Let $f:X\longrightarrow Y$ be a finite, flat morphism of schemes. Then, we know that $f_*\mathcal{O}_X$ is flat over $Y$, and also that $f_*\mathcal{O}_X$ is a coherent $\mathcal{O}_Y$ module.

We know from commutative algebra that, a finitely generated module $M$ over a local ring $R$ is a flat $R$-module if and only if $M$ is a free module.

From this, since $f_*\mathcal{O}_X$ is flat over $Y$ and coherent, $f_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$ module.

But the book I am reading (Ueno), claims that $f_*\mathcal{O}_X$ is free as well. Why is this? I am not able to get it.

Image from the book Ueno

Any help will be appreciated!

  • 1
    $\begingroup$ Are your schemes noetherian (since you write "coherent" and usually then the people mean "of finite type over noetherian scheme" although the general definition of "coherent" means something else)? Also, what is the precise reference to Ueno's book and has $f$ further properties (such as étale)? $\endgroup$ – Martin Brandenburg Nov 4 '14 at 16:47
  • $\begingroup$ I have added the text from Ueno's book. He doesn't seem to require noetherian-ness. $\endgroup$ – gradstudent Nov 4 '14 at 17:27
  • $\begingroup$ The noetherian-assumption is essential. A flat finitely generated module doesn't have to finitely presented if the base ring is not noetherian. $\endgroup$ – Martin Brandenburg Nov 4 '14 at 18:52

I think it's just a typo in Ueno's book. In general, the finite flat morphisms to a noetherian scheme $Y$ correspond 1:1 to quasi-coherent $\mathcal{O}_Y$-algebras whose underlying $\mathcal{O}_Y$-module is locally free of finite rank, and there is no reason to expect this module to be (globally) free.

Example: Let $I$ be a non-principal fractional ideal in a Dedekind domain $A$. Then the $A$-module $I \oplus A$ is not free (May's notes, Cor. 6.8), but locally free since $I$ is (Theorem 5.1), and carries the structure of an $A$-algebra with $I^2=0$. Now take $\mathrm{Spec}(A \oplus I) \to \mathrm{Spec}(A)$.

  • $\begingroup$ Thank you! I have been struggling with this all day! $\endgroup$ – gradstudent Nov 4 '14 at 18:13

The sheaf $f_*\mathcal{O}_X$ needn't be free even for $f:X\to Y$ a finite flat morphism of smooth varieties over a field $k$.

For example, to a smooth hypersurface $B\subset \mathbb P^n_k$ of degree $d$ one associates a finite flat morphism $f:X\to Y=\mathbb P^n_k$ ( a so called n-cyclic covering of $\mathbb P^n$) with branch-locus $B$ ( meaning that the restricted morphism $X\setminus f^{-1}(B)\to Y\setminus B$ is an étale covering of degree $d$) .
One then has the formula $$ f_*\mathcal{O}_X=\oplus_{j=0}^{d-1}\mathcal O_{\mathbb P^n} (-j) $$ A reference is the book Compact Complex surfaces by Barth, Hulek, Peters, Van de Ven, Chapter I, Section 17.

  • $\begingroup$ Thank you! I wish I could accept more than one answer! $\endgroup$ – gradstudent Nov 4 '14 at 18:14
  • $\begingroup$ Dear poorna, you are welcome. Martin's answer was posted one hour before mine and has a different flavour, which it is perfectly legitimate to prefer. $\endgroup$ – Georges Elencwajg Nov 4 '14 at 18:25

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