A theorem on the structure of the ring \(A_{\text{cris}}\) (Q2644364)

In Fontaine's theory of \(p\)-adic representations, the rings \(A_{\text{cris}}\) and \(B_{\text{cris}}\) play a prominent role. In his ``Grothendieck Festschrift'' paper [The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 249--309 (1990; Zbl 0743.11066)] \textit{J.-M. Fontaine} constructed a continuous \(W(R)\)-algebra isomorphism \(W(R){\widehat\otimes}_{{\mathbb Z}_p[\pi_0]}\;{\mathbb Z}_p\{\langle {\pi_0 \over p}\rangle \}\buildrel\sim\over \to A_{\text{cris}}.\) Without explaining all notations, let us say that \(W(R)\) is the Witt ring of a certain complete valuation ring \(R\) with characteristic \(p\) and \(\pi_0 = -p +\sum_{a \in {\mathbb F}_p} [\varepsilon]^a,\) where \([\varepsilon]\) is a canonical element of \(W(R)\) ``coming from'' the cyclotomic extension of \({\mathbb Q}_p.\) In this note, the author gives a general definition of \(\pi_0\) ``coming from'' any Lubin-Tate formal group over \({\mathbb Z}_p\) and shows that there is an isomorphism \(A_{\text{cris}} \buildrel\sim\over \to W(R)\{\langle {\pi_0 \over p} \rangle\}\) as subrings of \(B_{dR}.\) NB: The definition of \(\varepsilon\) on p. 1334 is not correct.