Uniformly rigid spaces
Let \(R\) be a complete discrete valuation ring and let \(K\) be its fraction field. Let \(\mathfrak{X}\) be the formal spectrum of a quotient \(R\langle T_{1},\dots,T_{n}\rangle/I\). In [Bull. Soc. Math. Fr., Suppl., Mém. 39--40, 319--327 (1974; Zbl 0299.14003)], \textit{M. Raynaud} defined its generic fiber \(\mathfrak{X}^{\mathrm{rig}}\) as the maximal spectrum of \(R\langle T_{1},\dots,T_{n}\rangle/I \otimes_{R} K\). This construction extends to a functor from the category of formal \(R\)-schemes locally of topologically finite type to the category of rigid \(K\)-analytic spaces, which is well understood: adding finiteness conditions and localizing by admissible blowups at the source, we may turn it into an equivalence of categories.NEWLINENEWLINELet us now consider formal \(R\)-schemes locally of formally finite type, i.e. locally formal spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle\). Berthelot extended the functor rig to this setting at the cost of losing some of its nice properties: some morphisms between rigid spaces in the image fail to come from morphisms between models. In the text under review, Kappen explains how to recover the desired properties by considering uniformly rigid spaces (a notion that he defines) instead of rigid spaces.NEWLINENEWLINEThe building blocks of uniformly rigid spaces are semiaffinoid spaces, i.e. spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle \otimes_{R} K\). Loosely speaking, this means that uniformly rigid spaces (and their functions) are constructed using bounded functions on products of open and closed discs. Those spaces are endowed with a Grothendieck topology and Kappen proves the analogues of the classical results from rigid geometry in this setting: the presheaf of semiaffinoid functions is a sheaf and coherent ideals on semiaffinoid spaces are associated to their ideals of global sections.NEWLINENEWLINEWe refer to the author's dissertation [Uniformly rigid spaces and Néron models of formally finite type. Univ. Münster (2009; Zbl 1219.14002)] for a more detailed review as well as applications to the notion of Néron models of formally finite type.
scientific article; zbMATH DE number 6064713