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.

arXiv commentv5: 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

References in corpus (12)

• Higher Topos Theory
• Triangulated categories of mixed motives
• Global homotopy theory
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• Enhanced six operations and base change theorem for higher Artin stacks
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• Equivariant cycles and cancellation for motivic cohomology
• Equivariant Motivic Homotopy Theory
• Equivariant motivic homotopy theory

Cited by in corpus (54)

• Triangulated categories of mixed motives
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• The Generalized Slices of Hermitian K-Theory
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• The étale local structure of algebraic stacks
• Cdh descent in equivariant homotopy K-theory
• The intersection motive of the moduli stack of shtukas
• The six-functor formalism for rigid analytic motives
• A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• Aspects of enumerative geometry with quadratic forms
• The universal six-functor formalism
• The Morel-Voevodsky localization theorem in spectral algebraic geometry
• K-theory and G-theory of derived algebraic stacks
• On Modules Over Motivic Ring Spectra
• Stable ${\mathbb{A}}^1$-connectivity over Dedekind schemes
• Framed transfers and motivic fundamental classes
• The intrinsic stable normal cone
• The motivic Satake equivalence
• On the infinite loop spaces of algebraic cobordism and the motivic sphere
• Categorical Milnor squares and K-theory of algebraic stacks
• Towards the dual motivic Steenrod algebra in positive characteristic
• Motivic Homotopy Theory of Algebraic Stacks
• A^1-homotopy invariance in spectral algebraic geometry
• Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)
• Involutions of Azumaya algebras
• Notes on motivic infinite loop space theory
• Real topological Hochschild homology of schemes
• Chiral Magnetic-Vortical Wave
• K-Motives and Koszul Duality
• Hermitian $K$-theory, Dedekind $\zeta$-functions, and quadratic forms over rings of integers in number fields
• Towards conservativity of ${\mathbb{G}}_m$-stabilization
• The Nisnevich Motive of an Algebraic Stack
• Cohomological correspondence categories
• Algebraic elliptic cohomology and flops II: $SL$-cobordism
• Nilpotence in normed MGL-modules
• The zeroth P^1-stable homotopy sheaf of a motivic space
• The geometric Satake equivalence for integral motives
• Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra
• Compactly supported ${\mathbb{A}}^1$-Euler characteristic and the Hochschild complex
• Yoneda's lemma for internal higher categories
• K-theory Soergel Bimodules
• On the Whitehead theorem for nilpotent motivic spaces
• Motivic Pontryagin classes and hyperbolic orientations
• $T$-equivariant motives of flag varieties
• The quadratic Euler characteristic of a smooth projective same-degree complete intersection
• An introduction to six-functor formalisms
• An axiomatization of six-functor formalisms
• Atiyah-Segal completion for the Hermitian K-theory of Symplectic Groups
• On the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve
• The Adams isomorphism revisited
• From Six Functors Formalisms to Derived Motivic Measures
• Modules over the de Rham cohomology spectrum
• An Introduction to ${\mathbf{A}}^1$-Enumerative Geometry