Let $f: X\to Y$ be a finite (surjective) morphism between two algebraic varieties. I know when $X$ and $Y$ are non-singular and $\dim Y =1$, $f$ is flat. But in general, is it true that $f$ is a flat morphism?
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$\begingroup$ See this question also: mathoverflow.net/questions/7559/… $\endgroup$– Hailong DaoCommented Apr 9, 2010 at 2:45
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If $X$ and $Y$ are both regular, then this is true. In fact, it's true more generally if $Y$ is regular and $X$ is Cohen-Macaulay (Eisenbud, Commutative Algebra, Corollary 18.17). In general it's certainly false.
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2$\begingroup$ This fact goes by the name of "miracle flatness" in Vakil's lecture notes. $\endgroup$ Commented Jul 4, 2017 at 10:17