In This question following result is stated:
If $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each point of $X$ the dimension of the fiber it belongs to is upper semicontinuous on $X$.
I assume that here the author considered by a variety a separate scheme, of finite type over a field.
Question: I'm looking for a reference for a proof of this claim. Especially my focus lies on the aspect if the assuption that $k$ is algebraically closed really neccessary or dispensable and I hope that a close inspection of the proof's details would reveal if & how the assumption on algebraic closedness of $k$ in essential way flows in the proof.