Given a scheme $X$, we know that there is a bijection between the points in $X$ and closed, irreducible subsets of $X$. Does that mean that every such point defines a closed subscheme of $X$ as it is the case for affine schemes? (Even if we don't hit all closed subschemes)
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I'm not sure what you mean by your question. Are you essentially just asking whether every closed subset of $X$ can be turned into a closed subscheme? (You say you know that each point of $X$ corresponds to a closed irreducible set, so the only remaining thing would be to turn that closed irreducible set into a closed subscheme...) – Eric Wofsey Feb 16 '21 at 22:08
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@EricWofsey Yes, what I meant is basically if I can turn any irreducible closed subset into a subscheme (in a unique way)? I was thinking about a point $p$ in an affine scheme defining a closed subscheme $V(p)$ having the closure of $p$ as underlying space and if that construction also works for general schemes? – Nicolas Brauch Feb 16 '21 at 22:22
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Potential duplicate of https://math.stackexchange.com/questions/374944/questions-on-reduced-induced-closed-subscheme – KReiser Feb 16 '21 at 22:24
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I think the notion you're looking for is the induced reduced structure on a closed subset of a scheme. – Tabes Bridges Feb 17 '21 at 00:33
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Yes, thank you, that's right :) – Nicolas Brauch Feb 17 '21 at 09:58