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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.

user267839
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    This is true much more often: if $f:X\to Y$ is a locally finite type morphism of schemes, then the function $x\mapsto \dim_x (X_{f(x)})$ is upper semi-continuous. This is EGA IV 13.1.3. – Hank Scorpio Dec 28 '21 at 18:11
  • but this statement works only with local dimension $\dim_x(X_{f(x)}) := \inf_{U \subset X_{f(x)} \text{ open nbhd of x }} \dim U$. so that's not a generalization of the claim above. – user267839 Dec 28 '21 at 22:53

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