Acyclic complexes and regular rings

TL;DR

This paper broadens the 2009 characterization of noetherian regular rings by showing that core properties of complexes of projective, flat, and injective modules imply equivalent regularity criteria without assuming regularity a priori, and it identifies coherent regular and von Neumann regular rings through these complex-theoretic conditions. It develops parallel cotorsion-pair frameworks for flat-cotorsion and fp-injective modules, establishing a network of equivalences among semi-projective/flat/injective, contractible, and pure-acyclic conditions in complexes. By separating regularity from the homological behavior of complexes, the authors provide new criteria for regularity that apply under coherence, and extend the duality between projective/injective and flat/cotorsion perspectives within the derived-category setting.

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.

Theorems & Definitions (31)

• Proposition 1.3
• proof : Proof
• Proposition 1.5
• proof : Proof
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• proof : Proof
• Example 2.2
• Remark 2.3