Reflexive modules over rational double points. (English) Zbl 0553.14001

Introduction: Let \(0\) denote a complete local algebra of dimension 2 with maximal ideal \({\mathfrak m}\) over an algebraically closed field k, such that \(0/{\mathfrak m}\simeq k\). Assume that the spectrum of \(0\) is a rational double point. The object of this note is to describe the isomorphism classes of reflexive \(0\)-modules which are indecomposable. In characteristic zero, such a ring is the ring of invariants of a finite subgroup \(G\subset SL_ 2(k)\), operating linearly on a power series ring k[[x,y]]. In this case, the McKay correspondence [cf. J. McKay, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026) and G. Gonzalez-Sprinberg and J.-L. Verdier, Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033)] establishes bijective correspondences between the three sets: (i) isomorphism classes of indecomposable reflexive modules; (ii) vertices of the extended Dynkin diagram associated to \(0\) and to G, (iii) isomorphism classes of irreducible representations of G. - Since the group G is not always available in characteristic p, it seems of some interest to derive the correspondence (i)\(\leftrightarrow (ii)\) directly, without assumption on the residue characteristic.


14B05 Singularities in algebraic geometry
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