Let $K$ be a finitely generated extension of $\mathbb {Q}$. We consider the family of $\ell $-adic representations ($\ell $ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\ell $-adic cohomology of a separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.