A generalization of formal schemes and rigid analytic varieties

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point of view there is a certain similarity in constructing formal schemes and rigid analytic varieties. In both cases one starts with a certain class of topological rings (the adic rings in formal geometry and Tate algebras in rigid geometry), defines to every topological ring of this class a locally and topologically ringed space, and glueing of such spaces give formal schemes or rigid analytic varieties. There is a natural class of topological rings which contains both the noetherian adic rings and the Tate algebras and which suggests itself. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring f-adic. The points of the formal scheme SpfA associated with an adic ring A are the open prime ideals of A, and the points of the rigid analytic variety SpA associated with a Tate algebra A are the maximal ideals of A. In both cases one can consider the points as continuous valuations of A. (A valuation v: A ~ F~ U {0} of a topological ring A is called continuous if the mapping v is continuous with respect to the ring topology of A and the order-induced topology of Fv U {0}.) Namely, if p is an open prime ideal of an adic ring A then the trivial valuation vp of A with vp (a) = 0 iff a C p is continuous, and if p is a maximal ideal of a Tate algebra A over a valued field k then the