Continuous valuations

0 Introduction In this paper we study, for a certain type of topological rings A, the topological space Cont A of all equivalence classes of continuous valuations of A. The space ContA is defined as follows. Let v: A-+Fw{0} be a valuation of A, where F is an ordered multiplicative group generated by im(v)\{0}. On Fw [0} we introduce the topology such that U _~ F ~ {0} is open iff 0 q~ U or {xeFlx < 7} c_ U for some yeF. We call v continuous if the mapping v: A-~Fw{0} is continuous with respect to the ring topology on A and the topology on Fw {0} just defined. Two continuous valuations v: A ~Fw {0} and w: A ~ A u {0} are called equivalent if there exists an isomorphism f: r~{0I~A~{0} of ordered monoids such that w=fov. Then Cont A is the set of all equivalence classes of continuous valuations of A equipped with the topology generated by the sets {veCont A [ v(a) < v(b) + O} (a, be A). Our study of the topological spaces Cont A is motivated by the following result. Let A be a Tate algebra over a complete, non-archimedean, valued field [2, 4, 13]. One can associate with A a topological space X A which is uniquely determined up to homeomorphism. Namely, let ,~ be the Grothendieck topolo-gy of the rigid analytic variety Sp A associated with A [4, III.2.1]. Then it is easily seen that the topos Shv(JA) of JA is spatial, i.e. there exists a sober topological space X A such that Shv(.~) is equivalent to the topos Shv(XA) of XA. By [6, IV.4.2.4] XA is uniquely determined up to homeomorphism. In this paper we will show that XA is homeomorphic to the topological subspace Spa(A, A ~ = { v e Cont A I v(a) < 1 for every a e A ~ } of Cont A (A ~ denotes the set of power bounded elements of A). We will even show that Shv(,~A) is canonically equivalent to Shy (Spa (A, A~ Having seen that, for Tate algebras A, the topological space Cont A occurs very naturally in rigid analytic geometry, one can ask for applications of Cont A for more general topological rings A. In this paper we restrict ourselves to a class of topological rings which I call f-adic rings: A topological ring is f-adic if …