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Rational singularities and almost split sequences. (English) Zbl 0594.20030
Let $$G\subseteq SL(2,{\mathbb{C}})$$ be a finite group and let V be a two- dimensional representation of G. Then V gives an action of G on $$S={\mathbb{C}}[[ X,Y]]$$ as a group of $${\mathbb{C}}$$- algebra automorphisms. It is proved that if V is faithful having no pseudoreflections then the category $$R_ S$$ of reflexive modules over the invariant $${\mathbb{C}}$$-algebra $$S^ G$$ has almost split sequences and they can be derived from a fundamental exact sequence. Moreover, the Auslander-Reiten quiver of $$R_ S$$ is the McKay graph of (G,V) and it is isomorphic to the desingularization graph of the associated singularity. Some of the results above remain valid in more general situations.
Reviewer: D.Simson

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 14J17 Singularities of surfaces or higher-dimensional varieties 13H05 Regular local rings 16Gxx Representation theory of associative rings and algebras 15A72 Vector and tensor algebra, theory of invariants
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