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Let $X$ and $Y$ be smooth scheme over a Dedekind domain (or over a field if you prefer). Let $f \colon X \to Y$ be a finite and flat morphism and let $D$ be a divisor on $X$. Since $f$ is finite flat, we have a divisor $f_\ast D$ on $Y$ and moreover the sheaf $f_\ast \mathcal O_X(D)$ is invertible (this is not true, see below).

Is it true that $f_\ast \mathcal O_X(D)$ is the invertible sheaf associated to $f_\ast D$?

Edit The question doesn't make sense, since $f_\ast \mathcal O_X(D)$ is locally free but not invertible, as Bruno Joyal pointed out. Indeed its rank is the degree of $f$, let me say $n$. So the new question is the following:

is it true that $\bigwedge^n \left ( f_\ast \mathcal O_X(D) \right )$ is the invertible sheaf associated to $f_\ast D$?

John
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    The sheaf $f_*\mathcal O_X(D)$ is not invertible, but locally free of rank equal to the degree of $f$. – Bruno Joyal Feb 25 '14 at 13:32
  • Oh, sorry for the stupid question... – John Feb 25 '14 at 13:39
  • No, it is a good question nevertheless. Perhaps one should ask how to relate these two things, and it is quite likely that they are indeed related. In fact, I think that $f_D$ is the divisor associated to the top exterior power of $f_\mathcal O_X(D)$ (which is an invertible sheaf). – Bruno Joyal Feb 25 '14 at 13:44
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    For a finite map of smooth curves, $\det(f_* \mathscr{L}(D)) \cong (\det f_* \mathcal{O}X) \otimes \mathscr{L}(f* D)$: see Hartshorne ch. IV, exercise 2.6 – zcn Feb 25 '14 at 19:08
  • @user115654 You should post that as an answer! – Bruno Joyal Feb 26 '14 at 12:16

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Converting my comment to an answer as suggested above:

If $f : X \to Y$ is a finite map of smooth curves over an algebraically closed field (which is necessarily flat), and $D$ is a divisor on $X$, $\mathscr{L}(D)$ the associated line bundle, then $\det(f_* \mathscr{L}(D)) \cong (\det f_* \mathcal{O}_X) \otimes \mathscr{L}(f_*D)$, where $\det$ is the top exterior power of a locally free sheaf. In particular $\det(f_* \mathscr{L}(D)) \not \cong \mathscr{L}(f_* D)$ in general.

zcn
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    is this true just for morphisms of curves? I'd like to know under which hypothesis the aforementioned relation holds. – manifold Oct 18 '17 at 14:44