Topological rings in rigid geometry (Q2900492)

The text under review is a survey of K. Fujiwara and F. Kato's approach to rigid analytic geometry. The author begins by reviewing M. Raynaud's viewpoint on rigid spaces as formal models up to admissible blow-ups. Then he turns to classical birational geometry and recalls the construction of the Zariski-Riemann space as a projective limit of admissible blow-ups. He also gives an overview, with enough details, of its features: quasi-compactness, description of points using valuation rings, etc. Next, the two techniques are blended in order to define Zariski-Riemann spaces associated to rigid spaces. Those are endowed with integral structure sheaves and may be thought of as a canonical formal models of rigid spaces.NEWLINENEWLINELater, the author introduces several classes of topological rings that enjoy nice properties related to Artin-Rees type results, coherence or local criteria of flatness, for instance. The more elaborate notion is that of topologically universally adhesive (t.u.a.) rings. A theorem due to O. Gabber gives a large class of examples: valuation rings of non-zero height that are \(a\)-adically complete for some non-zero \(a\) inside the maximal ideal. Some formal geometry over such rings follows and GFGA type theorems are mentioned.NEWLINENEWLINEIn the last sections of the text, a proper definition of rigid space is given. It has an associated Zariski-Riemann space, from whose underlying topological space one may recover its topos.NEWLINENEWLINELet \(A\) be an adic ring with a finitely generated ideal of definition \(I\). One defines an analytification functor for schemes over \(\mathrm{Spec}(A) \setminus V(I)\). In case \(A\) is t.u.a., it enjoys the usual GAGA properties.NEWLINENEWLINEFor the entire collection see [Zbl 1241.14001].