Recovering function fields from their integral $\ell$-adic cohomology with the Galois action

In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.