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For affine schemes, there are contravariant maps V(-) and I(-) that create a bijection between irreducible closed subsets of the affine scheme and prime ideals of the ring that is the ring of global sections of that scheme (i.e. points on the affine scheme).

What happens to V(-) and I(-) when we take an arbitrary scheme (not assuming it is Noetherian or quasicompact)?

In particular, given a point p of a scheme X, can we find an irreducible closed subset of X that corresponds to p, such that the correspondence is a bijection, the way we could with affine schemes?

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    $\begingroup$ Yes, schemes are sober spaces. The closed irreducible subset corresponding to a point is its closure. A more interesting bijection is the one between closed subschemes and quasicoherent sheaves of ideals. $\endgroup$
    – Denis Nardin
    Mar 18, 2018 at 15:09

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Given any scheme $X$, there is a bijective correspondence between quasi-coherent ideal sheaves on $X$ and closed subschemes of $X$. (See Hartshorne's Algebraic Geometry, Proposition II.5.9.) When we restrict to affine schemes, this correspondence is given by the usual $V(-)$ and $I(-)$ once we identify quasi-coherent ideal sheaves on an affine scheme with their corresponding ideals (there is also an equivalence of categories between these; see Hartshorne, Corollary II.5.5 for the general case of $A$-modules and quasi-coherent $\mathcal{O}_{\operatorname{Spec}(A)}$-modules, and note that this sends ideals of $A$ to quasi-coherent ideal sheaves on $\operatorname{Spec}(A)$ and vice versa).

All schemes are sober spaces, meaning that every irreducible closed subset is the closure of a unique point, exactly like for affine schemes.

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    $\begingroup$ Thank you. That's a great answer. I'm working my way through Ravi Vakil's notes, and I'm still on Chapter 5 (well before closed subschemes are defined, let alone quasi-coherent ideal sheaves), but this gives me something to focus on as I come across these concepts. $\endgroup$
    – Jeremy Gross
    Mar 18, 2018 at 22:45
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    $\begingroup$ For reference, closed subschemes and ideal sheaves are defined in section 8.1 of Vakil's notes (as of the November 2017 version), and the correspondence between closed subschemes and quasicoherent ideal sheaves is discussed in section 13.5.4. $\endgroup$
    – Daniel Hast
    Mar 19, 2018 at 5:31

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