## 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 $$\text{Sp}A$$.

### MSC:

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

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