# Stratify

I just added some generic flatness results to the stacks project (only for morphisms of schemes so far). There are two interesting features of the presentation in the stacks project:

1. Assuming only the morphism is of finite type the conclusion is that the morphism is flat and finite presentation over a dense open of the base, and
2. it suffices to assume the base is reduced.

Using these results we can discuss “flattening stratifications”. But I want to discuss this in a maximally general setting. Reader beware!

Let f : X —> S be a morphism of schemes of finite type. I want to find a stratification of S by reduced locally closed subschemes S_i such that X_{S_i} —> S_i is flat. If f is of finite presentation we can reduce to S Noetherian and there is (locally) a finite stratification that does the job; so what I am interested in here is the case where S is not Noetherian.

Step 0: Find the open stratum. Just replace S by its reduction S_{red} and let S_0 be the open dense U ⊂ S you get from generic flatness. Step 1: Let S_1 be the dense open of (S – S_0)_{red} you get from generic flatness. Step 2: Let S_2 be the dense open of (S – S_0 – S_1)_{red} you get from generic flatness. Etc.

Now we get S_0, S_1, … but it may not be the case that S = \bigcup S_i. For example the last post contains an example. So then you start all over again. Namely, note that the complement of S_0 ∪ S_1 ∪ S_2 ∪… is closed in S hence a scheme. So we restrict our family to this closed subset and we continue. Doesn’t it feel like we can just continue forever using transfinite induction? And moreover, the process does really have to stop as S has an underlying topological space which has a finite cofinality. Thus we do get our desired stratification of S.

But this is madness! Surely there are at most countably many strata…!?!