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This comes up in the proof of Prop. 10.30 in Görtz-Wedhorn: if $f:X\rightarrow Y$ is a quasi-compact morphism of schemes, why do we have $\operatorname{Supp}(\mathcal{O}_Y/\mathscr{K}_f)=\overline{f(X)},$ where $\mathscr{K}_f$ denotes the kernel of $f^{\flat}:\mathcal{O}_Y\rightarrow f_*\mathcal{O}_X$?

So far, I only think that I can embed $\mathcal{O}_Y/\mathscr{K}_f$ into $f_*\mathcal{O}_X$ and then get an inlcusion $\operatorname{Supp}(\mathcal{O}_Y/\mathscr{K}_f)\subseteq\overline{f(X)}.$ But I don't see at the moment why we have an equality.

Thanks.
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  • Related to/potentially a duplicate of https://math.stackexchange.com/questions/3897642/exercise-3-11-d-of-chapter-ii-in-hartshorne-how-to-describe-the-scheme-theoreti/ – KReiser Jul 05 '21 at 20:56

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