The Stacks project

Remark 51.8.6. 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. If there exists an $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ such that $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) \leq 1$, then $j_*\mathcal{F}$ is not coherent. To prove this we can do a flat base change to the spectrum of $\mathcal{O}_{X, z}$. Let $X' = \overline{\{ x\} }$. The assumption implies $\mathcal{O}_{X' \cap U} \subset \mathcal{F}$. Thus it suffices to see that $j_*\mathcal{O}_{X' \cap U}$ is not coherent. This is clear because $X' = \{ x, z\} $, hence $j_*\mathcal{O}_{X' \cap U}$ corresponds to $\kappa (x)$ as an $\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not a closed point.

In fact, the converse of Lemma 51.8.4 holds true: given an open immersion $j : U \to X$ of integral Noetherian schemes and there exists a $z \in X \setminus U$ and an associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge $ with $\dim (\mathcal{O}_{X, z}^\wedge /\mathfrak p) = 1$, then $j_*\mathcal{O}_ U$ is not coherent. Namely, you can pass to the local ring, you can enlarge $U$ to the punctured spectrum, you can pass to the completion, and then the argument above gives the nonfiniteness.

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