10.46298/epiga.2020.volume4.5570
van Dobben de Bruyn, Remy
Remy
van Dobben de Bruyn
The equivalence of several conjectures on independence of $\ell$
episciences.org
2020
Mathematics - Algebraic Geometry
14F20 (Primary) 14F30, 14C15, 14G15 (Secondary)
contact@episciences.org
episciences.org
2019-06-12T17:53:57+02:00
2021-02-08T12:04:29+01:00
2020-11-30
eng
Journal article
https://epiga.episciences.org/5570
arXiv:1808.00119
2491-6765
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Épijournal de Géométrie Algébrique ; Volume 4 ; 2491-6765
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.
Comment: Published version. 27 pages