Geometric monodromy - Semisimplicity and maximality

Anna Cadoret, Chun Yin Hui, Akio Tamagawa

Research output: Contribution to JournalArticleAcademicpeer-review

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*(Yx; F) are the reduction modulo-ℓ of those of π1(X, x) acting on H*(Yx, ℤ) 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-coefficients of the Grothendieck-Serre semisimplicity conjecture - namely, that π1(X, x) acts semisimply on H*(Yx, F) for ℓ ≫ 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.

Original languageEnglish
Pages (from-to)205-236
Number of pages32
JournalAnnals of Mathematics
Volume186
Issue number1
DOIs
Publication statusPublished - 1 Jul 2017

Keywords

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

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