Let $f:X\rightarrow Y$ be a finite morphism between Noetherian integral schemes that is surjective on the underlying topological spaces. Does there exist an integer $n>0$ and a coherent $O\_X$-module $F$ such that there is a morphism of $O\_Y$-modules $O\_Y^n\rightarrow f\_\*F$ inducing an isomorphism on the stalks at the generic point of $Y$? 


It is pretty easy to see that there must be a quasi-coherent $O\_X$-module like this. Pick an affine open $V\subset Y$, then its inverse image [is](https://stacks.math.columbia.edu/tag/01WI) an affine open $U\subset X$. It is easy to see that $O\_U$ answers the question for the morphism $f|\_{U}$. Therefore, if $i:U\rightarrow X$ denotes the inclusion, $i\_\* O\_U$ is a quasi-coherent $O\_X$-module ([because](https://stacks.math.columbia.edu/tag/01LC) $i$ is [qcqs](https://stacks.math.columbia.edu/tag/01OX)) answering the question (because the generic stalk can be computed on any non-empty open). 


It appears, however, that this way we will rarely get a coherent sheaf. [EGA](http://www.numdam.org/article/PMIHES_1967__32__5_0.pdf#page=306) IV, partie quadrieme, Cor. 21.12.7 says that if $X$ is separated, the complement of $U$ has codimension 1 so $i\_\*$ has little chance to preserve coherence. 


P.S. This is Hartshorne Ch. III Ex. 4.2 (a). I have consulted some solutions online and could not convince myself they really produce coherent (rather than quasi-coherent) sheaves. The same question has been asked on [Math Stackexchange](https://math.stackexchange.com/q/3197266/251240) but we have different perspective on the direction of the attack. 



Map of coherent sheaves inducing isomorphism on the stalks at the generic point
The following is basically a copy of dhy's comments with links (any mistakes are mine).


First, assume you know that for a finite surjective morphism $f:X\rightarrow Y$ between Noetherian integral affine schemes, there exists a positive integer $n$ and a morphism of $\mathcal{O}\_Y$-modules $\mathcal{O}\_Y^{\oplus n}\rightarrow f\_\*\mathcal{O}\_X$ inducing an isomorphism at the generic point. This is shown in the linked question.


Now, we have a finite surjective morphism $f:X\rightarrow Y$ between Noetherian integral schemes. Pick a non-empty affine open $j:V\rightarrow Y$ whose inverse image is a non-empty affine open $i:U\rightarrow X$. The restriction $f|\_U:U\rightarrow V$ is a finite surjective map ([by definition](https://stacks.math.columbia.edu/tag/01WH)). We know therefore that there is a positive integer $n$ and a morphism of $\mathcal{O}\_Y$-modules $j\_\*\mathcal{O}\_V^{\oplus n}\rightarrow f\_\* i\_\* \mathcal{O}\_U$ inducing an isomorphism at the generic point. This can be composed with the natural map $\mathcal{O}\_Y\rightarrow j\_\*\mathcal{O}\_V$ to get a morphism of $\mathcal{O}\_Y$-modules $\mathcal{O}\_Y^{\oplus n}\rightarrow f\_\* i\_\* \mathcal{O}\_U$ inducing an isomorphism at the generic point.


On any qcqs scheme any quasi-coherent sheaf [is](https://stacks.math.columbia.edu/tag/01PG) the directed colimit of its quasi-coherent subsheaves of finite type. Apply this to $i\_\*\mathcal{O}\_U$. [Since](https://stacks.math.columbia.edu/tag/01XZ) $X$ is locally Noetherian, these subsheaves are actually coherent.


We claim that the pushforward by a qcqs morphism, considered as a functor on the categories of quasi-coherent modules, commutes with directed colimits. First note that the functor of restriction to an open set commutes with colimits so the question is local on the target. Assume that the target is affine, then [express](https://stacks.math.columbia.edu/tag/01LC) the pushforward as a kernel. The conclusion follows from the fact that kernels commute with directed colimits and the fact that the pushforward by a morphism between affine schemes commutes with directed colimits.


Therefore, we can pushforward the the directed colimit to get a colimit of coherent sheaves on $Y$ (they will be coherent because $f$ is proper) coconverging to $f\_\*i\_\*\mathcal{O}\_U$.


Finally, use Lemma 34 and Proposition 75 [here](http://therisingsea.org/notes/ModulesOverAScheme.pdf) (the word "concentrated" means qcqs) to conclude that any coherent sheaf on a Noetherian scheme is a finitely presented object of the category of quasi-coherent sheaves (i.e. $\mathrm{Hom}(F, -)$ preserves directed colimits). Therefore, the map $\mathcal{O}\_Y^{\oplus n}\rightarrow f\_\* i\_\*\mathcal{O}\_U$ factors through a coherent $\mathcal{O}\_Y$-module $G\subset f\_\* i\_\*\mathcal{O}\_U$. To see that $\mathcal{O}\_Y^{\oplus n}\rightarrow G$ induces an isomorphism at the generic point, first notice that it must induce an injection at the generic point (because its composition with the inclusion is an isomorphism) and second notice that the generic rank of $G$ does not exceed that of $f\_\* i\_\*\mathcal{O}\_U$. An injective map between finite-dimensional vector spaces such that the dimension of the target does not exceed the dimension of the source must be an isomorphism. Game over.


