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We know crystalline cohomology is calculated by taking an inverse limit:

$$H_{cris}^i:=\varprojlim_nH_{cris}^i(X/W_n(k))$$

provided $X$ projective smooth over a perfect field $k$ of char $p$.

I want to ask, is there a wider cohomology theory that can calculate the crystalline cohomology directly (without taking the limit)?

Since pro-$\mathrm{\acute etale}$ cohomology generalizes $\mathbb Q_\ell$-cohomology, so I wonder if there is also a such thing for cris.

Could you provide references or give some reasons why it's difficult to generalize (if it doesn't exist)?

Thanks.

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    $\begingroup$ Crossposted at MSE: math.stackexchange.com/questions/4651875/… $\endgroup$
    – Noah Schweber
    Commented Mar 6, 2023 at 5:53
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    $\begingroup$ I’m voting to close this question because it is crossposted. $\endgroup$
    – Steven Landsburg
    Commented Mar 6, 2023 at 11:27
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    $\begingroup$ If some question should be closed, it's the one on MSE. This question is surely research-level. $\endgroup$
    – Gabriel
    Commented Mar 12, 2023 at 14:08
  • $\begingroup$ @Gabriel Thanks. I have deleted it. $\endgroup$
    – Richard
    Commented Mar 13, 2023 at 5:11
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    $\begingroup$ In the definition of the crystalline site, there are some nilpotent PD thickenings around. If you replace those by $p$-complete PD thickenings (so that modulo every $p^n$ you get a nilpotent PD thickening as before), you get a site that directly computes the limit over $n$ of crystalline cohomology modulo $p^n$. Is that what you are asking? $\endgroup$
    – Peter Scholze
    Commented Mar 13, 2023 at 6:51

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