Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there reasonable conditions (e.g. finite type) that we can put on $\mathcal{X}$ so that there does exist such a universal map to a scheme?
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I think the answer is almost certainly yes, an algebraic space can fail to have a universal map to a scheme (a "schemification"). I don't have a proof, but I think I know the right place to look for one (besides David Rydh's immediate surroundings). If we can find two maps of schemes which do not have a coequalizer in the category of schemes, but do have a coequalizer in the category of algebraic spaces, then the coequalizer algebraic space will not have a schemification. Consider Hironaka's example of a non-projective proper variety (see page 15 of Knutson's Algebraic Spaces). It has an action of ℤ/2 for which there is an algebraic space quotient. But in the category of schemes, there is no geometric quotient for this action. The question is whether there is a categorical quotient in this case. |
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