Finite morphisms between algebraic varieties are flat? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:02:00Z http://mathoverflow.net/feeds/question/20802 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20802/finite-morphisms-between-algebraic-varieties-are-flat Finite morphisms between algebraic varieties are flat? Fei YE 2010-04-09T01:43:52Z 2010-04-09T02:23:44Z <p>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? </p> http://mathoverflow.net/questions/20802/finite-morphisms-between-algebraic-varieties-are-flat/20806#20806 Answer by JT for Finite morphisms between algebraic varieties are flat? JT 2010-04-09T02:21:47Z 2010-04-09T02:21:47Z <p>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.</p>