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If we have a scheme and a locally closed subset of the underlying topological space, then there is a canonical way to put a scheme structure on it so that the inclusion map can be upgraded to a morphism of schemes with the map on the structure sheaves $\mathcal{O}_X\rightarrow f_*\mathcal{O}_U$ being surjective.

I think this is not true for e.g. finite disjoint unions of locally closed subsets (take the affine plane and the complement of the punctured axes); correct me if I am wrong.

However, in the example above there is a scheme structure on the subset under consideration. We just can not choose it compatibly with the scheme structure on the affine plane.

Which brings me to the question: are locally closed subsets somehow "the most general" class of subsets that can be functorially endowed with a scheme structure so that the inclusion map can be upgraded to a morphism of schemes with the map on the structure sheaves being surjective? Or are there more general subsets with this property?

EDIT: the examples given in the comments (at the time of the edit) are all monomorphisms in the category of schemes, I think. Is there an example that is not a monomorphism?

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    $\begingroup$ No, locally closed subsets are not the most general subsets. For instance, singleton subsets of non-closed points are (typically) not locally closed subsets, yet the Spec of the residue field is a scheme structure compatible with the subspace topology and the inclusion map. $\endgroup$
    – Jason Starr
    Commented May 20, 2019 at 10:15
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    $\begingroup$ Spec(local ring of a point) is another example. $\endgroup$
    – Laurent Moret-Bailly
    Commented May 20, 2019 at 10:58
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    $\begingroup$ The common feature of all these examples is that the scheme structure in the subset $S$ of the scheme $X$ is a quotient of the inverse image sheaf of rings $i^{-1}\mathcal{O}_X.$. For every locally ringed space $(i,i^*):(S,\mathcal{O}_S)\to (X,\mathcal{O}_X),$ this satisfies the universal property that a morphism to $S$ is a morphism to $X$ whose image subset is contained in $S$ and whose pullback map of sheaves of rings factors through $\mathcal{O}_S.$. Thus, the morphism $(i,i^*)$ is a universal mono morphism. $\endgroup$
    – Jason Starr
    Commented May 20, 2019 at 15:18
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    $\begingroup$ * “For every locally ringed space ...” -> “For every such locally ringed space ...” $\endgroup$
    – Jason Starr
    Commented May 20, 2019 at 15:20

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