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Generalized cohomology theories for algebraic stacks

Adeel A. Khan, Charanya Ravi

Abstract

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer-Vietoris. For example, we deduce that homotopy K-theory satisfies cdh descent on scalloped stacks. We also prove a fixed point localization formula for torus actions. Finally, the construction is contrasted with a "lisse-extended" stable motivic homotopy category, defined for arbitrary stacks: we show for example that lisse-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin-Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism. Moreover, the lisse-extended motivic homotopy type is shown to recover all previous constructions of motives of stacks.

Generalized cohomology theories for algebraic stacks

Abstract

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology theories for stacks like algebraic K-theory, as well as new examples like genuine motivic cohomology and algebraic cobordism. These cohomology theories admit Gysin maps and satisfy homotopy invariance, localization, and Mayer-Vietoris. For example, we deduce that homotopy K-theory satisfies cdh descent on scalloped stacks. We also prove a fixed point localization formula for torus actions. Finally, the construction is contrasted with a "lisse-extended" stable motivic homotopy category, defined for arbitrary stacks: we show for example that lisse-extended motivic cohomology of quotient stacks is computed by the equivariant higher Chow groups of Edidin-Graham, and we also get a good new theory of Borel-equivariant algebraic cobordism. Moreover, the lisse-extended motivic homotopy type is shown to recover all previous constructions of motives of stacks.

Paper Structure

This paper contains 115 sections, 80 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Motivic homotopy theory of stacks
  3. Genuine cohomology theories
  4. Fixed point localization
  5. Lisse-extended cohomology theories
  6. Derived stacks and virtual functoriality
  7. Homotopy invariant K-theory
  8. Outline of Part I
  9. Outline of Part II
  10. Outline of Part III
  11. Related work
  12. Conventions
  13. Acknowledgments
  14. Scalloped algebraic stacks
  15. Nice group schemes
  16. ...and 100 more sections

Key Result

Theorem 1

The assignment $\mathcal{X}\xspace \mapsto {\mathbf{SH}\xspace}(\mathcal{X}\xspace)$, together with the formalism of six operations, extends from qcqsquasi-compact and quasi-separated algebraic spaces to scalloped algebraic stacks. More precisely, we have the following operations: Moreover, these satisfy various identities including the base change and projection formulas, homotopy invariance, pu

Theorems & Definitions (238)

  • Theorem 1
  • Corollary 2
  • Theorem 3: Concentration
  • Theorem 4
  • Example 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • ...and 228 more