On the definition of rigid analytic spaces (Q2900491)

The text under review is devoted to rigid analytic spaces. They were introduced by \textit{J. Tate} as analogues of complex analytic spaces over non-archimedean fields (see [``Rigid analytic spaces'', Invent. Math. 12, 257--289 (1971; Zbl 0212.25601)]). The basic objects are the so-called affinoid spaces: maximal spectra of algebra of convergent power series and their quotients. Problems arise when one wants to glue such spaces and define analytic functions on them, due to the lack of connectivity of non-archimedean fields. To rigidify the situation, Tate introduced a notion of \(h\)-structure on a space \(X\): a family of structural morphisms from affinoid spaces to \(X\), subject to some conditions.NEWLINENEWLINEIn a more modern point of view (see [\textit{S. Bosch, U. Güntzer} and \textit{R. Remmert}, Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften, 261. Berlin etc.: Springer Verlag (1984; Zbl 0539.14017)], for instance), rigid analytic spaces are constructed as a site, via the definition of a Grothendieck topology: roughly speaking, some open subsets and some coverings of a space \(X\) are distinguished and used to define the structure sheaf.NEWLINENEWLINEAfter a detailed introduction to the subject, including a reminder of the definitions and basic properties of the spaces (Tate's acyclicity theorem, for instance), the author compares the two points of view mentioned above and shows that the two categories of rigid analytic spaces are indeed equivalent.NEWLINENEWLINEFor the entire collection see [Zbl 1241.14001].