## Abstract

Let X be a connected scheme, smooth and separated over an alge- braically 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*(Y_{x}; F_{ℓ}) are the reduction modulo-ℓ of those of π_{1}(X, x) acting on H*(Y_{x}, ℤ_{ℓ}) for ℓ greater than a constant depending only on f: Y → X, d. We apply this result to show that the geometric variant with F_{ℓ}-coefficients of the Grothendieck-Serre semisimplicity conjecture - namely, that π_{1}(X, x) acts semisimply on H*(Y_{x}, F_{ℓ}) for ℓ ≫ 0-is equivalent to the condition that the image of π_{1}(X, x) acting on H*(Y_{x};Q_{ℓ}) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of Q_{ℓ}-points of its Zariski closure. Ultimately, we prove the geometric variant with F_{ℓ}-coefficients of the Grothendieck-Serre semisimplicity conjecture.

Original language | English |
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Pages (from-to) | 205-236 |

Number of pages | 32 |

Journal | Annals of Mathematics |

Volume | 186 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jul 2017 |

## Keywords

- Algebraic groups
- Big monodromy
- Positive characteristic
- Semisimplicity
- Tate conjecture
- Tensor in- variants
- Étale cohomology
- Étale fundamental group