Rings that are homologically of minimal multiplicity

TL;DR

This paper explores whether local Cohen-Macaulay rings with canonical modules exhibiting polynomial growth in Betti numbers are Gorenstein, extending known results to a broader class called homologically of minimal multiplicity.

Contribution

It introduces the class of rings homologically of minimal multiplicity, characterizes them, and proves ascent and descent properties related to their Gorenstein status.

Findings

Characterizations of rings homologically of minimal multiplicity

Polynomial growth of Betti numbers implies Gorenstein property in this class

Establishment of ascent and descent results for these rings

Abstract

Let R be a local Cohen-Macaulay ring with canonical module \omega_R. We investigate the following question of Huneke: If the sequence of Betti numbers \{\beta_i^R(\omega_R)\} has polynomial growth, must R be Gorenstein? This question is well-understood when R has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide several characterizations of the rings in this class and establish a general ascent and descent result.