Ding injective modules, and $\mathcal{PGF}$ for projectively coresolved Gorenstein flat modules (see section 2 for definitions). We prove that the following are equivalent over any ring $R$: (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. We show that the analogue result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. if moreover, the ring is $n$-perfect for some integer $n \ge 0$, then the three equivalent statements for flat and Gorenstein flat modules are also equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings: Let $R$ be a commutative coherent ring. The following statements are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules. (2) every exact complex of injectives has all its cycles Ding injective modules and every $R$-module M such that $M^+$ is Gorenstein flat is Ding injective. If moreover the ring $R$ has finite Krull dimension then statements (1), (2) above are also equivalent to (3) $R$ is a Gorenstein ring (in the sense of Iwanaga).