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It is well known that for $f:X\rightarrow Y$ an affine morphism of schemes, the direct image functor $\mathscr{F}\mapsto f_*\mathscr{F}$ induces an equivalence of categories between the category of quasi-coherent $\mathscr{O}_X$-modules and the category of quasi-coherent $f_*\mathscr{O}_X$-modules. (See, for example, Lemma 29.11.6 at Stacks Project).

If $f:X\rightarrow Y$ is flat and finite then it is also true that for every locally free sheaf $\mathscr{F}$ of $\mathscr{O}_X$-modules the direct image $f_*\mathscr{F}$ is a locally free sheaf of $f_*\mathscr{O}_X$-modules.

If $\mathscr{E}$ is a locally free sheaf of $f_*\mathscr{O}_X$-modules, what can we say about the corresponding quasi-coherent sheaf $\mathscr{F}$ such that $f_*\mathscr{F}=\mathscr{E}$?

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    $\begingroup$ I don't think you can say much, consider for example the case that $Y$ is the spectrum of a field. $\endgroup$
    – naf
    Commented Dec 18, 2019 at 3:59
  • $\begingroup$ It seems to me that $f_*\mathscr{F}$ is locally free as a $f_*\mathscr{O}_X$-module if and only if $\mathcal F$ is locally free as an $\mathcal O_X$-module. $\endgroup$
    – Angelo
    Commented Dec 18, 2019 at 8:09

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