Rational singularities and almost split sequences. (English) Zbl 0594.20030

Auslander, Maurice

Trans. Am. Math. Soc. 293, 511-531 (1986).

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

Keywords:

action; group of \({\mathbb{C}}\)-algebra automorphisms; reflexive modules; almost split sequences; Auslander-Reiten quiver; McKay graph; desingularization graph; singularity

References:

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