** Weak analogue of Hartogs' Theorem: On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent. **

See [k-coherent] and see [IV, Proposition 7.2.2, EGA] for a special case.

Proposition 51.8.7 (Kollár). Let $j : U \to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. The following are equivalent

$j_*\mathcal{F}$ is coherent,

for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{ x\} }, z}^\wedge $ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}^\wedge /\mathfrak p) \geq 2$.

**Proof.**
If (2) holds we get (1) by a combination of Lemmas 51.8.1, Remark 51.8.5, and Lemma 51.8.4. If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_ x}$ is not finite for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in Remark 51.8.6 (and Remark 51.8.5). Thus $j_*\mathcal{F}$ is not coherent by Lemma 51.8.1.
$\square$

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