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The equivalence of several conjectures on independence of $\ell$

Remy van Dobben de Bruyn

Abstract

We consider several conjectures on the independence of $\ell$ of the étale cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main result is that independence of $\ell$ of the Betti numbers $h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of $\ell$ of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type $k$-schemes.

The equivalence of several conjectures on independence of $\ell$

Abstract

We consider several conjectures on the independence of of the étale cohomology of (singular, open) varieties over . The main result is that independence of of the Betti numbers for arbitrary varieties is equivalent to independence of of homological equivalence for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type -schemes.

Paper Structure

This paper contains 7 sections, 21 theorems, 76 equations.

Table of Contents

  1. Pure motives and Weil cohomology theories
  2. Simplicial topoi and mapping cones
  3. All-or-nothing motives and additional axioms
  4. Independence of Weil cohomology theory
  5. Cycle classes and ranks
  6. Ranks and dimensions
  7. Dimensions and cycle class maps

Key Result

Theorem 1

Let $k$ be an algebraically closed field. If $k = \bar{\mathbf F}_p$, then the following are equivalent: Moreover, if these hold when $k = \bar{\mathbf F}_p$ for some prime $p$ (resp. for every prime $p$), then they hold over any algebraically closed field of characteristic $p$ (resp. any algebraically closed field).

Theorems & Definitions (76)

  • Theorem 1
  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Example 1.8
  • Remark 1.9
  • ...and 66 more