I know that smoothness is fppf local on the base, but this is not enough because taking algebraic closures is not finitely presented. The reason I'm asking this is because I want an easy/quick argument or a reference (which I did not find).
If it helps, I'm looking at a surjective homomorphism of Abelian varieties $f:A \to B$ over some field $k$, and I know that $f_{\overline{k}}$ is smooth, but I want to show that $f$ is smooth.
