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Let $f: X \to Y$ be a finite surjective morphism of nonsingular varieties over a field $k$. Exercise III 9.3. in Hartshorne's Algebraic Geometry sais that if $k$ is algebraically closed, then $f$ is flat.

Is this still true, if $k$ is no longer algebraically closed, but still a perfect field?

In particular, I am interested in the case where $k=\mathbb{R}$.

Hans
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    This is true without any hypothesis on $k$, just because $X, Y$ are regular, see http://en.wikipedia.org/wiki/Flat_morphism#Flatness_and_dimension – Cantlog Aug 04 '14 at 16:13
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    If you need a reference, this is Remark 3.11 in Chapter 4 of Qing Liu's book Algebraic Geometry and Arithmetic Curves. – Jesko Hüttenhain Aug 04 '14 at 16:45

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