The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity
Volume 93 / 2011
Abstract
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.
