A Study of quasi-Gorenstein rings II: Deformation of quasi-Gorenstein property

TL;DR

This paper explores whether the quasi-Gorenstein property of a local ring is preserved under deformation, providing positive results under certain conditions and counterexamples in general.

Contribution

It offers new insights into the deformation behavior of quasi-Gorenstein rings, especially without Cohen-Macaulay assumptions.

Findings

Positive deformation results under specific assumptions

Counterexamples showing failure without additional conditions

Analysis of the role of Cohen-Macaulay property in deformation

Abstract

In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires some delicate studies.