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Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is connected, show that $f$ is an isomorphism.

Here's another exercise I encountered. I'm not exactly sure how to use the properties ascribed to $Y$ to prove this. Any help would be appreciated!

user 3462
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Separatedness of $Y$ over $S$ together with properness of $X$ over $S$ implies that the set-theoretic image of $f$ is closed (see my answer here The image of a proper scheme is closed). The image is open by assumption, so (assuming $X$ is non-empty) connectedness of $Y$ forces it to be all of $Y$. An open immersion which is surjective is an isomorphism.

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Keenan Kidwell
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