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 $p$, $B^{+}_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{\mathrm{cris}}$.

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?

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?

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 $p$, $B^{+}_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{\mathrm{cris}}$.

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 by it.

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?

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 $p$, $B^{+}_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{\mathrm{cris}}$.

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?

I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{cris}$.usually when $R$ is a perfect ring of characteristic $p$, $B^{+}_{cris}(R)$ defines as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{cris}$.

but in these note when R is ring of valuation of an algebraically closed perfectoid field $B$ defines as completion of $frac(W(R))$ with respect to all gauss norms and defined Fargues-Fontaine Curve by it.

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

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 $p$, $B^{+}_{\mathrm{cris}}(R)$ is defined as $p$-adic completion of divided power envelope of the map $W(R)\to R$ and $B_{cris}=B^{+}_{\mathrm{cris}}$.

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 by it.

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?