# Confusion about Exercise II .5.15 in Hartshorne

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?

(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 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. – red_trumpet Mar 5 at 6:43
• (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) – red_trumpet Mar 5 at 8:41