Is it possible to prove Serre's Problem (the Quillen-Suslin theorem) via Motivic Homotopy Theory?

Tuesday, January 24th, 2012 | Author:

These days I read Akhil Mathew's post on Vaserstein's proof of the Quillen-Suslin theorem, once known as Serre's Problem. This inspired the following.

Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

[UPDATE 2013-12-28] I have notes from a talk about Quillen's proof of Quillen-Suslin available now. [/UPDATE]

In the topological case, one classical proof goes like this:
Denote by Ψ the functor of rank r vector bundles up to isomorphism on topological spaces. This functor is representable in the homotopy category, in particular Ψ(X) ≅ [X,Gr(r)], where Gr(r) is the infinite Grassmannian of r-planes in an arbitrarily sized vector space and the parentheses [,] mean "homotopy classes of maps". To see this, embed any rank r vector bundle in a trivial one to get a map X->Gr(r), where different embeddings yield homotopic maps. Now observe that the pullback of the tautological bundle over Gr(r) along such a map gives a bundle isomorphic to the one you started with.

If X is contractible, i.e. homotopy equivalent to a point, [X,Gr(r)]=[pt,Gr(r)]={pt}, so there is nothing but one isomorphy-class of vector bundles on X, which must be the class of the trivial bundle. If X is affine, it is in particular contractible.

In the algebraic case, one might try to use this proof idea to proceed similarly. The first question arising here would be: which homotopy classes? Since the easiest example is affine space, one should choose a homotopy theory where affine space is contractible, which is the case for A¹-local simplicial homotopy theory, e.g. motivic homotopy theory.
The next question is now, whether we can represent the functor as in the topological setting.
This is certainly not the case: if G would be a classifying space (a representing object) for the functor Ψ, in the motivic homotopy theory, we'd have Ψ(X) ≅ [X,G], but the right-hand side is A¹-invariant by construction, while the left hand side is not! This means, [XxA¹,G]=[X,G] (along the projection morphism) but Ψ(XxA¹)≠Ψ(X) (this fails for example even in the case X=P¹).

One could stop here, given that there are nice proofs of the Quillen-Suslin theorem. But wait, Quillen-Suslin is only about affine space and we just tried to prove something about all schemes. Let's try to look at something weaker. What about (A¹-)contractible spaces?

There is a nice paper of Asok and Doran on vector bundles on contractible schemes which explains that there are lots of A¹-contractible schemes (over a field) with lots of vector bundles that are not trivial. These vector bundles are somewhat invisible, since they are indistinguishable by cohomology or K-theory: since the base is A¹-contractible and motivic cohomology as well as K-theory are representable in the A¹-homotopy category, cohomology and K-theory of these schemes are those of a point.

Well, it just seems the approach to Quillen-Suslin via A¹-homotopy theory is doomed. Now let me tell you that I think that is the case, but nevertheless, using the Quillen-Suslin theorem (and stronger results on the more general Bass-Quillen conjecture), Morel claims that the representability works like in the topological case, as long as we only look at affine schemes:

Let X be a smooth affine k-scheme, then Ψ(X) ≅ [X,Gr(r)].

For now, the proof didn't appear in any journal, but you can take a look at Morel's book-in-progress "A¹-algebraic topology over a field" here. The parts related to this discussion are mostly in section 7 and 8 and these sections were previously contained in an earlier paper draft called "A¹-homotopy classification of vector bundles over smooth affine schemes".

For some more info about the history and various approaches to Serre's problem I like to recommend Lam's excellent book "Serre's Problem on projective modules".

Category: English, Mathematics

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