Geodesic
The equivalence of several conjectures on independence of $\ell$
Épijournal de Géométrie Algébrique, Tome 4 (2020)

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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. 
@article{10_46298_epiga_2020_volume4_5570,
     author = {de Bruyn, Remy van Dobben},
     title = {The equivalence of several conjectures on independence of $\ell$},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
     publisher = {mathdoc},
     volume = {4},
     year = {2020},
     doi = {10.46298/epiga.2020.volume4.5570},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2020.volume4.5570/}
}
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de Bruyn, Remy van Dobben. The equivalence of several conjectures on independence of $\ell$. Épijournal de Géométrie Algébrique, Tome 4 (2020). doi: 10.46298/epiga.2020.volume4.5570

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