Timeline for Motivating the coefficient field of $\ell$-adic cohomology
Current License: CC BY-SA 4.0
11 events
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| Jul 5, 2020 at 13:40 | comment | added | D.-C. Cisinski | @lemiller If $k$ is an algebraically closed field, then any $k$-linear tannakian category is neutral. That is due to Deligne for $k$ of char. $0$ and to Coulembrier in general as Theorem 6.4.1 in this paper: arxiv.org/pdf/1812.02452.pdf Since the standard conjectures imply that pure motives form a tannakian category (see e.g. the proceedings of the Seattle conference of Yves André's book on motives, or Milne's papers on motives), they imply the existence of a $\bar{\mathbf{Q}}$-linear Weil cohomology. | |
| Jul 5, 2020 at 4:24 | comment | added | lemiller | @Denis-CharlesCisinski Is there a place the Tannakian argument based on the standard conjectures is written? | |
| Jul 4, 2020 at 20:48 | comment | added | R. van Dobben de Bruyn | @Denis-CharlesCisinski agreed (but note that this imprecision was already present in the original question). | |
| Jul 4, 2020 at 20:47 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Added missing assumption.
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| Jul 4, 2020 at 9:25 | comment | added | D.-C. Cisinski | Therefore, if, as you wrote, we only need one Weil cohomology, we may take a $\mathbf{Q}_p$-linear one defined over $\mathbf{F}_p$... I still think your post is not completely accurate. | |
| Jul 4, 2020 at 7:38 | comment | added | D.-C. Cisinski | Crystalline/rigid cohomology for $k$-varieties take values in $W(k)$. So $\mathbf{Q}_p$ is a possible field of coefficient for $k=\mathbf{F}_p$. | |
| Jul 3, 2020 at 21:43 | comment | added | R. van Dobben de Bruyn | Also, the original question is about "historically, why $\mathbf Q_\ell$", and my point is that if you're trying to prove the Weil conjectures you only need one cohomology theory. (Of course I am aware of the interesting "independence of $\ell$" questions ― I even have a paper on it!) | |
| Jul 3, 2020 at 21:41 | comment | added | R. van Dobben de Bruyn | @Denis-CharlesCisinski thanks for your comments. Serre's argument depends on representations of the quaternion algebra over $\mathbf Q$ that splits at $p$ and $\infty$, so it equally rules out $\mathbf Q_p$- and $\mathbf R$-valued Weil cohomology theories (as soon as $k \supseteq \mathbf F_{p^2}$). | |
| Jul 3, 2020 at 12:20 | comment | added | D.-C. Cisinski | It might also be worth mentionning that the standard conjectures together with tannakian arguments imply the existence of a $\bar{\mathbf{Q}}$-linear Weil cohomology. | |
| Jul 3, 2020 at 12:17 | comment | added | D.-C. Cisinski | Serre only shows that there is no $\mathbf{Q}$-linear Weil cohomology, at least if we restrict to algebraic varieties over a field which contains $\mathbf{F}_{p^2}$. If we work over $\mathbf{F}_p$, crystalline cohomology actually is a $\mathbf{Q}_p$-linear Weil cohomology, and I am not aware of an argument which rules out the existence of $\mathbf{Q}$-linear one. Your assertion that "it does not matter" can also be discussed: independence of $\ell$ problems have a long history which seem to point in a rather different direction. | |
| Jul 3, 2020 at 3:12 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |