Lemma 27.25.3. 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 27.25.1. 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 27.25.1 (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 $s|_{W_ j}$. 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 $W_{jj'l}$. 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. $\square$

Comment #946 by correction_bot on

In the proof, there are references to "(1)" and "(2)", but these aren't labeled in the statement of the lemma.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).