Temkin, M. On local properties of non-Archimedean analytic spaces. (English) Zbl 0972.32019 Math. Ann. 318, No. 3, 585-607 (2000). In Berkovich theory of non-Archimedean spaces over a field \(k\) (which is complete w.r.t. a non-trivial non-Archimedean valuation) there are several notions of \(k\)-analytic space. The main result of this paper gives a criterion for the property : a point \(x\) of a “general” \(k\)-analytic space has an affinoid neighbourhood. A \(k\)-analytic space \(X\) is called “good”, if every point \(x\) has an affinoid neighbourhood. The translation of “good space \(X\)” in terms of the associated rigid space \(X_0\) is the following: \(X_0\) has two admissible coverings \(\{U_0^i\}_i\) and \(\{V_0^i\}_i\) such that \(U^i_0\subset \subset _{X_0}V^i_0\) for all \(i\). The above criterion is formulated in terms of the reduction of the germ \(X_x\) which is some object \(\widetilde{X}_x\) over the residue field \(\widetilde{k}\) of \(k\). More precisely, it is a topological space \(T\) and a local homeomorphism \(f\) between \(T\) and the Riemann-Zariski space \({\mathbf P}_K\) of all valuations of a field extension \(K\) of \(\widetilde{k}\) which are trivial on \(\widetilde{k}\). The criterion reads: \(x\) has an affinoid neighbourhood if and only if there are elements \(f_1,\dots ,f_n\in K\) such that \(f(T)=\{v\in {\mathbf P}_K\mid f_1,\dots ,f_n\in v\}\). As an application of this criterion two theorems of W. Lütkebohmert are extended. Reviewer: M.van der Put (Groningen) Cited in 2 ReviewsCited in 30 Documents MSC: 32P05 Non-Archimedean analysis Keywords:Berkovich space; non-Archimedean space × Cite Format Result Cite Review PDF Full Text: DOI