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Gauss-Manin stratification and stratified fundamental group schemes. (Stratification de Gauss-Manin et groupes fondamentaux stratifiés.) (English. French summary) Zbl 1298.14022
Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(X/k\) be a smooth connected scheme and \(S/k\) a smooth scheme. The category \(\text{str}(X/k)\) of stratified bundles, equipped with the fiber functor at a \(k\)-value point \(x\) of \(X\) is a Tannaka category, which yields a pro-algebraic group scheme \(\pi^{\text{str}}(X, x)\), called the stratified group scheme of \(X\) at \(x\). The commutative quotient and the solvable quotient are \(\pi^{\text{str}}_{\text{comm}}(X,x)\) and \(\pi^{\text{str}}_{\text{sol}}(X,x)\) respectively.
For a smooth proper map \(f: X \to S\) with connected fibers, we fix \(x \in X(k)\) and let \(s = f(x)\). Denote \(X_s\) the fiber of \(f\) at \(s\) and let \(i: X_s \to X\) be the natural morphism. The map \(f\) yields a tensor functor \(f^*:\text{str}(S/k) \to \text{str}(X/k)\) and the map \(i\) yields a tensor functor \(i^*:\text{str}(X/k) \to \text{str}(X_s/k)\). Thus we get a sequence \[ \pi_{\text{str}}(X_s,x) \to \pi_{\text{str}}(X,x) \to \pi_{\text{str}}(S,s) \to 1 \tag{*} \] and two similar sequences \[ \pi^{\text{str}}_{\text{comm}}(X_s,x) \to \pi^{\text{str}}_{\text{comm}}(X,x) \to \pi^{\text{str}}_{\text{comm}}(S,s) \to 1 \tag{**} \] and \[ \pi^{\text{str}}_{\text{sol}}(X_s,x) \to \pi^{\text{str}}_{\text{sol}}(X,x) \to \pi^{\text{str}}_{\text{sol}}(S,s) \to 1. \tag{***} \] The main theorem of the paper is that the sequence (**) and (***) are exact, whether the sequence (*) is exact or not is unknown.
Reviewer: Xiao Xiao (Utica)

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14L17 Affine algebraic groups, hyperalgebra constructions
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