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Local monodromy in non-Archimedean analytic geometry. (English) Zbl 1111.14012

The paper considers the behavior of local systems on punctured discs over non-Archimedean fields. Let \(\left( K,\left| \cdot\right| \right) \) be an algebraically closed valued field of mixed characteristic \((0,p)\), complete for its rank one valuation \(\left| \cdot\right| :K\rightarrow\Gamma_{K}\cup\left\{ 0\right\} \). Let \(D(r)^{\ast}\) denote the punctured disc over \(K\) of radius \(r\). Let \(\Lambda\) be a local ring which is a filteredunion of finite rings on which \(p\) is invertible. Let \(\mathcal{F}\) be a locally constant and locally free sheaf of \(\Lambda \)-modules of finite rank on the étale site of \(D(1)^{\ast}\).
The main result of the paper (see Theorem 4.2.40) states that if \(H^{1}\left( D(r)_{\text{ét}}^{\ast},\mathcal{F}\right) \) is a \(\Lambda\)-module of finite type, then there exists a connected open subset \(U\subset D(1)^{\ast}\) such that \(U\cap D(\varepsilon)^{\ast}\neq\varnothing\) for every \(\varepsilon>0\), and the restriction of \(\mathcal{F}\) to \(U_{\text{ét}}\) admits a break decomposition as direct sum of locally constant subsheaves of the form \(\mathcal{F\mid}_{U}{\simeq} \bigoplus_{\gamma\in\Gamma_{0}} \mathcal{F}\left( \gamma\right) \), where \(\Gamma_{0}\) is the product of the ordered groups \(\mathbb{Q}\times\mathbb{R}\), endowed with the lexicographic ordering. This decomposition is compatible with tensor products and Hom functors. Furthermore, a Hasse-Arf type result holds. The paper also contains a new proof of the \(p\)-adic Riemann existence theorem for finite étale coverings of annuli.

MSC:

14G22 Rigid analytic geometry
12J25 Non-Archimedean valued fields
14F20 Étale and other Grothendieck topologies and (co)homologies
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