The six operations in equivariant motivic homotopy theory
Abstract
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 and 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.
Keywords
Cite
@article{arxiv.1509.02145,
title = {The six operations in equivariant motivic homotopy theory},
author = {Marc Hoyois},
journal= {arXiv preprint arXiv:1509.02145},
year = {2024}
}
Comments
v5: fixed the statement of Prop. 3.4 (homotopy localization) and the proofs in sections 4.2 (exactness of pushforwards) and 4.3 (gluing); v4: added subsection 2.5 to fix a mistake