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Let $f: Y \to X$ be a morphism of varieties (proper if necessary).

I read from a paper that if all the fibres of $f$ are of the same dimension then $f$ is flat. This seems skeptical for me, and I was wondering more conditions are need to have the flatness

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up vote 7 down vote accepted

You are right to be skeptical because the result is false.
For example if $X\subset \mathbb A^2$ is the cusp defined by $y^2=x^3$, its normalization $f:\mathbb A^1=Y\to X: t\mapsto (t^2,t^3)$ is not flat although $f$ is proper and all of its fibers consist of exactly one point and thus have the same dimension, namely $0$ .

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This does hold in a broad class of varieties though, like when X and Y are both regular. –  Alex Youcis Feb 3 at 23:01
    
@Georges Elencwajg Thank you! But is there any case the above claim to be true? I might miss something while reading the paper... –  Li Yutong Feb 3 at 23:03
    
@AlexYoucis What do you mean by this? –  Li Yutong Feb 3 at 23:04
    
@LiYutong: Dear Li Yutong, See my answer. Regards, –  Matt E Feb 4 at 4:21
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This is true provided that the source is Cohen--Macaulay (e.g. regular) and the base is regular. This is sometimes called miracle flatness. (It can be found as Thm. 23.1 of Matsumura's CRT.)

It generalizes the fact that a non-constant map from a reduced curve to a smooth curve is necessarily flat.

(In the case when the fibre dimension is zero, it is related to the Auslander--Buchsbaum theorem, which says that a f.g. faithful Cohen--Macaulay module over a regular local ring is necessarily free.)

Added: As Georges Elencwajg points out in a comment below, in the context of miracle flatness I should require that we are working with locally finite type schemes over a field.

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Dear Matt, aren't the schemes in question required to be locally algebraic over a field ? ( The references I checked, including Matsumura, make that hypothesis). –  Georges Elencwajg Feb 4 at 9:56
    
@GeorgesElencwajg: Dear Georges, Quite possibly, if that's what the references suggest. (I've never carefully thought about this result or its proof; even my remark about the relative dimension zero case was a bit off-the-cuff, although I do think there's a connection.) Best wishes, Matt –  Matt E Feb 4 at 23:56
    
@GeorgesElencwajg: P.S. I should have added: thanks for pointing this out! I've made a correction to this effect. –  Matt E Feb 4 at 23:58
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