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In this paper we generalize Artin-Verdier, Esnault and Wunram construction ofMcKay correspondence to arbitrary Gorenstein surface singularities. The keyidea is the definition and a systematic use of a degeneracy module, which is anenhancement of the first Chern class construction via a degeneracy locus. Westudy also deformation and moduli questions. Among our main result we quote: afull classification of special reflexive MCM modules on normal Gorensteinsurface singularities in terms of divisorial valuations centered at thesingularity, a first Chern class determination at an adequate resolution ofsingularities, construction of moduli spaces of special reflexive modules, acomplete classification of Gorenstein normal surface singularities inrepresentation types, and a study on the deformation theory of MCM modules andits interaction with their pullbacks at resolutions. For the proof of thesetheorems we prove several isomorphisms between different deformation functorsthat we expect that will be useful in further work.

Reflexive modules on normal Gorenstein Stein surfaces, their deformations and moduli