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The six operations in equivariant motivic homotopy theory

Marc Hoyois

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.

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.

Paper Structure

This paper contains 37 sections, 74 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Equivariant homotopy theory
  3. The formalism of six operations
  4. Summary of the construction
  5. Some applications
  6. Motivic Wirthmüller and Adams isomorphisms
  7. An equivariant Lefschetz trace formula
  8. Cdh descent for the homotopy $K$-theory of tame stacks
  9. Acknowledgements
  10. Notation and terminology
  11. Some equivariant geometry
  12. Invariant subschemes
  13. Quasi-affine and quasi-projective morphisms
  14. Linearly reductive groups
  15. Affine resolutions
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $B$ be a qcqs scheme and $G$ a tame group scheme over $B$. If $G$ is not finite, we assume that $B$ has the $G$-resolution property. Then the six operations satisfy the following properties on finitely presented $G$-quasi-projective $B$-schemes (or on all qcqs $G$-schemes if $G$ is discrete), whenever the exceptional functors are defined.

Theorems & Definitions (169)

  • Theorem 1.1: Theorem \ref{['thm:main']} and Proposition \ref{['prop:constructible']}
  • Theorem 1.2: Proposition \ref{['prop:cdh']}
  • Theorem 1.3: Theorem \ref{['thm:wexcisive']}
  • Theorem 1.4: Theorem \ref{['thm:gluing']}
  • Theorem 1.5: Theorem \ref{['thm:stableduality']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 159 more