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In algebraic geometry, one is usually interested in morphisms relative to a base scheme, say $S$. A closed subscheme in Hartshorne is defined as an equivalence class of closed immersions. But what happens if we take a closed immersion which is not a morphism over $S$? Could it be that this immersion induces a closed subscheme which does not come from a closed immersion over $S$? If that's so, when dealing with closed subschemes shouldn't we specify the base scheme, e.g. say "closed $S$-subschemes"?

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If we have a scheme over a base $X\to S$ and a closed immersion $Z\to X$, then $Z$ acquires the structure of an $S$-scheme from the composition $Z\to X\to S$, so every closed immersion is equivalent to one over $S$. If $Z\to S$ comes with a specified structure of an $S$-scheme, then it may be that there are closed immersions $Z\to X$ which are not $S$-morphisms, so one might have to worry about it. Most of the time one doesn't have to worry, but you're right that it's a possibility.

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