This is a continuation of previous post on flattening stratifications. The experts reading this blog could probably tell that I hadn’t really understood what is going on at all. I still haven’t mastered the subject but I think I know a little bit more now.

Let f : X —> S be a morphism of schemes. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates the set of morphisms T —> S such that the base change X_T is flat over T. Clearly the map F —> S is a monomorphism. We propose to introduce the following

Definition: We say the *flattening stratification of f exists* if F is an algebraic space.

What I added to the stacks project last Friday is the following: Assume S is the spectrum of a Noetherian complete local ring and f is of finite type. Then there exists a biggest closed subscheme Z of S such that X_Z —> Z is flat at all the points of the closed fibre. Moreover, Z satisfies a universal property which is formulated in terms of local morphisms of local schemes and flatness at points of the special fibre. If in addition X —> S is closed, then it follows that X_Z —> Z is flat as the set of points where X_Z —> Z is flat is an open set.

Assume S Noetherian and f of finite type and proper. In terms of Artin’s axioms for F the result in the previous paragraph takes care of the existence of a formal versal deformation. I think there is a straightforward little argument which takes care of openness of versality (but I did not write this out completely). Since f is of finite presentation, it follows that F is of finite presentation by the usual arguments on limits and flatness. Relative representability is OK too. Hence, if S is excellent then F is an algebraic space by Artin’s theorem. But of course we can descend X —> S to a situation of finite type over Z and hence we get the result in general (with same hypotheses). In fact, using limit arguments we may be able to prove the same thing when S is arbitrary and f proper and of finite presentation.

Still, my answer to Jason’s question here was a bit premature. Some of the above may work exactly as stated in the generality of Jason’s question. But I was trying to prove flattening stratifications exist without using Artin’s theorem. In particular, it should be possible to avoid using general N\’eron desingularization.

The reason I started looking at flattening stratifications was to construct Quot and Hilbert schemes/spaces/stacks. And the reason to discuss those was that Artin’s trick uses Hilbert spaces. However, it only uses the Hilbert space parametrizing closed subschemes of length n on a space. Of course I could take the easy way out and just use one of the explicit constructions of Hilb^n. But once I started looking at the problem of constructing flattening stratifications (which is related to descent of flat modules) I just couldn’t stop myself.

I think you can use Raynaud-Gruson, Theorem 4.1.2 to avoid using general N\’eron desingularization. This theorem basically says that the result of the 4th paragraph above works over a henselian local ring. But I think there is a much easier way in the proper finite presentation case.

Pingback: Raynaud-Gruson « Stacks Project Blog

Pingback: Universal flattening « Stacks Project Blog