The six operations in equivariant motivic homotopy theory

TL;DR

This paper develops an equivariant motivic homotopy theory for quotient stacks with reductive groups, extending foundational theorems and establishing a six operations formalism for equivariant spectra.

Contribution

It introduces an equivariant framework for motivic homotopy theory, extending key theorems and constructing a six operations formalism for equivariant motivic spectra.

Findings

Extended purity and gluing theorems to the equivariant setting.

Proved ambidexterity theorem in the equivariant context, with a novel proof.

Established descent properties for cohomology theories represented by motivic spectra.

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.