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Smooth \(p\)-adic analytic spaces are locally contractible. II. (English) Zbl 1060.32010
Adolphson, Alan (ed.) et al., Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter (ISBN 3-11-017478-2/hbk). 293-370 (2004).
The paper is a continuation of the author’s article [V. G. Berkovich, Invent. Math. 137, 1–84 (1999; Zbl 0930.32016)]. Let \(k\) be a complete non-Archimedean nontrivially valued field. The author considers smooth \(k\)-analytic spaces [see V. G. Berkovich, Publ. Math. Inst. Hautes Étud. Sci. 78, 5–161 (1993; Zbl 0804.32019)]. It was shown in the first mentioned paper that any strictly analytic subdomain \(X\) of a smooth \(k\)-analytic space is locally contractible. The paper under review is devoted mainly to the proof of the following stronger result.
Each point \(x\in X\) has a fundamental system of open neighbourhoods \(V\), such that: (a) there is a contraction \(\Phi\) of \(V\) to a point \(x_0\in V\); (b) there is an increasing sequence of compact strictly analytic domains \(X_1\subset X_2\subset \cdots \subset V\) which exhaust \(V\) and are preserved under \(\Phi\); (c) for any bigger non-Archimedean field \(K\), \(V\widehat \otimes K\) has a finite number of connected components and \(\Phi\) lifts to a contraction of each of them to a point over \(x_0\); (d) there is a finite separable extension \(L\) of \(k\), such that if the above \(K\) contains \(L\), then the map \(V\widehat \otimes K\to V\widehat \otimes L\) induces a bijection between the sets of connected components.
In addition, the author develops, for general \(k\)-analytic spaces, in the case of zero characteristic, a Coleman type of \(p\)-adic integration theory. For the case of smooth \(k\)-analytic curves, see R. Coleman [Invent. Math. 69, 171–208 (1982; Zbl 0516.12017)] and R. Coleman and E. de Shalit [Invent. Math. 93, 239–266 (1988; Zbl 0655.14010)].
For the entire collection see [Zbl 1047.14001].

32P05 Non-Archimedean analysis
32C35 Analytic sheaves and cohomology groups
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