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

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 b...

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Bibliographic Details
Published in:	Journal of algebra Vol. 634; pp. 667 - 697
Main Authors:	Shimomoto, Kazuma, Tavanfar, Ehsan
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 15-11-2023
Subjects:	Ample [formula omitted]-Weil divisors Buchsbaum rings Frobenius direct images Frobenius pushforward Graded rings/modules Maximal Cohen-Macaulay modules Quasi-Gorenstein rings Small Cohen-Macaulay conjecture Unique factorization domain Small Cohen-Macaulay conjecture Quasi-Gorenstein rings Buchsbaum rings Unique factorization domain Maximal Cohen-Macaulay modules Frobenius direct images 13D22 13D45 Ample Q-Weil divisors 13A35 Graded rings/modules Frobenius pushforward
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Summary:	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.
ISSN:	0021-8693 1090-266X
DOI:	10.1016/j.jalgebra.2023.06.045