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Let $f: X \rightarrow Y$ be a morphism of irreducible projective varieties, that is both finite and surjective.

Does this mean that it is flat?

I have tried the following:

By finiteness, the map is locally $\text{Spec}(B) \rightarrow \text{Spec}(A)$, i.e. $A \rightarrow B$ in rings, where $B$ is a finitely generated $A$-module. By surjectiveness, A goes injectively into $B$. Since we are dealing with varieites, $A$ and $B$ are finitely generated $k$-algebras with no nilpotents. In fact since the varieties are irreducible, they have no zero divisors.

Since a module is flat over a ring if and if for all prime ideals in the ring, the localization of the module is flat over the localized ring, we would be done if we could show that $B$ is flat over $A$.

This is where my knowledge gets too sketchy. We cannot assume $A$ to be a PID right? Is it true that we exactly need to show that $B$ is an acyclic object in $A$-Mod for $\text{Tor}$? Finally, can someone tell me how to show that $B$ is flat over $A$, or tell me that this is wrong?

Thanks!

Joachim
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  • $\text{Tor}_1$ sounds better. –  Feb 02 '13 at 21:37
  • So, you have a finite ring extension $A\subset B$ of $k$-affine domains (projectivity should implies that $A,B$ are also graded, right?) and want to prove that $B$ is flat over $A$? I guess you need something more. –  Feb 02 '13 at 22:14
  • This is true if $X$ CM and $Y$ regular. References are given in https://math.stackexchange.com/q/887218/52299 – ggg Jan 29 '21 at 17:14

1 Answers1

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No. Let $C$ be a projective curve which has only one singularity, a node. Then the normalization map $\tilde{C} \rightarrow C$ is not flat, but it is finite and surjective.

Surjectivity is clear. Finiteness is by the finiteness of normalization.

If the map is flat, then as it is flat and finite, all fibers should have the same size, which is false here.

I believe that more generally, normalization is typically not flat; but I don't know what hypothesis you need to say this. Perhaps it's true whenever the variety isn't already normal.

EDIT: I found https://mathoverflow.net/questions/64776/flatness-of-normalization, so apparently this is true whenever the variety isn't already normal.

Community
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Nehsb
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  • I thought of this example, but I was not sure that flatness implied fibers of the same size. Do you know a reference for this? – Brenin Feb 03 '13 at 15:01
  • Here's one reference: Vakil's notes, http://math.stanford.edu/~vakil/216blog/FOAGjan1513public.pdf, see section 24.4, in particular Exercise 24.4.G. For a more general version, see section 24.7. – Nehsb Feb 03 '13 at 15:08
  • Thanks! In the meanwhile i indeed find out that under my hypothesis, the result is true if the varieties are smooth and may not be true if the varieties are not smooth: Hartshorne exercise III.9.3.. – Joachim Feb 03 '13 at 17:30