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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).

MSC:

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

Citations:

Zbl 0972.32019
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References:

[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
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