## finite surjective l.c.i morphism is flat

X,Y are locally Noetherian schemes. f:X-->Y is finite, surjective, and locally complete intersection, i.e., locally it can decompose into regular immersion and smooth morphism. Recall an immersion X-->Y is called a regular immersion at point x if O_{X,x} is O_{Y,y} modulo an ideal generated by a regular sequence. Prove that f is flat. In particular, f will be a simultaneously open and closed morphism.

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Sorry, very late answer to your comment below: l.c.i implies finite flat dimension essentially because a quotient by a regular sequence has finite flat resolution (the Koszul complex) – Hailong Dao Apr 30 2010 at 0:47
This is worded like a homework problem. – Angelo Jan 15 2012 at 6:25

Work locally, assume that $f: R\to S$ is a local homomorphism. Let $\operatorname{cmd}(R) =\operatorname{dim} R-\operatorname{depth} R$ (this is the so-called Cohen-Macaulay defect of $R$). Claim: $\operatorname{cmd}$ is preserved by l.c.i maps (easy, essentially because both depth and dimension drop by one when you kill a regular element).

Now since the map $\phi: \operatorname{Spec}(S) \to \operatorname{Spec} (R)$ is finite and surjective, $\operatorname{dim} R= \operatorname{dim} S$, which combines with the last claim to show that $\operatorname{depth} R = \operatorname{depth} S$. But since l.c.i also implies finite flat dimension, we have $\operatorname{depth} R -\operatorname{depth} S = pd_RS$, so $S$ is flat over $R$.

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 Do either of you know any good references for this stuff, other than Liu? I like the topics and style of his book, so now I'm greedy and looking for more :) – Andrew Critch Dec 2 2009 at 9:22 Andrew: Unfortunately I do not know any. If you want all the technical details of recent results on this kind of stuff, look at Lucho Avramov publications page (especially the lower half): math.unl.edu/~lavramov2/papers.html – Hailong Dao Dec 2 2009 at 16:52 Thanks, but I am still not sure why l.c.i impies finite flat dimension – Taisong Jing Dec 3 2009 at 7:35 @Andrew Critch: I'm pretty sure this is covered in the commutative algebra section of Stacks-GIT. – Harry Gindi Apr 9 2010 at 2:51