# Geometric monodromy -- semisimplicity and maximality

## TL;DR

This paper investigates the properties of monodromy actions on étale cohomology in algebraic geometry, establishing connections between semisimplicity, maximality, and invariants across different coefficients, ultimately proving a geometric variant of the Grothendieck-Serre conjecture.

## Contribution

It proves that tensor invariants of bounded length are preserved modulo- for large  and establishes the geometric variant of the Grothendieck-Serre semisimplicity conjecture.

## Key findings

- Tensor invariants are preserved modulo- for large .
- Semisimplicity of -coefficient actions is equivalent to 'almost maximal' image conditions.
- The geometric variant of the Grothendieck-Serre semisimplicity conjecture is proved.

## Abstract

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the \'etale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture -- namely that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,F_\ell)$ for $\ell\gg 0$ -- is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,Q_\ell)$ is `almost maximal' (in a precise sense; what we call `almost hyperspecial') with respect to the group of $Q_\ell$-points of its Zariski closure. Ultimately, we prove the geometric variant with $F_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture.