The equivalence of several conjectures on independence of $\ell$
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866914755451551744 |
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| author | de Bruyn, Remy van Dobben |
| author_facet | de Bruyn, Remy van Dobben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_00119 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The equivalence of several conjectures on independence of $\ell$ de Bruyn, Remy van Dobben Algebraic Geometry 14F20 (Primary) 14F30, 14C15, 14G15 (Secondary) 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. |
| title | The equivalence of several conjectures on independence of $\ell$ |
| topic | Algebraic Geometry 14F20 (Primary) 14F30, 14C15, 14G15 (Secondary) |
| url | https://arxiv.org/abs/1808.00119 |