Acyclic complexes and regular rings

TL;DR

This paper explores the properties of acyclic complexes over noetherian rings, demonstrating their equivalence in characterizing coherent regular rings and von Neumann regular rings, independent of the ring's regularity.

Contribution

It generalizes previous characterizations by showing that properties of complexes can characterize various regular rings without assuming regularity.

Findings

Acyclic complex properties are equivalent across different module types.

Characterization of coherent regular rings via complex properties.

Identification of von Neumann regular rings through complex conditions.

Abstract

A 2009 paper by Iacob and Iyengar characterizes noetherian regular rings in terms of properties of complexes of projective modules, flat modules, and injective modules. We show that the relevant properties of such complexes are equivalent without reference to regularity of the ring and that they characterize coherent regular rings and von Neumann regular rings.