I'm a bit confused about Exercise II 5.15 in Hartshorne's Algebraic Geometry, especially part (b) and (c) which are

(b) Let $X$ be an affine noetherian scheme, $U$ an open subset, and $\mathscr{F}$ coherent on $U$. Then there exists a coherent sheaf $\mathscr{F}'$ on $X$ with $\mathscr{F}'|_U \cong \mathscr{F}$. [Hint: Let $i: U \to X$ be the inclusion map. Show that $i_*\mathscr{F}$ is quasi-coherent, then use (a).]

(c) With $X, U, \mathscr{F}$ as in (b), suppose furthermore we are given a quasi-coherent sheaf $\mathscr{G}$ on $X$ such that $\mathscr{F} \subset \mathscr{G}|_U$. Show that we can find $\mathscr{F}'$ a coherent subsheaf of $\mathscr{G}$, with $\mathscr{F}'|_U \cong \mathscr{F}$. [Hint: Use the same method, but replace $i_*\mathscr{F}$ by $\rho^{-1}(i_*\mathscr{F})$, where $\rho$ is the natural map $\mathscr{G} \to i_*(\mathscr{G}|_U)$.]

Part (a) is to show that any quasi-coherent sheaf on an affine noetherian scheme is the union of its coherent subsheaves, which is easy.

So my guess at part b. is to cover $U$ finitely by open affines $D(f_i)$, over which $\Gamma(D(f_i), \mathscr{F})$ is finitely generated as an $A_{f_i}$-module. Those generators come from $\Gamma(X, i_*\mathscr{F})$, if multiplied by appropriate powers of $f_i$. After all, $\Gamma(D(f_i), \mathscr{F})$ is just the localisation of $\Gamma(X, i_*\mathscr{F})$ at $f_i$ ($i_*\mathscr{F}$ is quasi-coherent, because $U$ is noetherian, by Prop. II 5.8). Then we take the finitely generated $A$-module, which contains all the preimages of the generators over $D(f_i)$. This defines a coherent sheaf $\mathscr{F}'$ on $X$, which restricts to $\mathscr{F}|_{D(f_i)}$, hence $\mathscr{F}'|_U = \mathscr{F}$.

As far as I see this method works identically for c. by taking local generators of $\mathscr{F}|_{D(f_i)} \subset \mathscr{G}|_{D(f_i)}$ in $\Gamma(X, \mathscr{G})$.

I just don't see how I should use (a), or why I should use natural map $\rho: \mathscr{G} \to i_*(\mathscr{G}|_U)$. Do I have any mistake in my reasoning?


Here are outlines to solutions for (b) and (c) which use (a):

(b) $i_*\mathcal{F}$ is quasicoherent by proposition II.5.8.c, so it can be written as $\bigcup \mathcal{G}_\alpha$ for coherent subsheaves $\mathcal{G}_\alpha$ each isomorphic to $\widetilde{N}_\alpha$ for $A$-modules $N_\alpha$. Since $A$ is Noetherian, every directed set of submodules has a maximal element, which is the union of the $N_\alpha$. Write $N=\bigcup N_\alpha$ and then $\mathcal{F}'=\widetilde{N}$, which is coherent by construction. Then $\mathcal{F}'|_U\cong i^*\mathcal{F}'\cong i^*i_*\mathcal{F}\cong \mathcal{F}$.

(c) $\rho^{-1}(i_*\mathcal{F})$ is the pullback of a quasicoherent sheaf under a map of quasicoherent sheaves and thus quasicoherent. Since $\rho^{-1}(i_*\mathcal{F})|_U=\mathcal{F}$, we can now apply (b).

I don't see an obvious error in your reasoning, but personally I like these proofs a little better than constructing the sheaf on an open cover. Whatever floats your boat.

  • $\begingroup$ I don't think that $N$ is fin. gen. That would imply that $i_*\mathcal{F}$ is itself coherent, because it is the union of the directed set of it's coherent subsheaves. $\endgroup$ – red_trumpet Mar 5 at 6:43
  • $\begingroup$ (Using the fact that $\widetilde{}$ and $\Gamma(X,-)$ give an equivalence of categories of quasi-coherent sheaves and $A$-modules, under which coherent sheaves are mapped to finitely generated modules) $\endgroup$ – red_trumpet Mar 5 at 8:41

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