Let f : X —> Y be a morphism over a base B. Let P be a property of morphisms. We often want to know that<\/p>\n
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.<\/p>\n
A well known and useful case is the following result<\/p>\n
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. <\/p><\/blockquote>\n
I recently added this to the stacks project for relative maps of algebraic spaces, see Lemma Tag 05XD<\/a>. When you analyze the proof you find two more basic results that lead to the above. The first is that<\/p>\n
If X is proper over B, Y is separated over B, and P = “being a closed immersion”, then 1 and 2 hold. <\/p><\/blockquote>\n
see Lemma Tag 05XA<\/a>. This first result is in some sense elementary (although its proof in the current exposition is not). The second is that<\/p>\n
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. <\/p><\/blockquote>\n
see Lemma Tag 05XB<\/a>. 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?<\/p>\n","protected":false},"excerpt":{"rendered":"
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 : … Continue reading