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

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.