A valuative criterion

Let f : X —> S be a quasi-compact morphism of schemes. Then f is universally closed when it satisfies the existence part of the valuative criterion. This is a straightforward application of the following cute fact: If A —> B is a ring map and the image of Spec(B) in Spec(A) is closed under specialization, then it is closed.

It turns out that if S is locally Noetherian and f of finite type, then it suffices to check the existence part of the valuative criterion for discrete valuation rings. Jarod Alper told yesterday how to prove this based on Lemma Tag 05BD which I initially introduced to study impurities (more about this in a future post).

Here is the lemma: Let f : X —> S be quasi-compact. Suppose that g : T —> S is a morphism of schemes, Z a closed subset of X_T and t a point of T not in the image of Z. Then one can find, after shrinking T to a neighborhood of t, a factorization T — a –> T’ — b –> S of g such that b is locally of finite presentation and such that there exists a closed Z’ ⊂ X_{T’} which contains the image of Z and whose image in T’ does not contain a(t).

In particular, if the image of Z in T is not closed and t is a point witnessing the non-closedness of the image, then a(t) is a point of T’ witnessing the non-closedness of the image of Z’. In other words, if f is not universally closed, then there exists a base change which is locally of finite presentation which is not closed. By some straightforward argument we deduce that it suffices to check that f crossed with A^n is closed in order to prove that f is universally closed. This is Lemma Tag 05JX. In particular, if we now assume that f is of finite type and S is locally Noetherian, then it is easy to see that it suffices to check the existence part of the valuative criterion for discrete valuation rings in order to be able to conclude that f is universally closed. See Lemma Tag 05JY for a precise statement.

A key observation is that we do not assume that f is separated. (In the separated case there is a proof of the criterion using Chow’s lemma, see Lemma Tag 0208.) Proving things for non-separated schemes is a testing ground for proving results in the setting of algebraic stacks (since the non Deligne-Mumford ones are rarely separated). Jarod really made his suggestion in the setting of finite type morphisms of  locally Noetherian algebraic stacks and I think the above goes through (mutatis mutandis), although I have not written out all the details (Jarod and I worked it out on the blackboard though).

[A word of caution: Points of an algebraic stack X are defined as equivalence classes of morphisms from spectra of fields. There is a natural topology on the set |X| of points. But it need no longer be true that |X| is a sober topological space; this can already be false for algebraic spaces. Moreover if U —> X is a presentation it need not be the case that you can lift generalizations along the map |U| —> |X|; there is a counter example for algebraic spaces already due to David Rydh I think. I do think we should define closedness of morphisms of algebraic stacks in terms of these topological spaces, but as you can see from the above you have to be very careful when you try to think about what that means.]

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