## The field of $$p$$-adic periods. (Le corps des périodes $$p$$-adiques.)(French)Zbl 0707.11082

Let $$p$$ be a prime number, $${\overline{\mathbb Q}}_ p$$ the algebraic closure of $${\mathbb Q}_ p$$, $$O$$ the ring of integers of $${\mathbb Q}_ p$$, $$C_ p$$ the $$p$$-adic completion of $${\overline{\mathbb Q}}_ p$$ with the ring of integers denoted by $$\hat O$$. Let $$R$$ be the set of sequences $$X=(X^{(n)})_{n\in {\mathbb N}}$$ of elements from $$\hat O$$ such that $$(X^{n+1)})=X^{(n)}$$. $$R$$ is a complete ring of characteristic $$p$$, with valuation, by the following $(x+y)^{(n)}=\lim_{m\to \infty}(x^{(n+m)}+y^{(n+m)}),\quad (x\cdot y)^{(n)}=x^{(n)}\cdot y^{(n)},\quad v_ R(x)=v_ p(x^ 0).$ Denote by $$W(R)$$ the ring of Witt vectors with coefficients in $$R$$ and by $$[x]$$ we denote the Teichmüller representative in $$W(R)$$ of $$x\in R$$. We shall define $$\theta\colon W(R)[p^{-1}]\to C_ p$$ the homomorphism such that $\theta\left (\sum^{\infty}_{n=0}p^ n[x_ n^{p^{- n}}]\right)=\sum^{\infty}_{n=0}p^ nx_ n^{(n)}.$ The ring $$B^+_{\text{DR}}$$ defined by J.-M. Fontaine [Ann. Math. (2) 115, 529–577 (1982; Zbl 0544.14016)] is the following: $B^+_{\text{DR}}=\lim_{\leftarrow n}W(R)[p^{-1}]/(\ker \theta)^ n.$ Denote again by $$\theta$$ the homomorphism obtained on $$B^+_{\text{DR}}$$ by use of continuity and let $$I$$ be the kernel of $$\theta$$. Put $$O_ n={\overline{\mathbb Q}}_ p\cap (W(R)+I^{n+1}).$$
The main results are the following: $${\mathbb Q}_ p$$ is dense in $$B^+_{\text{DR}}$$ and $$B^+_{\text{DR}}=\lim_{\leftarrow n}({\mathbb Q}_ p\otimes \lim_{\leftarrow k}(O_ n/p^ kO_ n)),$$ it means that $$B^+_{\text{DR}}$$ is the completion of $${\mathbb Q}_ p$$ for the topology defined by the fundamental system of neighbourhood of $$0$$ given by $$V_{nk}=p^ kO_ n$$.

### MSC:

 11S25 Galois cohomology 13F35 Witt vectors and related rings 13J10 Complete rings, completion 14F30 $$p$$-adic cohomology, crystalline cohomology 14L05 Formal groups, $$p$$-divisible groups

Zbl 0544.14016