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.