## Remy van Dobben de Bruyn - The equivalence of several conjectures on independence of $\ell$

epiga:5570 - Épijournal de Géométrie Algébrique, November 30, 2020, Volume 4 - https://doi.org/10.46298/epiga.2020.volume4.5570
The equivalence of several conjectures on independence of $\ell$

Authors: Remy van Dobben de Bruyn

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.

Volume: Volume 4
Published on: November 30, 2020
Accepted on: November 30, 2020
Submitted on: June 12, 2019
Keywords: Mathematics - Algebraic Geometry,14F20 (Primary) 14F30, 14C15, 14G15 (Secondary)