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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
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