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 $F_{\ast }^e(M)$ of an $R$-module $M$ comprises a maximal Cohen-Macaulay direct summand in both local and graded cases.

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References in corpus (2)

• Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristic
• On the existence of birational maximal Cohen-Macaulay modules over biradical extensions in mixed characteristic