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For a closed immersion $i$ of schemes, the coherent sheaf $i_\ast\mathcal{O}_Y$ of $\mathcal{O}_X$-algebras is generated by a (single) global section. Now, let $f:Y\to X$ be a surjective finite morphism between Noetherian schemes.

Is the coherent sheaf $f_\ast\mathcal{O}_Y$ of $\mathcal{O}_X$-algebras generated by its global sections? If that is not satisfied in general, is it known when it is?

References or counterexamples would be appreciated.

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It is almost never the case for example when both $X,Y$ are projective varieties. Of course, everything is globally generated if $X$ (and then $Y$) is affine. In the projective case, if $\deg f>1$ and assuming both smooth, we have $f_*\mathcal{O}_Y=\mathcal{O}_X\oplus E$ where $E$ is a rank $\deg f-1$ vector bundle on $X$ and $H^0(E)=0$, so it can not be globally generated.

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  • $\begingroup$ Thank you! My motivation for the question is an attempt to understand a proof in a paper, as detailed here. $\endgroup$ – user24453 Apr 5 '17 at 15:07
  • $\begingroup$ @Mohan Do we know higher cohomology of $E$? and is $E$ always locally free? Thanks $\endgroup$ – Feng Hao Oct 8 '17 at 4:24
  • $\begingroup$ For a finite map, there is no higher cohomology, since those maps are affine. No, $E$ may not be locally free in general, but in the above case I was thinking of smooth curves. $E$ is a vector bundle if and only if $f$ is in addition flat. $\endgroup$ – Mohan Oct 8 '17 at 13:14

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