The six operations in equivariant motivic homotopy theory

We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing the...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2017-01, Vol.305, p.197-279
Main Author: Hoyois, Marc
Format: Article
Language:English
Subjects:
Algebraic stacks
Equivariant homotopy theory
Motivic homotopy theory
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Summary:We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel–Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoub's and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2016.09.031