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ISSN 1088-4165

 
 

 

Cherednik algebras and Hilbert schemes in characteristic $p$


Authors: Roman Bezrukavnikov, Michael Finkelberg and Victor Ginzburg; with an Appendix by Pavel Etingof
Journal: Represent. Theory 10 (2006), 254-298
DOI: https://doi.org/10.1090/S1088-4165-06-00309-8
Published electronically: April 17, 2006
MathSciNet review: 2219114
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Abstract | References | Additional Information

Abstract: We prove a localization theorem for the type $\mathbf {A}_{n-1}$ rational Cherednik algebra $\mathsf {H}_c=\mathsf {H}_{1,c}(\mathbf {A}_{n-1})$ over $\overline {\mathbb {F}}_p$, an algebraic closure of the finite field. In the most interesting special case where $c\in \mathbb {F}_p$, we construct an Azumaya algebra $\scr H_c$ on $\operatorname {Hilb}^n{\mathbb {A}}^2$, the Hilbert scheme of $n$ points in the plane, such that $\Gamma (\operatorname {Hilb}^n{\mathbb {A}}^2, \scr H_c)=\mathsf {H}_c$. Our localization theorem provides an equivalence between the bounded derived categories of $\mathsf {H}_c$-modules and sheaves of coherent $\scr H_c$-modules on $\operatorname {Hilb}^n{\mathbb {A}}^2$, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.


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Additional Information

Roman Bezrukavnikov
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139
MR Author ID: 347192
Email: bezrukav@math.mit.edu

Michael Finkelberg
Affiliation: Independent University of Moscow, 11 Bolshoy Vlasyevskiy per., 119002 Moscow, Russia
MR Author ID: 304673
Email: fnklberg@mccme.ru

Victor Ginzburg
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: ginzburg@math.uchicago.edu

Pavel Etingof
Affiliation: Department of Mathematics, M.I.T., 77 Massachusetts Ave, Cambridge, Massachusetts 02139
MR Author ID: 289118
Email: etingof@math.mit.edu

Received by editor(s): May 4, 2005
Received by editor(s) in revised form: February 19, 2006
Published electronically: April 17, 2006
Dedicated: To David Kazhdan with admiration
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.