Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be morphisms of irreducible varieties such that $f$ is surjective on scheme points (equivalently geometric points) and $gf$ is flat. Is it necessarily true that $g$ is flat?
The fibres of $g$ at least are equidimensional, so for many cases it is automatic by miracle flatness that this map is flat. I haven't seen this written anywhere which makes me nervous.