Sections of quasi-coherent sheaves have only meromorphic singularities at infinity.

Lemma 28.17.1. Let $X$ be a scheme. Let $f \in \Gamma (X, \mathcal{O}_ X)$. Denote $X_ f \subset X$ the open where $f$ is invertible, see Schemes, Lemma 26.6.2. If $X$ is quasi-compact and quasi-separated, the canonical map

$\Gamma (X, \mathcal{O}_ X)_ f \longrightarrow \Gamma (X_ f, \mathcal{O}_ X)$

is an isomorphism. Moreover, if $\mathcal{F}$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules the map

$\Gamma (X, \mathcal{F})_ f \longrightarrow \Gamma (X_ f, \mathcal{F})$

is an isomorphism.

Proof. Write $R = \Gamma (X, \mathcal{O}_ X)$. Consider the canonical morphism

$\varphi : X \longrightarrow \mathop{\mathrm{Spec}}(R)$

of schemes, see Schemes, Lemma 26.6.4. Then the inverse image of the standard open $D(f)$ on the right hand side is $X_ f$ on the left hand side. Moreover, since $X$ is assumed quasi-compact and quasi-separated the morphism $\varphi$ is quasi-compact and quasi-separated, see Schemes, Lemma 26.19.2 and 26.21.13. Hence by Schemes, Lemma 26.24.1 we see that $\varphi _*\mathcal{F}$ is quasi-coherent. Hence we see that $\varphi _*\mathcal{F} = \widetilde M$ with $M = \Gamma (X, \mathcal{F})$ as an $R$-module. Thus we see that

$\Gamma (X_ f, \mathcal{F}) = \Gamma (D(f), \varphi _*\mathcal{F}) = \Gamma (D(f), \widetilde M) = M_ f$

which is exactly the content of the lemma. The first displayed isomorphism of the lemma follows by taking $\mathcal{F} = \mathcal{O}_ X$. $\square$

Comment #1144 by Simon Pepin Lehalleur on

Suggested slogan: Sections of quasi-coherent sheaves have only meromorphic singularities.

Comment #1145 by on

The last sentence of the proof is a bit dodgy. I'm not quite sure what you intend to say. Something like "The first displayed isomorphism of the lemma follows by taking $\mathcal{F}=\mathcal{O}_X$"?

Comment #4579 by Andy on

Pardon my ignorance, what does the slogan have to do with this theorem?

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