Two theorems about maximal Cohen-Macaulay modules

Craig Huneke, Graham J. Leuschke

Research output: Contribution to journalArticle

68 Scopus citations

Abstract

This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen-Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macanlay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen-Macaulay local ring of finite Cohen-Macaulay type is again of finite Cohen-Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of R1/pe divided by pde has a positive limit. This result relates to work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties.

Original languageEnglish (US)
Pages (from-to)391-404
Number of pages14
JournalMathematische Annalen
Volume324
Issue number2
DOIs
StatePublished - 2002

ASJC Scopus subject areas

  • Mathematics(all)

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