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.


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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