Lemma 28.25.3. Let X be a quasi-compact scheme. Let \mathcal{I} \subset \mathcal{O}_ X be a quasi-coherent sheaf of ideals of finite type. Let Z \subset X be the closed subscheme defined by \mathcal{I} and set U = X \setminus Z. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. The canonical map
\mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}) \longrightarrow \Gamma (U, \mathcal{F})
is injective. Assume further that X is quasi-separated. Let \mathcal{F}_ n \subset \mathcal{F} be subsheaf of sections annihilated by \mathcal{I}^ n. The canonical map
\mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}/\mathcal{F}_ n) \longrightarrow \Gamma (U, \mathcal{F})
is an isomorphism.
Proof.
Let \mathop{\mathrm{Spec}}(A) = W \subset X be an affine open. Write \mathcal{F}|_ W = \widetilde{M} for some A-module M and \mathcal{I}|_ W = \widetilde{I} for some finite type ideal I \subset A. Restricting the first displayed map of the lemma to W we obtain the first displayed map of Lemma 28.25.1. Since we can cover X by a finite number of affine opens this proves the first displayed map of the lemma is injective.
We have \mathcal{F}_ n|_ W = \widetilde{M_ n} where M_ n \subset M is defined as in Lemma 28.25.1 (details omitted). The lemma guarantees that we have a bijection
\mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ W}( \mathcal{I}^ n|_ W, (\mathcal{F}/\mathcal{F}_ n)|_ W) \longrightarrow \Gamma (U \cap W, \mathcal{F})
for any such affine open W.
To see the second displayed arrow of the lemma is bijective, we choose a finite affine open covering X = \bigcup _{j = 1, \ldots , m} W_ j. The injectivity follows immediately from the above and the finiteness of the covering. If X is quasi-separated, then for each pair j, j' we choose a finite affine open covering
W_ j \cap W_{j'} = \bigcup \nolimits _{k = 1, \ldots , m_{jj'}} W_{jj'k}.
Let s \in \Gamma (U, \mathcal{F}). As seen above for each j there exists an n_ j and a map \varphi _ j : \mathcal{I}^{n_ j}|_{W_ j} \to (\mathcal{F}/\mathcal{F}_{n_ j})|_{W_ j} which corresponds to s|_{U \cap W_ j}. By the same token for each triple (j, j', k) there exists an integer n_{jj'k} such that the restriction of \varphi _ j and \varphi _{j'} as maps \mathcal{I}^{n_{jj'k}} \to \mathcal{F}/\mathcal{F}_{n_{jj'k}} agree over W_{jj'k}. Let n = \max \{ n_ j, n_{jj'k}\} and we see that the \varphi _ j glue as maps \mathcal{I}^ n \to \mathcal{F}/\mathcal{F}_ n over X. This proves surjectivity of the map.
\square
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