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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.

##### MSC:
 14B05 Singularities in algebraic geometry
##### Citations:
Zbl 0451.05026; Zbl 0538.14033
Full Text:
##### References:
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