Tamely ramified forms of closed polydisks and laces
This paper deals with the description of some forms of Berkovich spaces, including polydiscs and annuli. For the case of polydiscs, the study started with the following result of \textit{A. Ducros} [Math. Z. 273, No. 1--2, 331--353 (2013; Zbl 1264.14035)]: Let \(k\) be a complete non-archimedean valued field, \(X\) a \(k\)-analytic space (in the sense of Berkovich) and \(L\) a finite, separable, tamely ramified extension of \(k\). Then, if \(X_L\) is an open \(L\)-polydisc, \(X\) is an open \(k\)-polydisc. Briefly, every tamely ramified form of an open polydisc is trivial. The proof of this theorem relies on \textit{M. Temkin}'s graded reduction theory (see [Isr. J. Math. 140, 1--27 (2004; Zbl 1066.32025)]) and this strategy led \textit{T. Schmidt} to adapt Hilbert theorem 90 to this context and prove a similar result for closed discs (see [Ann. Inst. Fourier 65, No. 3, 1301--1347 (2015; Zbl 1329.14058)]). In this paper, the author extends the graded Hilbert theorem 90 to the multidimensional case on the assumption that the Galois action is residually affine. Consequently, the following fact is shown: every tamely ramified form of a closed polydisc on which the Galois action is residually affine is trivial. It can be noted that, for every trivial form of a polydisc \(X\), the Galois action on \(X\) is always residually affine. For the case of annuli, the author suggests studying the more general notion of \textit{lace}, which is equivalent in dimension one. To a lace \(X\), one can associate a free \(\mathbb{Z}\)-module called \textit{lattice} and, for every form of \(X\), there is a natural Galois action on the lattice of \(X\). If this action is trivial, the author shows that the form is a lace and describes it. The article ends by an example of a non-trivial tamely ramified form of an annulus.
scientific article; zbMATH DE number 7362104
scientific article