Independence of l-adic representations of geometric Galois groups

TL;DR

This paper proves that the family of l-adic Galois representations associated with a scheme over a finitely generated field is almost independent, meaning their kernels' fixed fields are linearly disjoint over a finite extension, revealing new insights into their images and ramification.

Contribution

It establishes the almost independence of l-adic Galois representations over finitely generated fields, extending understanding of their structure and ramification properties.

Findings

Family of l-adic representations is almost independent.

Fixed fields of kernels are linearly disjoint over a finite extension.

Provides new results on images and ramification of these representations.

Abstract

Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations varying over ell is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ell become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and ramification of this family of representations.