Timeline for Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya
Current License: CC BY-SA 4.0
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| Nov 3, 2019 at 8:45 | comment | added | user20948 | (continued) where $X$ is a smooth $\mathbb F_p$-scheme. I don't know a good reference. Bhatt's lecture notes about prismatic cohomology covered this in Corollary 1.8 of lecture 6. | |
| Nov 3, 2019 at 8:40 | comment | added | user20948 | I would guess that the motivation to lift $\mathbb F_p$ to $\mathbb Z_p$ is that, the objects that we are really interested are $\mathbb Z$-schemes. In fact, the derived $\mathbb Z$-de Rham cohomology $\operatorname{dR}_{R/\mathbb Z}$ is not that bad (I was also misled. The $\mathbb F_p$-de Rham cohomology that you have mentioned is $\operatorname{dR}_{R/\mathbb F_p}$). On the other hand, a precise comparison of $\mathbb F_p$-de Rham and crystalline cohomologies is: $R\Gamma_{\mathrm{cris}}(X/\mathbb Z_p)\otimes_{\mathbb Z_p}^{\mathbb L}\mathbb F_p\simeq R\Gamma_{\mathrm dR}(X/\mathbb F_p)$. | |
| Nov 3, 2019 at 8:26 | comment | added | user20948 | I would not call these gadgets "a lift to char 0" - they lives over $\operatorname{Spec}\mathbb Z_p$, which is usually called of mixed char. After all, crystalline cohomology is an integral cohomology theory. | |
| May 6, 2018 at 20:33 | history | answered | Luca Ghidelli | CC BY-SA 4.0 |