Totally acyclic complexes and homological invariants

Abstract: In this paper, we study equivalent characterizations of the condition that every acyclic complex of projective (resp., injective and flat) modules is totally acyclic over a general ring R. This line of inquiry was initiated by Iyengar and Krause in 2006 for commutative Noetherian rings with dualizing complexes. We demonstrate that certain equivalent conditions are closely related to the invariants silp(R) and spli(R) defined by Gedrich and Gruenberg, as well as to the invariant sfli(R) defined by Ding and Chen. We also examine some sufficient conditions for the equality spli(R) = silp(R), that leads to a generalization of a result by Ballas and Chatzistavridis that was originally proved in the case that R is a left (and right) coherent ring which is isomorphic with its opposite ring. Finally, we provide examples to illustrate relations among these conditions.