Geodesic
$DG$-models of projective modules and Nakajima quiver varieties
Homology, homotopy, and applications, Tome 9 (2007) no. 2, pp. 177-208.

Voir la notice de l'article provenant de la source International Press of Boston

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 $O^{\tau}$.
DOI : 10.4310/HHA.2007.v9.n2.a8
Classification : 16S38, 18E30, 55U35
Keywords: noncommutative deformation of Kleinian singularities, DG category, small models, Nakajima quiver variety 
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     author = {Farkhod Eshmatov},
     title = {$DG$-models of projective modules and {Nakajima} quiver varieties},
     journal = {Homology, homotopy, and applications},
     pages = {177--208},
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     number = {2},
     year = {2007},
     doi = {10.4310/HHA.2007.v9.n2.a8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2007.v9.n2.a8/}
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Farkhod Eshmatov. $DG$-models of projective modules and Nakajima quiver varieties. Homology, homotopy, and applications, Tome 9 (2007) no. 2, pp. 177-208. doi : 10.4310/HHA.2007.v9.n2.a8. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2007.v9.n2.a8/

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