Graded quiver varieties and derived categories
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Bernhard Keller
und
Sarah Scherotzke
Abstract
Inspired by recent work of Hernandez–Leclerc and Leclerc–Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc–Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category 𝒮. We determine the quiver of 𝒮 and the number of minimal relations between any two given vertices. We construct a δ-functor Φ taking each finite-dimensional representation of 𝒮 to an object of the derived category of Q. We show that the functor Φ establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Φ. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of `line bundles over the non-commutative curve given by Q'. We show that the degeneration order between strata corresponds to Jensen–Su–Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category 𝒮 is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.
A large part of the work on this article was done during the cluster algebra program at the MSRI in fall 2012. The authors are grateful to the MSRI for financial support and ideal working conditions. They are indebted to Bernard Leclerc and Pierre-Guy Plamondon for informing them about the main results of [`Nakajima varieties and repetitive algebras', preprint 2013] prior to its appearance on the archive. They are obliged to Osamu Iyama for pointing out reference [Algebr. Represent. Theory 8 (2005), no. 3, 297–321] and to Harold Williams for asking a question that lead to Theorem 2.8. They thank Giovanni Cerulli Irelli, David Hernandez, Osamu Iyama, Bernard Leclerc, Pierre-Guy Plamondon, Fan Qin and Markus Reineke for stimulating conversations and for helpful comments on a preliminary version of this article.
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Congruence kernels around affine curves
- On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
- On the splitting of quasilinear p-forms
- Graded quiver varieties and derived categories
- On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
- Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
- Linear stability of algebraic Ricci solitons
- Dynamical stability of algebraic Ricci solitons
- Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
- Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
Artikel in diesem Heft
- Frontmatter
- Congruence kernels around affine curves
- On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
- On the splitting of quasilinear p-forms
- Graded quiver varieties and derived categories
- On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
- Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
- Linear stability of algebraic Ricci solitons
- Dynamical stability of algebraic Ricci solitons
- Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
- Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
