On an Instance of the Small Cohen-Macaulay Conjecture

TL;DR

This paper offers a simplified proof of a theorem demonstrating that certain 3-dimensional quasi-Gorenstein local rings admit small Cohen-Macaulay modules, advancing understanding in commutative algebra.

Contribution

It provides a more straightforward proof of a known theorem regarding small Cohen-Macaulay modules in specific quasi-Gorenstein rings.

Findings

Simplified proof of the theorem by Tavanfar and Shimomoto.

Establishment that certain 3-dimensional quasi-Gorenstein rings admit small Cohen-Macaulay modules.

Clarification of conditions under which these modules exist.

Abstract

We provide a simplified proof of a theorem proved by Tavanfar and Shimomoto which states that a quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein local ring $(A,m,k)$ with $H^2_m(A)=k$ admits a small Cohen-Macaulay module.