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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 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 $i$ is qcqs) 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 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 but we have different perspective on the direction of the attack.

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  • $\begingroup$ Idea how to approach this: take $f_* \mathcal{O}_X$, on a dense affine open $U\subseteq Y$ this is free, fix trivialization $f:\mathcal{O}_U^n \simeq (f_*\mathcal{O}_X)|_U$. If $Y\setminus U$ supported a Cartier divisor $D$, then there exists a line bundle $L= \mathcal{O}_Y(-mD)$ ($m\gg 0$) on $Y$ such that $f$ extends to a morphism $g:L^{\oplus n}\to f_*\mathcal{O}_X$. Now tensor this by $L^{-1}$ and use the projection formula to get $g'\colon \mathcal{O}_Y^n \to (f_*\mathcal{O}_X)\otimes L^{-1} = f_* F$ where $F = f^* L^{-1}$. The problem is ensuring the existence of $L$. $\endgroup$ – Piotr Achinger Apr 24 at 21:58
  • $\begingroup$ @PiotrAchinger do you need something like ampleness of $L^\vee$ for your argument to work? I believe any coherent sheaf on an integral Noetherian scheme is a direct sum of a locally free sheaf and a torsion sheaf so without loss of generality assume that $f_* O_X$ is a vector bundle. If we want a morphism $L^{\otimes n}\rightarrow f_* O_X$ that induces an isomorphism at the generic point, we need $n=\mathrm{rank}f_*O_X$ independent maps $L\rightarrow f_*O_X$, which is the same as $n$ sections of $L^\vee\otimes f_*O_X$. $\endgroup$ – user138661 Apr 25 at 14:31
  • $\begingroup$ If $L^\vee \otimes f_* O_X$ were globally generated, we would have these sections (but you need ampleness of $L^\vee$ for this). I think on an arbitrary proper integral scheme over a field you won't find an ample line bundle. Or does your argument work for $O_X(D)$, $D$ an arbitrary effective Cartier divisor? $\endgroup$ – user138661 Apr 25 at 14:32
  • $\begingroup$ I think ampleness is really needed if you want the extension $g$ to be surjective. Here, the sections $f_i$ have poles of some order along the components of $D$, and if $m$ is bigger than those orders then you get an extension $g$, at least generically along the components of $D$. If $X$ is normal ($S_2$) along $D$ then this $g$ extends further to $X$. $\endgroup$ – Piotr Achinger Apr 25 at 16:35
  • $\begingroup$ Maybe the following could work: $L=K(X)$ is a finite extension of $K=K(Y)$, so $L = K l_1 + \cdots + K l_n$. Now take $\mathcal{F}$ be the $\mathcal{O}_X$--module $\mathcal{O}_X l_1 + \cdots + \mathcal{O}_X l_n$ which is an $\mathcal{O}_X$--submodule of the flasque $\mathcal{O}_X$--Module $\tilde{L}$. $\endgroup$ – Jürgen Böhm Apr 25 at 17:53
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I think this can also be solved by taking the "coherent subsheaf generated by finitely many global sections of $i_* \mathcal{O}_U$". I'm not sure this is fundamentally different from Aknazar's solution, but it avoids the direct mention of (co)limits, and might be what Hartshorne had in mind.

As Aknazar, suppose we can solve the problem for affine schemes, and choose an open affine $j: V \hookrightarrow Y$ and let $i: U \hookrightarrow X$ be its preimage. Suppose we have a morphism $$ \alpha_V: \mathcal{O}_V^n \to (f|_U)_*\mathcal{O}_U$$ which is an isomorphism at the generic point $\eta$ of $Y$.

The problem is that $i_* \mathcal{O}$ is not in general coherent on $X$. But $\alpha_V$ chooses $n$ global sections $s_1,\dotsc,s_n \in \Gamma(X, i_* \mathcal{O}_U) = \Gamma(U, \mathcal{O}_U)$, which can be used to define a morphism $\alpha_X: \mathcal{O}_X^n \to i_* \mathcal{O}_U$. Let $\mathscr{G}$ be the image of this morphism. Then $\mathscr{G}$ is coherent, because for every open affine $\text{Spec }A = W \subset X$, $\mathscr{G}(W) \subset (i_*\mathcal{O}_U)(W)$ is the $A$-submodule generated by $s_1|_W,\dotsc,s_n|_W$.

This allows us to define the morphism $$ \alpha_Y: \mathcal{O}_Y^n \to f_* \mathscr{G} \subset (f i)_* \mathcal{O}_U$$ by takting the same global sections $s_1,\dots,s_n \in \Gamma(Y, f_* \mathscr{G})$. At the generic point this yields $$ \mathcal{O}^n_{Y, \eta} \xrightarrow{\alpha_{Y, \eta}} (f_*\mathscr{G})_\eta \hookrightarrow ((fi)_* \mathcal{O}_U)_\eta, $$ and the composition is $\alpha_{U, \eta}$ which is an isomorphism. Hence $\alpha_{Y, \eta}$ is an isomorphism as well.

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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). 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 the directed colimit of its quasi-coherent subsheaves of finite type. Apply this to $i_*\mathcal{O}_U$. Since $X$ 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 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 (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.

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