$B\_{\mathrm{cris}}\subseteq B\_{\mathrm{dR}}$ 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!



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


