Two theorems about maximal Cohen–Macaulay modules

By Craig Huneke, Graham J. Leuschke. 2002

176 citations recorded by the source corpus.

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 lo cal ring of finite Cohen–Macaulay 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 $R^{1/p^e}$ divided by $p^{de}$ 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9d20bcc27fe720818e6bcb56cb6211a8a27aaa5f

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Citations in the corpus

The structure of F-pure rings (2003, background): “Moreover, Huneke and Leuschke proved that if R is local, reduced and Gorenstein, then the F -signature exists.”
The F-Signature and Strong F-Regularity (2002, methodology): “Our formulation differs slightly from the original definition in [7], where it is assumed that k is perfect, or equivalently that α(R) = 0.”
Finite Gorenstein representation type implies simple singularity (2007, methodology): “R) is ﬁnite, then Ris Gorenstein or every module in G(R) is free. As this theorem does not require Rto be complete, we considerably strengthen Theorem A using work of Huneke, Leuschke, and R. Wiegand [19, 22, 30]; this occurs in (4.5). Theorem B was conjectured by R. Takahashi [29], who proved it fo”
Quantifying singularities with differential operators (2018, background): “r of properties that relate to diﬀerent aspects of the singularity of R. For example, (i) s(R) ∈[0,1]. QUANTIFYING SINGULARITIES WITH DIFFERENTIAL OPERATORS 3 (ii) Ris regular if and only if s(R) = 1 [HL02]. (iii) Ris strongly F-regular if and only if s(R) >0 [AL03]. Unfortunately, computing the ”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “In this article, we will be primarily concerned with two important numerical invariants that measure the failure of ﬂatness for the iterated Frobenius: the Hilbert–Kunz multiplicity [Monsky 1983] and the F -signature [Smith and Van den Bergh 1997; Huneke and Leuschke 2002].”