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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

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

  2. 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. $\square$

Comments (5)

Comment #2598 by Rogier Brussee on

Suggested slogan: (weak analogue of Hartogh's 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.

Comment #3409 by Kestutis Cesnavicius on

One could include a reference to a special case: EGAIV2, 7.2.2.

Comment #3468 by on

Yes, you are right. I have added the reference. Strangely the proof of the result in EGA shows something more general than what they state (this almost never happens in EGA). See changes here.

Comment #4932 by Kollar on

This is now Thm 2 in

@ARTICLE{k-coherent, AUTHOR = {Koll{\'a}r, J{\'a}nos}, TITLE = {Coherence of local and global hulls}, JOURNAL = {Methods Appl. Anal.}, FJOURNAL = {Methods and Applications of Analysis}, VOLUME = {24}, YEAR = {2017}, NUMBER = {1}, PAGES = {63--70}, ISSN = {1073-2772}, MRCLASS = {14F05 (13B22 13E05)}, MRNUMBER = {3694300}, URL = {https://doi.org/10.4310/MAA.2017.v24.n1.a5}, }

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