Siegfried Bosch

Formal and rigid geometry. I. Rigid spaces.

In 1974 Raynaud proposed a program (Raynaud in Mem. Soc. Math. Fr. 39–40:319–327, 1974), where he introduced groundbreaking ideas to rigid geometry by interpreting a rigid analytic space as the generic fiber of a formal schemes over \(\operatorname{Spf}R\). Here \(\operatorname{Spf}R\) is always the formal spectrum of a complete valuation ring \(R\) of height 1, where its topology is given by an ideal \((\pi)\) for some element \(\pi\in R\) with \(0<|\pi|<1\). Due to results on flat modules (Raynaud and Gruson in Invent. Math. 13:1–89, 1971) his approach also works in the non-Noetherian case of formal schemes of topologically finite presentation over \(\operatorname{Spf}(R)\).

Lectures on formal and rigid geometry

Classical Rigid Geometry.- Tate Algebras.- Affinoid Algebras and their Associated Spaces.- Affinoid Functions.- Towards the Notion of Rigid Spaces.- Coherent Sheaves on Rigid Spaces.- Formal Geometry.- Adic Rings and their Associated Formal Schemes.- Raynauds View on Rigid Spaces.- More Advanced Stuff.- Appendix.- References.- Index.

Orthonormalbasen in der nichtarchimedischen Funktionentheorie

AbstractIn this paper one finds a new method to calculate problems concerning affinoid algebras. The method which uses orthonormal bases in normed vector spaces is developed in the first two paragraphs and is applied to affinoid algebras later on. In a simple way there are obtained nearly all results about affinoid algebras which are already known. Further this method gives new information about the functor F which associates to each affinoid space X an affine algebraic variety

Rational points of the group of components of a Néron model

\tilde X

What Is a Néron Model

. In detail: F is compatible with extensions of the field k (if affinoid spaces

Towards the Notion of Rigid Spaces

X \subset k^{ \circ n}