The Stacks project

Lemma 30.10.4. Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be coherent $\mathcal{O}_ X$-module. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Denote $Z \subset X$ the corresponding closed subscheme and set $U = X \setminus Z$. There is a canonical isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}|_ U, \mathcal{F}|_ U). \]

In particular we have an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, \mathcal{F}) \longrightarrow \Gamma (U, \mathcal{F}). \]

Proof. We first prove the second map is an isomorphism. It is injective by Properties, Lemma 28.25.3. Since $\mathcal{F}$ is the union of its coherent submodules, see Properties, Lemma 28.22.3 (and Lemma 30.9.1) we may and do assume that $\mathcal{F}$ is coherent to prove surjectivity. Let $\mathcal{F}_ n$ denote the quasi-coherent subsheaf of $\mathcal{F}$ consisting of sections annihilated by $\mathcal{I}^ n$, see Properties, Lemma 28.25.3. Since $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots $ we see that $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \ldots $ for some $n \geq 0$ by Lemma 30.10.1. Set $\mathcal{H} = \mathcal{F}_ n$ for this $n$. By Artin-Rees (Lemma 30.10.3) there exists an $c \geq 0$ such that $\mathcal{I}^ m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^{m - c}\mathcal{H}$. Picking $m = n + c$ we get $\mathcal{I}^ m\mathcal{F} \cap \mathcal{H} \subset \mathcal{I}^ n\mathcal{H} = 0$. Thus if we set $\mathcal{F}' = \mathcal{I}^ m\mathcal{F}$ then we see that $\mathcal{F}' \cap \mathcal{F}_ n = 0$ and $\mathcal{F}'|_ U = \mathcal{F}|_ U$. Note in particular that the subsheaf $(\mathcal{F}')_ N$ of sections annihilated by $\mathcal{I}^ N$ is zero for all $N \geq 0$. Hence by Properties, Lemma 28.25.3 we deduce that the top horizontal arrow in the following commutative diagram is a bijection:

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, \mathcal{F}') \ar[r] \ar[d] & \Gamma (U, \mathcal{F}') \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, \mathcal{F}) \ar[r] & \Gamma (U, \mathcal{F}) } \]

Since also the right vertical arrow is a bijection we conclude that the bottom horizontal arrow is surjective as desired.

Next, we prove the first arrow of the lemma is a bijection. By Lemma 30.9.1 the sheaf $\mathcal{G}$ is of finite presentation and hence the sheaf $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ is quasi-coherent, see Schemes, Section 26.24. By definition we have

\[ \mathcal{H}(U) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}|_ U, \mathcal{F}|_ U) \]

Pick a $\psi $ in the right hand side of the first arrow of the lemma, i.e., $\psi \in \mathcal{H}(U)$. The result just proved applies to $\mathcal{H}$ and hence there exists an $n \geq 0$ and an $\varphi : \mathcal{I}^ n \to \mathcal{H}$ which recovers $\psi $ on restriction to $U$. By Modules, Lemma 17.21.1 $\varphi $ corresponds to a map

\[ \varphi : \mathcal{I}^{\otimes n} \otimes _{\mathcal{O}_ X} \mathcal{G} \longrightarrow \mathcal{F}. \]

This is almost what we want except that the source of the arrow is the tensor product of $\mathcal{I}^ n$ and $\mathcal{G}$ and not the product. We will show that, at the cost of increasing $n$, the difference is irrelevant. Consider the short exact sequence

\[ 0 \to \mathcal{K} \to \mathcal{I}^ n \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{I}^ n\mathcal{G} \to 0 \]

where $\mathcal{K}$ is defined as the kernel. Note that $\mathcal{I}^ n\mathcal{K} = 0$ (proof omitted). By Artin-Rees again we see that

\[ \mathcal{K} \cap \mathcal{I}^ m(\mathcal{I}^ n \otimes _{\mathcal{O}_ X} \mathcal{G}) = 0 \]

for some $m$ large enough. In other words we see that

\[ \mathcal{I}^ m(\mathcal{I}^ n \otimes _{\mathcal{O}_ X} \mathcal{G}) \longrightarrow \mathcal{I}^{n + m}\mathcal{G} \]

is an isomorphism. Let $\varphi '$ be the restriction of $\varphi $ to this submodule thought of as a map $\mathcal{I}^{m + n}\mathcal{G} \to \mathcal{F}$. Then $\varphi '$ gives an element of the left hand side of the first arrow of the lemma which maps to $\psi $ via the arrow. In other words we have proved surjectivity of the arrow. We omit the proof of injectivity. $\square$


Comments (1)

Comment #947 by correction_bot on

Second to last sentence, should say ``we have proved…''


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