UNIFORMLY RIGID SPACES 1. Introduction Let K be a non ...

6 CHRISTIAN KAPPENinternationale Forschungs- und Studienvorhaben; the author would like toextend his gratitude to these institutions.Contents1. Introduction 12. Uniformly rigid spaces 62.1. Formal schemes of formally finite type 62.2. Semi-affinoid algebras 72.3. Semi-affinoid spaces 132.4. Uniformly rigid spaces 263. Coherent modules on uniformly rigid spaces 323.1. Closed uniformly rigid subspaces 364. Comparison with the theories of Berkovich and Huber 39References 402. Uniformly rigid spacesLet R be a discrete valuation ring with residue field k and fraction field K,and let π ∈ R be a uniformizer.2.1. Formal schemes of formally finite type. A morphism of locallynoetherian formal schemes is said to be of locally formally finite (ff) typeif the induced morphism of smallest subschemes of definition is of locallyfinite type. Equivalently, any induced morphism of subschemes of definitionis of locally finite type. A morphism of locally noetherian formal schemesis called of ff type if it is of locally ff type and quasi-compact. If A is anoetherian adic ring and if B is a noetherian adic topological A-algebra,then Spf B is of ff type over Spf A if and only if B is a topological quotientof a mixed formal power series ring A[[S 1 , . . . , S m ]]〈T 1 , . . . , T n 〉, whereA[[S 1 , . . . , S m ]] carries the a + (S 1 , . . . , S m )-adic topology for any ideal ofdefinition a of A, cf. [2] Lemma 1.2. In this case, we say that the topologicalA-algebra B is of ff type. Morphisms of locally ff type are preserved undercomposition, base change and formal completion.We say that an R-algebra is of formally finite (ff) type if it admits a ringtopology such that it becomes a topological R-algebra of ff type in the abovesense, where R carries the π-adic topology. Equivalently, an R-algebra is