# Finite morphisms between algebraic varieties are flat?

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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See this question also: mathoverflow.net/questions/7559/… –  Hailong Dao Apr 9 '10 at 2:45

## 1 Answer

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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