Scheme theoretic image of quasi-compact morphism

Stream: toric

[Paul Lezeau]

Re the task @Michał Mrugała mentioned

Prove that if f : X ⟶ Y is quasi-compact and Z is the scheme-theoretic image of f, then f : X ⟶ Z is dominant.

do we actually have anything about the scheme-theoretic image of a morphism? I had a quick look but didn't find anything.

[Michał Mrugała]

@Andrew Yang has done some work on it

[Kevin Buzzard]

Isn't this a thorny notion? IIRC there's a false exercise in Hartshorne about this (or maybe it was fixed by now).

[Michał Mrugała]

It definitely exhibits some weird behavior, especially if the morphism is not quasicompact. My use case is that I need to take the closure of the image of a torus in affine space. It shouldn’t be too bad since everything is affine.

[Kevin Buzzard]

yeah, the bad examples are things like "disjoint union of infinitely many points maps into distinct points in affine 1-space" or "disjoint union of thicker and thicker points all map onto a point in affine 1-space" and these aren't qc

[Michał Mrugała]

@Andrew Yang and I were discussing whether the scheme theoretic image should be a subtype of a scheme for this reason, because the closure can behave strangely in these non-quasicompact situations