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The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity

Tom 93 / 2011

Trond Stølen Gustavsen, Runar Ile Banach Center Publications 93 (2011), 41-50 MSC: Primary 14B12; Secondary 14D20, 13C14, 14J17. DOI: 10.4064/bc93-0-3

Streszczenie

Let $X$ be a quotient surface singularity, and define $\mathbf{G}% ^{\mathop{\rm def}}(X,r)$ as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of $\mathbf{G}% ^{\mathop{\rm def}}(X,r)$ is equal to the order of the divisor class group of $X,$ and when $X$ is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity $X,$ we prove the conjecture under an additional cancellation assumption. We discuss the deformation relation in some examples, and in particular we give all deformations of an indecomposable MCM module on a rational double point.

Autorzy

  • Trond Stølen GustavsenDepartment of Education
    Buskerud University College
    P.O. Box 7053
    N-3007 Drammen, Norway
    e-mail
  • Runar IleDeparment of Mathematics
    University of Bergen
    P.O. Box 7803
    N-5020 Bergen, Norway
    e-mail

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