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The $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-cyclic McKay correspondence via motivic integration

Part of: Birational geometry Algebraic number theory: local and $p$-adic fields General commutative ring theory Representation theory of groups

Published online by Cambridge University Press:  10 June 2014

Takehiko Yasuda
Takehiko Yasuda*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan email [email protected]
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Abstract

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We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

Keywords

McKay correspondencemotivic integrationstringy invariantpositive characteristicArtin–Schreier extension

MSC classification

Secondary: 14E16: McKay correspondence 14E18: Arcs and motivic integration 13A50: Actions of groups on commutative rings; invariant theory 11S15: Ramification and extension theory 20C20: Modular representations and characters
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 7 , July 2014 , pp. 1125 - 1168
Copyright
© The Author 2014 

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