$B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

Question

$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B_{\mathrm{dR}}=B_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B_{\mathrm{cris}}$ where $t$ is Fontaine's $2\pi i$.

I want to ask if the natural map $B_{\mathrm{cris}}\rightarrow \frac{B_{\mathrm{dR}}}{B_{\mathrm{dR}}^+}$ induced by the inclusion is surjective. This is to say if $B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$. For example, if $x\in B_{\mathrm{dR}}^+$, then $\frac{x}{t}\in B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

Thanks!

Answer 1

Much more is true: the subring $B_{\mathrm{cris}}^{\varphi = 1}$ surjects onto $B_{\mathrm{dR}} / B_{\mathrm{dR}}^+$, so there is an exact sequence $$ 0 \to \mathbf{Q}_p \to B_{\mathrm{cris}}^{\varphi = 1} \to B_{\mathrm{dR}} / B_{\mathrm{dR}}^+ \to 0.$$ This is the Bloch--Kato fundamental exact sequence which is used to construct the Bloch--Kato exponential map; you can read all about it in the paper by Bloch and Kato in the Grothendieck Festschrift.