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A categorical Künneth formula for constructible Weil sheaves

Tamir Hemo, Timo Richarz, Jakob Scholbach

Abstract

We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p > 0$ for various coefficients, including finite discrete rings, algebraic field extensions $E \supset \mathbf Q_\ell$, $\ell \ne p$ and their rings of integers $O_E$. We also consider a variant for ind-construtible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.

A categorical Künneth formula for constructible Weil sheaves

Abstract

We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic for various coefficients, including finite discrete rings, algebraic field extensions , and their rings of integers . We also consider a variant for ind-construtible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.

Paper Structure

This paper contains 22 sections, 27 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Definitions and results
  3. Recollections on $\infty$-categories
  4. Monoidal aspects
  5. Fixed points of $\infty$-categories
  6. Lisse and constructible sheaves
  7. Weil sheaves
  8. The Weil-proétale site
  9. Weil sheaves on products
  10. Partial-Frobenius stability
  11. Lisse and constructible Weil sheaves
  12. Relation with the Weil groupoid
  13. Weil-étale versus étale sheaves
  14. The categorical Künneth formula
  15. A formulation in terms of prestacks
  16. ...and 7 more sections

Key Result

Proposition 1.2

The pullback of sheaves along $(X_{\mathbb F})_\mathrm{pro\acute et}\rightarrow X^{\operatorname{Weil}}_\mathrm{pro\acute et}$ induces an equivalence of $\Lambda_*$-linear symmetric monoidal stable $\infty$-categories for $\bullet\in \{\varnothing,{\rm lis},{\operatorname{cons}}\}$.

Theorems & Definitions (72)

  • Definition 1.1
  • Proposition 1.2: \ref{['presentation.Weil.prop']}, \ref{['prop.cons.weil.as.fixed.points']}
  • Theorem 1.3: \ref{['derived_Drinfeld.text']}, \ref{['limits.Drinfeld.rema']}
  • Example 1.4: compare SGA1
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 62 more