The Stacks project

Lemma 51.8.1. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. For $x \in U$ let $i_ x : W_ x \to U$ be the integral closed subscheme with generic point $x$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. The following are equivalent

  1. for all $x \in \text{Ass}(\mathcal{F})$ the $\mathcal{O}_ X$-module $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent,

  2. $j_*\mathcal{F}$ is coherent.

Proof. We first prove that (1) implies (2). Assume (1) holds. The statement is local on $X$, hence we may assume $X$ is affine. Then $U$ is quasi-compact, hence $\text{Ass}(\mathcal{F})$ is finite (Divisors, Lemma 31.2.5). Thus we may argue by induction on the number of associated points. Let $x \in U$ be a generic point of an irreducible component of the support of $\mathcal{F}$. By Divisors, Lemma 31.2.5 we have $x \in \text{Ass}(\mathcal{F})$. By our choice of $x$ we have $\dim (\mathcal{F}_ x) = 0$ as $\mathcal{O}_{X, x}$-module. Hence $\mathcal{F}_ x$ has finite length as an $\mathcal{O}_{X, x}$-module (Algebra, Lemma 10.62.3). Thus we may use induction on this length.

Set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_ x}$. This is a coherent $\mathcal{O}_ X$-module by assumption. We have $\mathcal{G}_ x = \kappa (x)$. Choose a nonzero map $\varphi _ x : \mathcal{F}_ x \to \kappa (x) = \mathcal{G}_ x$. By Cohomology of Schemes, Lemma 30.9.6 there is an open $x \in V \subset U$ and a map $\varphi _ V : \mathcal{F}|_ V \to \mathcal{G}|_ V$ whose stalk at $x$ is $\varphi _ x$. Choose $f \in \Gamma (X, \mathcal{O}_ X)$ which does not vanish at $x$ such that $D(f) \subset V$. By Cohomology of Schemes, Lemma 30.10.5 (for example) we see that $\varphi _ V$ extends to $f^ n\mathcal{F} \to \mathcal{G}|_ U$ for some $n$. Precomposing with multiplication by $f^ n$ we obtain a map $\mathcal{F} \to \mathcal{G}|_ U$ whose stalk at $x$ is nonzero. Let $\mathcal{F}' \subset \mathcal{F}$ be the kernel. Note that $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$, see Divisors, Lemma 31.2.4. Since $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) - 1$ we may apply the induction hypothesis to conclude $j_*\mathcal{F}'$ is coherent. Since $\mathcal{G} = j_*(\mathcal{G}|_ U) = j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent, we can consider the exact sequence

\[ 0 \to j_*\mathcal{F}' \to j_*\mathcal{F} \to \mathcal{G} \]

By Schemes, Lemma 26.24.1 the sheaf $j_*\mathcal{F}$ is quasi-coherent. Hence the image of $j_*\mathcal{F}$ in $j_*(\mathcal{G}|_ U)$ is coherent by Cohomology of Schemes, Lemma 30.9.3. Finally, $j_*\mathcal{F}$ is coherent by Cohomology of Schemes, Lemma 30.9.2.

Assume (2) holds. Exactly in the same manner as above we reduce to the case $X$ affine. We pick $x \in \text{Ass}(\mathcal{F})$ and we set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_ x}$. Then we choose a nonzero map $\varphi _ x : \mathcal{G}_ x = \kappa (x) \to \mathcal{F}_ x$ which exists exactly because $x$ is an associated point of $\mathcal{F}$. Arguing exactly as above we may assume $\varphi _ x$ extends to an $\mathcal{O}_ U$-module map $\varphi : \mathcal{G}|_ U \to \mathcal{F}$. Then $\varphi $ is injective (for example by Divisors, Lemma 31.2.10) and we find an injective map $\mathcal{G} = j_*(\mathcal{G}|_ V) \to j_*\mathcal{F}$. Thus (1) holds. $\square$

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