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
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