Timeline for Frobenius action on de Rham cohomology
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2020 at 22:43 | comment | added | Anonymous | Expanding on what was already said: the Frobenius map $H^*_{dR}(X) \to H^*_{dR}(X)$ factors as $H^*_{dR}(X) \to H^*(X,O_X) \to H^*(X,O_X) \to H^*_{dR}(X)$ where the first and last map are edge maps for the Hodge/conjugate spectral sequences, and the middle map is the Frobenius on structure sheaf cohomology. So understanding, e.g., the rank of the Frobenius on de Rham cohomology requires knowing that rank of the Frobenius on $O_X$-cohomology as well as behaviour of certain differentials in the Hodge/conjugate spectral sequences. | |
| Jun 11, 2020 at 22:04 | comment | added | Vitay | Thank you! Do you have a reference which explains the relation to de Rham–Witt cohomology? | |
| Jun 11, 2020 at 20:59 | comment | added | R. van Dobben de Bruyn | Another example: if $E$ is an elliptic curve, then the Frobenius action on $H^1(E,\mathcal O_E)$ is $0$ if and only if $E$ is supersingular. More generally, it is related to de Rham–Witt cohomology, which rationally gives the slope filtration on crystalline cohomology. So the answer depends on subtle arithmetic information. | |
| Jun 11, 2020 at 20:59 | comment | added | R. van Dobben de Bruyn | On $\mathbf P^n$, the map is just $[x_0:\ldots:x_n] \mapsto [x_0^p:\ldots:x_n^p]$. In arbitrary characteristic, the map $[x_0:\ldots:x_n] \mapsto [x_0^d:\ldots:x_n^d]$ acts on $H^i(\mathbf P^n,\Omega^i)$ as multiplication by $d^i$, so in this case its zero for $i > 0$. | |
| Jun 11, 2020 at 20:46 | history | edited | Vitay | CC BY-SA 4.0 |
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| Jun 11, 2020 at 18:26 | review | First posts | |||
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| Jun 11, 2020 at 18:19 | history | asked | Vitay | CC BY-SA 4.0 |