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I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=A_{\mathrm{cris}}[1/t]$.

But in these notes when $R$ is ring of valuation of an algebraically closed perfectoid field $B$ is defined as completion of $\mathrm{Frac}(W(R))$ with respect to all Gauss norms and defined Fargues-Fontaine curve using $B$ in place of $B_{\mathrm{cris}}$.

I want to know the relation between $B$ and $B_{\mathrm{cris}}$ in general. Is it true that they are isomorphic if R is the valuation ring of a perfectoid field?

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  • $\begingroup$ If somebody could fix English in "and defined Fargues-Fontaine curve by it"? I don't now the relevant math enough to do so. $\endgroup$
    – YCor
    Commented Feb 19, 2020 at 22:22
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    $\begingroup$ Some relations between analogous rings $B, B^+$ and $B^+_{\mathrm{cris}, \rho}$ are discussed in Section 1.2 of Fargues' and Fontaine's paper "Vector bundles on curves and p-adic Hodge theory", webusers.imj-prg.fr/~laurent.fargues/Durham.pdf . But it might require some care to see that/whether those agree with $B, B^+$ from Lurie's notes (and for which $\rho$ is $B_{\mathrm{cris}, \rho}^+=B^{+}_{\mathrm{cris}}$). $\endgroup$
    – Pavel Čoupek
    Commented Feb 20, 2020 at 4:18
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    $\begingroup$ It is not true that $B_{cris} = B_{cris}^+$. What is true is that $B_{cris} = B_{cris}^+[1/t]$. In addition, what you define in the first paragraph is not $B_{cris}^+$ but $A_{cris}$. $\endgroup$
    – Laurent Berger
    Commented Feb 20, 2020 at 9:50
  • $\begingroup$ @LaurentBerger you are right I edited the question. $\endgroup$
    – ali
    Commented Feb 20, 2020 at 9:59
  • $\begingroup$ @PavelČoupek thanks it's helpful. $\endgroup$
    – ali
    Commented Feb 20, 2020 at 10:00

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