Let f : X —> Y be a morphism over a base B. Let P be a property of morphisms. We often want to know that
- there is a maximal open W of B such that the restriction f_W : X_W —> Y_W of f has property P, and
- 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?