Lemma 51.8.3. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Assume

1. $X$ is Nagata,

2. $X$ is universally catenary, and

3. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) \geq 2$.

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

Proof. By Lemma 51.8.1 it suffices to prove $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent for $x \in \text{Ass}(\mathcal{F})$. Let $\pi : Y \to X$ be the normalization of $X$ in $\mathop{\mathrm{Spec}}(\kappa (x))$, see Morphisms, Section 29.54. By Morphisms, Lemma 29.53.14 the morphism $\pi$ is finite. Since $\pi$ is finite $\mathcal{G} = \pi _*\mathcal{O}_ Y$ is a coherent $\mathcal{O}_ X$-module by Cohomology of Schemes, Lemma 30.9.9. Observe that $W_ x = U \cap \pi (Y)$. Thus $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ factors through $i_ x : W_ x \to U$ and we obtain a canonical map

$i_{x, *}\mathcal{O}_{W_ x} \longrightarrow (\pi |_{\pi ^{-1}(U)})_*(\mathcal{O}_{\pi ^{-1}(U)}) = (\pi _*\mathcal{O}_ Y)|_ U = \mathcal{G}|_ U$

This map is injective (for example by Divisors, Lemma 31.2.10). Hence $j_*i_{x, *}\mathcal{O}_{W_ x} \subset j_*\mathcal{G}|_ U$ and it suffices to show that $j_*\mathcal{G}|_ U$ is coherent.

It remains to prove that $j_*(\mathcal{G}|_ U)$ is coherent. We claim Divisors, Lemma 31.5.11 applies to

$\mathcal{G} \longrightarrow j_*(\mathcal{G}|_ U)$

which finishes the proof. It suffices to show that $\text{depth}(\mathcal{G}_ z) \geq 2$ for $z \in Z$. Let $y_1, \ldots , y_ n \in Y$ be the points mapping to $z$. By Algebra, Lemma 10.72.11 it suffices to show that $\text{depth}(\mathcal{O}_{Y, y_ i}) \geq 2$ for $i = 1, \ldots , n$. If not, then by Properties, Lemma 28.12.5 we see that $\dim (\mathcal{O}_{Y, y_ i}) = 1$ for some $i$. This is impossible by the dimension formula (Morphisms, Lemma 29.52.1) for $\pi : Y \to \overline{\{ x\} }$ and assumption (3). $\square$

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AWA. Beware of the difference between the letter 'O' and the digit '0'.