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

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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Published 1 January 2007