  We prove that the Tate conjecture in codimension $1$ over a finitely
generated field follows from the same conjecture for surfaces over its prime
subfield. In positive characteristic, this is due to de Jong--Morrow over
$\mathbf{F}_p$ and to Ambrosi for the reduction to $\mathbf{F}_p$. We give a
different proof than Ambrosi's, which also works in characteristic $0$; over
$\mathbf{Q}$, the reduction to surfaces follows from a simple argument using
Lefschetz's $(1,1)$ theorem.

Number Theory;Algebraic Geometry