A construction of \(\mathbf B_{\text{dR}}^+\) (Q1947824)

Let \(K\) be a local field of characteristic \(0\) and residual characteristic \(p>0\). Let \(\mathbf B_{\text{dR}}^+\) be the ring containing the \(p\)-adic periods of the algebraic varieties defined over \(K\). This ring was defined by \textit{J. M. Fontaine} [Ann. Math. (2) 115, 529--577 (1982; Zbl 0544.14016); Fontaine, Jean-Marc (ed.), Périodes \(p\)-adiques. Astérisque 223, 59--111, Appendix 103--111 (1994; Zbl 0940.14012)]. It is proved that \(\mathbf B_{\text{dR}}^+\) is the completion of \(\bar{K}\), the algebraic closure of \(K\), with respect to a topology described intrinsically that involves semimultiplicative but not multiplicative norms on \(K\). This is given in Théorème 3.1. An explicit formula is found to describe an element of \(\bar{K}\) as an element of \(\mathbf B_{\text{dR}}^+\). This formula is not essential to prove that \(\bar{K}\) is dense in \(\mathbf B_{\text{dR}}^+\) but it is useful for visualizing the topology induced on \(K\) by \(\mathbf B_{\text{dR}}^+\). The first three sections of this paper are modifications of the appendix written by the author in [loc. cit]. Section 1 contains results on complete local algebras. Section 2 is dedicated to the formula for elements of \(\bar{K}\) as elements of \(\mathbf B_{\text{dR}}^+\). In Section 3 it is proved the density of \(\bar{K}\) in \(\mathbf B_{\text{dR}}^+\) and its consequences. In the last section, the results are extended to the case where the starting point is a Banach algebra, for instance, the Tate algebra \({\mathbb Q}_p\{T_1,\ldots,T_d\}\), instead of a local field.