Every tamely ramified form of an open polydisc is trivial
Let \(k\) be a complete non-Archimedean valued field and \(X\) be a \(k\)-analytic space. The main result of the paper is the following: if there exists a finite and tamely ramified extension \(L\) of \(k\) such that \(X \hat{\otimes}_{k} L\) is an open polydisc over \(L\), then \(X\) itself is an open polydisc over \(k\). Let us point out that the radii of the polydiscs may be arbitrary elements of \(\mathbb{R}_{>0}\) and do not need to belong to the value group \(|k^*|\). The paper is written in the setting of Berkovich spaces. In the introduction, the author explains his strategy in a simpler case: \(L/k\) is unramified and \(X \hat{\otimes}_{k} L\) is the unit polydisc. In this case, the result asserts that \(X\) is isomorphic to the unit polydisc over \(k\). The proof heavily relies on reduction arguments: we have \(\widetilde{\mathcal{O}(X) \hat{\otimes}_{k} L} = \widetilde{\mathcal{O}(X \hat{\otimes}_{k} L)} \simeq \tilde{L}[\![t_{1},\dots,t_{n}]\!]\). Using the \(t_{i}\)'s, one may construct a morphism from \(X\) to the unit polydisc over \(k\) that becomes an isomorphism over \(L\). The conclusion now follows from a descent argument. In the general case (tamely ramified extensions and arbitrary radii), reduction may behave badly and has to be replaced by \textit{M. Temkin}'s graded reduction (see [Isr. J. Math. 140, 1--27 (2004; Zbl 1066.32025)]). Otherwise, the overall strategy remains the same. In fact, Ducros claims that tamely ramified extensions are the analogues of unramified extensions in the graded setting. The author also adapts several classical notions and results from commutative algebra to the context of graded commutative algebra: Galois theory, étale algebras, etc. The paper ends by an example to show that the result may fail when wild ramification occurs.
scientific article; zbMATH DE number 6134684