Vanishing of cohomology over Cohen--Macaulay rings

TL;DR

This paper investigates the transfer of the AC property related to cohomology vanishing in Cohen-Macaulay rings, demonstrating its preservation under common algebraic operations and providing new examples of such rings.

Contribution

It establishes that the AC property is preserved under standard local algebra procedures in Cohen-Macaulay rings and introduces new examples of Cohen-Macaulay AC rings.

Findings

AC property is preserved by local homomorphisms in Cohen-Macaulay rings

New examples of Cohen-Macaulay AC rings are constructed

Results enhance understanding of homological properties of AC rings

Abstract

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings - colloquially called AC rings - that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen--Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen-Macaulay AC rings.