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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)

##### MSC:
 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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