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

TL;DR

This paper explores the relationships between various conjectures on the independence of the prime number ll in tale cohomology, establishing equivalences and implications across different classes of algebraic varieties.

Contribution

It proves the equivalence between independence of ll of Betti numbers and homological equivalence on smooth projective varieties, and shows implications for arbitrary varieties.

Findings

Independence of ll of Betti numbers for arbitrary varieties is equivalent to independence of ll of homological equivalence.

Independence of ll for smooth quasi-projective varieties implies the same for all separated finite type schemes.

Several other equivalent statements relating these conjectures are provided.

Abstract

We consider several conjectures on the independence of $\ell$ of the \'etale 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.