# Points of the fppf topos

Let S be the affine line over the complex numbers. Consider the big fppf site (Sch/S)_{fppf} of S. By a theorem of Deligne this site has enough points. How can we describe these points?

Here is one way to construct points. Write S = Spec(C[x]) and suppose that B is a local C[x]-algebra such that any faithfully flat, finitely presented ring map B —> C has a section. Then the functor which associates to an fppf sheaf F the value F(Spec(B)) is a stalk functor, hence determines a point. In fact, I think all points of (Sch/S)_{fppf} are of this form.

Actually, if B is henselian, then it suffices if finite free ring maps B —> C have a section; this uses the material discussed here. If B is a henselian domain, it suffices if its fraction field is algebraically closed. A specific example is the ring B = ∪ C[[x]][x^(1/n)].

Anyway, I was hoping to use this description to say something about question 4 of this post on exactness of pushfoward along closed immersions for the fppf topology. I still don’t know the answer to that question. Do you?

# General Neron Desingularization

Last weekend I spent some time going through my notes of a course I taught at Princeton in 1996(?) on General Neron desingularization. Before I tell you more about my notes, let me state the theorem (which was proved by… Popescu):

Let R —> Λ be a regular ring homomorphism of Noetherian rings. Then Λ is a filtered colimit of smooth R-algebras.

The proof of this theorem is fairly difficult and moreover the first time Popescu’s proof appeared in print it was doubted it was correct. Using papers by Ogama and notes by Andre, a definite account of this proof appears in a paper by Swan with the title “Neron-Popescu desingularization”.

In my lectures at Princeton I wasn’t able to get across even the basic structure of the proof. It seems difficult to break up the proof into manageable and meaningful chunks. In the end, I mostly focused on explaining and understanding two very interesting lemmas, one which is called the “lifting lemma” and the other is called the “desingularization lemma”.

At the time I thought I had found a way to simplify Swan’s exposition further. Happily I sent off a set of notes containing the idea as well as some other arguments to Professor Swan at Chicago. Unfortunately, two weeks later he wrote back to tell me he had found some mistakes. In fact, when I looked at it again I was unable to fix them.

However, as a footnote to his letter with the bad news, he mentioned that he had no problems with the other set of notes that I had included in my letter. These contain a proof of the following statement

If Neron-Popescu holds whenever R is a field, then it holds.

Moreover, the proof of this statement can be split into meaningful parts, and rests on (simplified versions) of the two lemmas mentioned above. It is probable that the experts at the time were aware of this intermediate step. On the other hand, working through the rest of Popescu’s proof (as written up by Swan) in the special case that R is a field does not, at first sight, seem to simplify the proof. In writing a paper for publication in a journal you would therefore discard this intermediate result.

In the next few days I’ll take another look to see if some kind of simplification isn’t possible. Let me know if you have any ideas!

# Clarification

Let me just clarify what I was trying to say in the previous post.

Setup. Let A be a Noetherian local ring. Set S = Spec(A). Denote U the punctured spectrum of A. Let A ⊂ C ⊂ A^* be a finite type A-algebra contained in the completion of A. Set V = Spec(C) ∐ U. Consider the functor F : (Sch/S)^{opp} —> Sets which to a scheme T/S assigns

1. F(T) = {*} = a singleton if there exists an fppf covering {T_i —> T} such that each T_i —> S factors through V.
2. F(T) = ∅ else.

I claim that all of Artin’s criteria are satisfied for this fppf sheaf (details omitted). Moreover, note that both F(Spec(A^*)) = {*} and F(U) = {*} are nonempty.

If Artin’s criteria imply that F is an algebraic space, then, choosing a surjective etale morphism X —> F where X is an affine scheme, we conclude that X is surjective and etale over A (this takes a little argument). Using the definition of F we find a faithfully flat, finite type A-algebra B and an A-algebra map C —> B.

Conversely, if there exists an A-algebra map C —> B with B a faithfully flat, finite type A-algebra, then Spec(B) —> F is a flat, surjective, finitely presented morphism and F is an algebraic space (by a result of Artin we blogged about recently).

This analysis singles out the following condition on a Noetherian local ring A: Every finite type A-algebra C contained in the completion of A should have an A-algebra map to a faithfully flat, finite type A-algebra B. But it isn’t necessary for B to also map into A^*! I missed this earlier when I was thinking about this issue. For example any dvr has this property (but there exist non-excellent dvrs).

Finally, if Artin’s criteria characterize algebraic spaces over Spec(R) for some Noetherian ring R then this property holds for any local ring of any finite type R-algebra. Likely this isn’t a sufficient condition.

# Update

Just this morning I finished revising the chapter on formal deformation theory that was written by Alex Perry. Next week I hope to put this chapter to use in the stacks project, and start writing about Artin’s criteria.

In particular, I hope to get back to what I said in this post. Put in another fashion, I want to prove an equivalence of the form “Artin’s criterion holds for stacks in groupoids over S” <=> “S is good” where good is a property of Noetherian base schemes to be determined. Goodness might be closely related to a condition like “completions of local rings are limits of flat finitely presented algebras”. I believe this could be fun!