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

$$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$$ 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!

## 1 Answer

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.

• Yes, when I tried to prove this sequence is exact, I thought of this question...Could you explain how to factor $\frac{x}{t}$ into a sum? Thanks! – user141691 Jan 7 '20 at 15:52
• This is the $r = 1$ case of Lemma 1.17.3 of the Bloch--Kato paper. – David Loeffler Jan 7 '20 at 20:07