Topological Examples of Projective Modules

Abstract

Abstract.A new and more elementary proof is given for Lonsted's theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of Lensted. In certain cases, methods associated with Hubert's 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If m = 2 mod 4, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of rank # m are free.In [29], Lonsted proved a remarkable theorem which shows that vector bundles over a finite CW complex can be represented by finitely generated projective modules over a noetherian ring. This means that by purely topological constructions one can produce examples of noetherian rings whose projective modules have certain specified properties. This method has the advantage that one can impose conditions on the totality of projective modules while the more elementary method of [42] only allows us to construct a finite number of such modules at a time.Lonsted's construction makes use of rather deep properties of analytic functions. In attempting to analyze his proof, I discovered a more elementary proof of the theorem which I will present here. This proof gives rings which are considerably smaller than the ones used by Lonsted. They are, in fact, localizations of algebras of finite type over R. In certain cases, one can even give a very simple and purely algebraic description of the ring. This will be done in §10 using methods associated with Hubert's 17th problem.The starting point for this work was a question of A. Geramita. He pointed out that the rings in [42] have nontrivial projective modules of low rank and asked whether, for all n, there are noetherian rings having nontrivial projective modules but such that all such modules of rank less than n are free. Three

Dense morphisms in commutative Banach algebras

“…It is obvious that sr(^4) < dsr(/l), but we do not know if equality holds for every commutative Banach algebra A. This question is related to a problem raised by Swan [23]. We will discuss this later.…”

“…Swan [23] asked if a dense full subalgebra A of B has the same stable rank. The last corollary shows that this is, in fact, the case when sr(A) = dsr(A).…”

Algebraic vector bundles over real algebraic varieties

“…On the other hand, V n is algebraically isomorphic to S n and hence every continuous C-vector bundle over V n admits an algebraic structure [4,16]. It follows from Theorem 1 that ifg_ al (V n ,Z) is nontrivial, provided that n is even.…”

“…Theorem 4 is interesting in view of the fact that every continuous F-vector bundle over S n admits an algebraic structure [4,16]. Let us also mention that every continuous stably trivial F-vector bundle admits an algebraic structure [16,Theorem 2.2].…”

“…Characterize continuous F-vector bundles over X which admit an algebraic structure. This is an old problem, but despite considerable effort, the situation is well understood only in a few special cases: when X is the unit sphere S n [4,16], when X is the real projective space RF n [5,7] and when dimX < 3 and F = R [8, 9] (cf. also [13] for a short survey).…”