Homology, Homotopy and Applications

Volume 9 (2007)

Number 2

$DG$-models of projective modules and Nakajima quiver varieties

Pages: 177 – 208

DOI: http://dx.doi.org/10.4310/HHA.2007.v9.n2.a8

Author

Farkhod Eshmatov (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Abstract

Associated to each finite subgroup $\Gamma$ of $\mathtt{SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O^{\tau}(\Gamma)$, which is a deformation of the coordinate ring of the Kleinian singularity $\mathbb{C}^{2}/\Gamma$. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over $O^{\tau}$ and a certain class of quiver varieties associated to $\Gamma$. We show that this bijection is naturally equivariant under the action of a “large” Dixmier-type automorphism group $G$. Our construction leads to a completely explicit description of ideals of the algebras $O^{\tau}$.

Keywords

noncommutative deformation of Kleinian singularities; DG category; small models; Nakajima quiver variety

2010 Mathematics Subject Classification

16S38, 18E30, 55U35

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