On local properties of non-Archimedean analytic spaces. II. (English) Zbl 1066.32025

The local models of \(k\)-analytic spaces in the sense of Berkovich are spectra of \(k\)-algebras of the form \[ k\langle r^{-1}_1X _1,\dots, r^{-1}_n X_n\rangle /I, \] where the radii of convergence \(r_v\) are arbitrary elements of \(\mathbb{R}^X_+\). If \(r_v\in \sqrt{|k^X|}\), such algebras are called strictly \(k\)-affinoid and the corresponding global spaces strictly \(k\)-analytic.
The author proves that the functor of spaces \[ \text{strictly }k\text{-An}\to k\text{-An} \] is fully faithful, in particular there is at most one strict model of a \(k\)-analytic space. His main tool is a generalized concept of reduction of an affinoid space: \[ \widetilde A:= \bigoplus_{r\in \mathbb{R}^X_+} A_r/A_{<r}, \] where \(A_r\) resp. \(A_{<r}\) denotes the set of elements with spectral norm \(\leq r\) resp. \(<r\).
The reduction \(\widetilde k\) is not a field, but as a graduated ring very much behaves like a field. Thus the author succeeds – not without some technical difficulties – in defining and studying thoroughly reductions of germs of analytic spaces and gets the desired result. A by-product is the statement that for \(k\)-analytic spaces the properties “closedness” and “properness” of morphisms are local with respect to the base space.
For part I of this paper, see the author, Math. Ann. 318, No. 3, 585–607 (2000; Zbl 0972.32019).


32P05 Non-Archimedean analysis
14G22 Rigid analytic geometry
12J20 General valuation theory for fields
13A18 Valuations and their generalizations for commutative rings


Zbl 0972.32019
Full Text: DOI


[1] Berkovich, V. G., Spectral Theory and Analytic Geometry Over Non-Archimedean Fields (1990), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0715.14013
[2] Berkovich, V. G., Étale cohomology for non-Archimedean analytic spaces, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 78, 5-161 (1993) · Zbl 0804.32019
[3] Bosch, S.; Güntzer, U.; Remmert, R., Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry (1984), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0539.14017
[4] Bosch, S., Orthonormalbasen in der nichtarchimedean Funktionentheorie, Manuscripta Mathematica, 1, 35-57 (1969) · Zbl 0164.21202
[5] Bourbaki, N., Algèbre commutative (1961), Paris: Hermann, Paris
[6] Huber, R., Étale Cohomology of Rigid Analytic Varieties and Adic Spaces (1996), Braunschweig: Vieweg, Braunschweig · Zbl 0868.14010
[7] H. Matsumura,Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1989. · Zbl 0666.13002
[8] Temkin, M., On local properties of non-Archimedean analytic spaces, Mathematische Annalen, 318, 585-607 (2000) · Zbl 0972.32019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.