Continuous valuations. (English) Zbl 0788.13010

A useful generalization of the class of adic rings with finitely generated ideals of definition is the class of \(f\)-adic rings: A topological ring is called \(f\)-adic if it has an open subring which is adic and has a finitely generated ideal of definition.
In this paper we study, for an \(f\)-adic ring \(A\), the topological space \(\text{Cont}A\) of all equivalence classes of continuous valuations of \(A\). In the first part of the paper we prove that \(\text{Cont}A\) is a spectral space and give an explicit description of the set of constructible subsets of \(\text{Cont}A\). The affinoid rings occurring in rigid analytic geometry are \(f\)-adic. – In the second part of the paper we explain, for an affinoid ring \(A\), the relation between the topological space \(\text{Cont}A\) and the rigid analytic variety \(\text{Sp}A\).


13J99 Topological rings and modules
13A18 Valuations and their generalizations for commutative rings
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