Totally acyclic complexes and homological invariants over arbitrary rings

TL;DR

This paper explores the properties of totally acyclic complexes over arbitrary rings, linking them to homological invariants and extending key results to non-commutative rings.

Contribution

It provides new characterizations of acyclic complexes, relates homological invariants, and generalizes existing theorems to non-commutative rings.

Findings

Conditions for equality spli(R) = silp(R) are established.

Characterizations of Iwanaga-Gorenstein rings are extended to non-commutative rings.

Relationships among acyclic complexes and homological invariants are clarified.

Abstract

In this paper, we investigate equivalent characterizations of the condition that every acyclic complex of projective, injective, or flat modules is totally acyclic over a general ring R. We provide examples to illustrate relationships among these conditions and show that several are closely tied to the homological invariants silp(R), spli(R) and sfli(R). We also give sufficient conditions for the equality spli(R) = silp(R), thereby refining results due to Ballas-Chatzistavridis and Wang-Yang. Further, we extend a result of Christensen-Foxby-Holm on characterizations of Iwanaga-Gorenstein rings to the non-commutative setting. This generalizes a theorem of Estrada-Fu-Iacob, offering additional equivalent characterizations under a general assumption while also yielding characterizations of the Nakayama conjecture.