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

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 $h_{\text{c}}^i(X,{\mathbf{Q}}_{\ell })$ 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.

Published version. 27 pages