Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules

TL;DR

This paper investigates conditions under which certain local rings admit maximal Cohen-Macaulay modules, extending known results and exploring the behavior of Frobenius pushforwards in relation to the small Cohen-Macaulay conjecture.

Contribution

It establishes that quasi-Gorenstein deformations of specific 3-dimensional rings admit maximal Cohen-Macaulay modules and examines Frobenius pushforwards in this context.

Findings

Quasi-Gorenstein deformations of certain rings admit maximal Cohen-Macaulay modules.

Includes examples of unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel.

Analyzes when Frobenius pushforwards contain maximal Cohen-Macaulay summands.

Abstract

We show that any quasi-Gorenstein deformation of a $3$-dimensional quasi-Gorenstein Buchsbaum local ring with $I$-invariant $1$ admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings includes two instances of unique factorization domains constructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart from this result, motivated by the small Cohen-Macaulay conjecture in prime characteristic, we examine a question about when the Frobenius pushforward $F^e_*(M)$ of an $R$-module $M$ comprises a maximal Cohen-Macaulay direct summand in both local and graded cases.