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Deforming syzygies of liftable modules and generalised Knörrer functors

Runar Ile — 2007

Collectanea Mathematica

Maps between deformation functors of modules are given which generalise the maps induced by the Knörrer functors. These maps become isomorphisms after introducing certain equations in the target functor restricting the Zariski tangent space. Explicit examples are given on how the isomorphisms extend results about deformation theory and classification of MCM modules to higher dimensions.

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

Trond Stølen GustavsenRunar Ile — 2011

Banach Center Publications

Let X be a quotient surface singularity, and define G d e f ( 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 G d e f ( 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...

Representation theory for log-canonical surface singularities

Trond Stølen GustavsenRunar Ile — 2010

Annales de l’institut Fourier

We consider the representation theory for a class of log-canonical surface singularities in the sense of reflexive (or equivalently maximal Cohen-Macaulay) modules and in the sense of finite dimensional representations of the local fundamental group. A detailed classification and enumeration of the indecomposable reflexive modules is given, and we prove that any reflexive module admits an integrable connection and hence is induced from a finite dimensional representation of the local fundamental...

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