## 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).

### MathOverflow Questions:

Enlightening examples of tropical skeletons of Berkovich spaces

### MSC:

 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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### References:

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