Geometric monodromy — semisimplicity and maximality
Résumé
Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p ≥ 0, let f : Y → X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded length ≤ d of π1(X, x) acting on the étale cohomology groups H * (Yx, F ) are the reduction modulo-of those of π1(X, x) acting on H * (Yx, Z ) for greater than a constant depending only on f : Y → X, d. We apply this result to show that the geometric variant with F -coefficients of the Grothendieck-Serre semisimplicity conjecture -namely that π1(X, x) acts semisimply on H * (Yx, F ) for 0 -is equivalent to the condition that the image of π1(X, x) acting on H * (Yx, Q ) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Q -points of its Zariski closure. Ultimately, we prove the geometric variant with F -coefficients of the Grothendieck-Serre semisimplicity conjecture.
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