Timeline for Are the Eigenvalues of the Frobenius on Crystalline cohomology bounded by degree?
Current License: CC BY-SA 4.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 5, 2018 at 8:43 | history | edited | SashaP | CC BY-SA 4.0 |
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| Sep 9, 2016 at 13:15 | comment | added | SashaP | @PaxKivimae I do not now the answer from the top of my head. Probably it is worth asking a separate question. | |
| Sep 5, 2016 at 13:53 | comment | added | Pax | Yes, that is what I meant, I seem to be getting worse at writing Tex. But since $H^{r-i}(W\Omega^i)[1/p]$ has sloops in $[i, i+1[$, there is an isogenous $(H^{r-i}(W\Omega^i )[1/p][-i])[i]$, with slopes in $[0, 1[$, and so is the $H^{r-i}(W\Omega^i )[1/p][-i]$ is a Dieudonne module, and can be written as $D(G_i)$. I was just wondering if the $G_i$ had interpretations, since I think that $G_0$ is just the Artin-Mazur formal group. | |
| Sep 5, 2016 at 9:48 | comment | added | SashaP | @PaxKivimae No, I do not claim that $H^{r-i}(W\Omega^i)[1/p]=H^r[1/p]$. There is a direct sum decomposition(exactly as in Hodge theory)$H^r[1/p]=\bigoplus_i H^{r-i}(W\Omega^i)[1/p]$ compatible with Frobenius action and summand $H^{r-i}(W\Omega^i)[1/p]$ has slopes in the interval $[i,i+1[$. | |
| Sep 5, 2016 at 9:44 | history | edited | SashaP | CC BY-SA 3.0 |
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| Sep 4, 2016 at 23:11 | comment | added | Pax | groups obtained from the shift operator and duality, $D^{-1}(C[-i][i])$ as remarked by Groethendieck (I forget where though). | |
| Sep 4, 2016 at 23:09 | comment | added | Pax | This answer is exactly what I was looking for! Glad to have an intuitive proof on hand before braving Illusie (especially since my french is horrendous). The equality $H^{r-i}(W\Omega^i)[1/p]=H^r[1/p]$ is very interesting. I believe we have that $H^r(W\Omega^0)=H^r(X, W)=H^r(X, D\mathbb{G}_m)=D(H^r(X, \mathbb{G}_m))$, for the last object being the formal group associated to $Ker(H^r(X\times Spec(A))\to H^r(X))$ (we might need some smoothness conditions here). Are there similar interpretations of the higher $H^{r-i}(W\Omega^i)[1/p]$ in terms of formal groups? I believe these are the formal | |
| Sep 4, 2016 at 22:57 | vote | accept | Pax | ||
| Sep 4, 2016 at 22:31 | history | answered | SashaP | CC BY-SA 3.0 |