Relative maps

Let f : X —> Y be a morphism over a base B. Let P be a property of morphisms. We often want to know that

  1. there is a maximal open W of B such that the restriction f_W : X_W —> Y_W of f has property P, and
  2. formation of W commutes with arbitrary base change B’ —> B.

Of course this usually isn’t the case without further assumptions on X,Y,f, and B. One of the reasons this type of result is useful, is that you can check whether a point b of B is in W by looking at the base change of the morphism f to a morphism f_b : X_b —> Y_b of schemes (or algebraic spaces or algebraic stacks) over the point b.

A well known and useful case is the following result

If X is proper, flat, of finite presentation over B, Y is proper over B, and P = “being an isomorphism”, then 1 and 2 hold. 

I recently added this to the stacks project for relative maps of algebraic spaces, see Lemma Tag 05XD. When you analyze the proof you find two more basic results that lead to the above. The first is that

If X is proper over B, Y is separated over B, and P = “being a closed immersion”, then 1 and 2 hold. 

see Lemma Tag 05XA. This first result is in some sense elementary (although its proof in the current exposition is not). The second is that

If X is proper, flat, of finite presentation over B, Y is locally of finite type over B, and P = “being flat”, then 1 and 2 hold. 

see Lemma Tag 05XB. The current proof of this second result uses the “critère de platitude par fibres” which is nontrivial. Does anybody know how to prove the result on the locus where f is an isomorphism without appealing to this criterion?

5 thoughts on “Relative maps

  1. Johan, just to clarify matters, is the reason you refer to the fibral flatness criterion as “nontrivial” because you have in mind to invoke Raynaud’s version from EGA IV_3, section 11 which avoids all noetherian hypotheses (in contrast with the version found in textbooks)? I always figured that if one is aiming to prove results in the “finite presentation” setting, all useful foundational stuff from sections 8, 9, 11 of EGA IV is to be regarded as fair game (since the point of those results is precisely to allow one to argue over a general base “as if” one were in the noetherian case). Of course, your question makes sense. I’m just not sure it is reasonable to avoid that technique since you’re not willing to assume Y is finitely presented. That being said, I have nothing useful to say in the direction of your question. Sorry.

    • Oops, even the case of finitely presented Y doesn’t seem to bypass the problem, and it is “easy” to reduce to that case anyway (so that reduction doesn’t appear to help) .

  2. Dear Johan,

    If you change the hypothesis from “proper” to “projective”, then I believe the result is easy to prove, cf. Lemma 4.7 in our joint paper, “Every rationally connected …”. Our proof invokes flatness of both morphisms, but I believe this is unnecessary.

    Best regards,

    • Yes, of course! It replaces the criterion by an appeal to cohomology and base change (this isn’t yet in the stacks project…). I think what you mean with your last sentence is that you only need X to be flat over the base for that argument to work (and you do really need that of course). Also we assume the base is a curve which is totally unnecessary also. Thanks, Jason!

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