Auslander, Maurice Rational singularities and almost split sequences. (English) Zbl 0594.20030 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 Cited in 4 ReviewsCited in 76 Documents 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 PDF BibTeX XML Cite \textit{M. Auslander}, Trans. Am. Math. Soc. 293, 511--531 (1986; Zbl 0594.20030) Full Text: DOI References: [1] M. Artin and J.-L. Verdier, Reflexive modules over rational double points, Math. Ann. 270 (1985), no. 1, 79 – 82. · Zbl 0553.14001 · doi:10.1007/BF01455531 · doi.org [2] Maurice Auslander, On the purity of the branch locus, Amer. J. Math. 84 (1962), 116 – 125. · Zbl 0112.13101 · doi:10.2307/2372807 · doi.org [3] Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Dekker, New York, 1978, pp. 1 – 244. Lecture Notes in Pure Appl. Math., Vol. 37. [4] Maurice Auslander and Jon F. Carlson, Almost-split sequences and group rings, J. Algebra 103 (1986), no. 1, 122 – 140. · Zbl 0594.20005 · doi:10.1016/0021-8693(86)90173-0 · doi.org [5] Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. III. Almost split sequences, Comm. Algebra 3 (1975), 239 – 294. · Zbl 0331.16027 · doi:10.1080/00927877508822046 · doi.org [6] G. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 409 – 449 (1984) (French). · Zbl 0538.14033 [7] Jürgen Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln, Math. Ann. 233 (1978), no. 1, 21 – 34 (German). · Zbl 0358.13009 · doi:10.1007/BF01351494 · doi.org [8] K. W. Roggenkamp and J. W. Schmidt, Almost split sequences for integral group rings and orders, Comm. Algebra 4 (1976), no. 10, 893 – 917. · Zbl 0361.16007 · doi:10.1080/00927877608822144 · doi.org This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.