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!