The Stacks project

Remark 51.8.5. Let $j : U \to X$ be an open immersion of locally Noetherian schemes. Let $x \in U$. Let $i_ x : W_ x \to U$ be the integral closed subscheme with generic point $x$ and let $\overline{\{ x\} }$ be the closure in $X$. Then we have a commutative diagram

\[ \xymatrix{ W_ x \ar[d]_{i_ x} \ar[r]_{j'} & \overline{\{ x\} } \ar[d]^ i \\ U \ar[r]^ j & X } \]

We have $j_*i_{x, *}\mathcal{O}_{W_ x} = i_*j'_*\mathcal{O}_{W_ x}$. As the left vertical arrow is a closed immersion we see that $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent if and only if $j'_*\mathcal{O}_{W_ x}$ is coherent.

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