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On the quasi-compact Stein spaces in rigid geometry. (Sur les espaces de Stein quasi-compacts en géométrie rigide.) (French) Zbl 0724.32018

Rigid analytic spaces defined over a complete ultrametric field are considered. Such a space is called a Stein space (quasi compact space), if \(H^ q(X,F)=0\) for each \(q\geq 1\) and each coherent sheaf \(F\) on \(X\) (if it has a finite admissible affinoid covering resp.). Beside others the following interesting results are proven.
(1) Let X be a quasi compact Stein space over \(K\) and \(U\) a rational subset of \(X\), i.e. there exists a generating system \(\{f_ 0,f_ 1,...,f_ n\}\) of the unit ideal of \({\mathcal O}_ X(X)\) such that \(U\) is equal to the open set \(\{x\in X;\quad | f_ i(x)| \leq | f_ 0(x)| \text{ for } i=1,...,n\}.\) Then \(U\) is a Stein space, too.
(2) \(X\times_{SpK}Y\) is a Stein space if \(X\) and \(Y\) are quasi compact Stein spaces over \(K\).
(3) There exists a regular, quasi compact Stein space which is not affinoid.
The main tool is the following useful theorem: A quasi compact separable space over \(K\) is a Stein space if and only if it is holomorphically separable and \(H^ q(X,{\mathcal O}_ X)=0\) for each \(q\geq 1\).
Reviewer: A.Duma (Hagen)

MSC:

32P05 Non-Archimedean analysis
32E10 Stein spaces
14G20 Local ground fields in algebraic geometry
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