The Stacks project

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. $\square$


Comments (1)

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.


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