Abstract
In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of \({{\rm GL}(2,\mathbb{C})}\), it means that the endomorphism ring of the special CM \({\mathbb{C}}\) [[x, y]]G-modules can be used to build the dual graph of the minimal resolution of \({\mathbb{C}^{2}/G}\), extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of \({{\rm SL}(2,\mathbb{C})}\) to all finite subgroups of \({{\rm GL}(2,\mathbb{C})}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Artin M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)
Auslander M.: Rational singularities and almost split sequences. Trans. Am. Math. Soc. 293(2), 511–531 (1986)
Auslander M., Reiten I.: McKay quivers and extended Dynkin diagrams. Trans. Am. Math. Soc. 293(1), 293–301 (1986)
Bridgeland T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002)
Brieskorn E.: Rationale singularitäten komplexer flächen. Invent. Math. 4, 336–358 (1968)
Buan, A., Iyama, O., Reiten, I., Smith, D.: Mutation of cluster-tilting objects and potentials. Am. J. Math. (to appear)
Esnault H.: Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362, 63–71 (1985)
Ishii A.: On the McKay correspondence for a finite small subgroup of \({{\rm GL}(2,\mathbb{C})}\). J. Reine Angew. Math. 549, 221–233 (2002)
Iyama O.: τ-categories I: Ladders. Algebr. Represent. Theory 8(3), 297–321 (2005)
Iyama O., Wemyss M.: The classification of special Cohen Macaulay modules. Math. Z. 265(1), 41–83 (2010)
McKay J.: Graphs, singularities, and finite groups. Proc. Symp. Pure Math. 37, 183–186 (1980)
Riemenschneider O.: Invarianten endlicher Untergruppen. Math. Z. 153, 37–50 (1977)
Tráng L.D., Tosun M.: Combinatorics of rational singularities. Comment. Math. Helv. 79(3), 582–604 (2004)
Van den Bergh M.: Three-dimensional flops and noncommutative rings. Duke Math. J. 122(3), 423–455 (2004)
Wemyss, M.: Reconstruction algebras of type A. Trans. AMS (to appear)
Wemyss, M.: Reconstruction algebras of type D (I). arXiv:0905.1154 (2009)
Wemyss, M.: Reconstruction algebras of type D (II). arXiv:0905.1155 (2009)
Wunram J.: Reflexive modules on quotient surface singularities. Math. Ann. 279(4), 583–598 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wemyss, M. The \({{\rm GL}(2,\mathbb{C})}\) McKay correspondence. Math. Ann. 350, 631–659 (2011). https://doi.org/10.1007/s00208-010-0572-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0572-9