Vector Bundles and Projective Modules

Abstract

The author gives a rigorous treatment of a correspondence between vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X, and then gives some of the examples which may be constructed in this way. Preliminaries. Let K be a real, complex, or quaternionic field and ξ={E(ξ),X,p,FX(ξ)=Kn} a K-vector bundle over a topological space X. In this section, the following proposition is proved. If X is paracompact, any subbundle η of a vector bundle ξ is a direct summand. For the proof he utilizes the fact that every K-vector bundle over a paracompact X has an inner product [J. Milnor, Notes on characteristic classes, Princeton Univ. Press, Princeton, N.J., 1957]. Modules of sections. Let C(X) be the ring of continuous K-valued functions on X and Γ(ξ) the C(X)-module of the set of all cross-sections of ξ over X. If ξ is the product bundle E(ξ)=X×Kn, then Γ(ξ) is obviously a free C(X)-module on n generations. In this section, the following theorem is established. If X is normal, the function Γ gives an isomorphism Hom(ξ,η)≈HomC(X)(Γ(ξ),Γ(η)). Projective modules. Let X be compact Hausdorff. Let ξ be any K-vector bundle over X. It is shown that there is a trivial vector bundle ζ (i.e., E(ζ)=X×Kn) and an epimorphism f:ζ→ξ. Therefore, ξ is a direct summand of a trivial bundle ζ, and Γ(ξ) is a direct summand of Γ(ζ) which is a finitely generated free C(X)-module, and so Γ(ξ) is a finitely generated projective (f.g. proj.) C(X)-module But the following converse also holds. Theorem: P is a f.g. proj. C(X)-module ⇔∃ξ, P=Γ(ξ) [cf. J.-P. Serre, Séminaire Dubreil, Dubreil-Jacotin et Pisot, 1957/58, Exp. 23, Secrétariat mathématique, Paris, 1958; MR0108506 (21 #7222)]. Examples. In place of the large rings C(X), the author gives examples of projective modules over affine rings. For this purpose he proceeds as follows. Find some affine ring Λ⊂C(X) and a finitely generated projective Λ-module P. Then C(X)⊗1P is a f.g. proj. C(X)-module and so is isomorphic to some Γ(ξ). To prove that P is nontrivial in some sense (not free, indecomposable, etc.), it suffices to show that ξ is nontrivial in the same sense. Example 1 (Kaplansky): A projective module with a free complement which is itself not free. Let τn be the tangent bundle of the n-sphere Sn and ν1 the normal bundle of Sn. Then τn⊕ν1 is trivial and ν1 is also trivial. Thus Γ(τn) is C(Sn)-projective and has a free complement. Γ(τn) cannot be free unless n=0,1,3,7 [cf. R. Bott and J. Milnor, Bull. Amer. Math. Soc. 64 (1958), 87–89; MR0102804 (21 #1590)]. Further, considering the Euler classes, it is shown that Γ(τn) is indecomposable for n even. Next, reducing C(Sn) to the affine ring Λn=R[x0,⋯,xn]/(x02+⋯+xn2−1), R being the real field, he proves the following theorem. Let P be the Λ-module with generators s0,⋯,sn and relation ∑xisi=0. Then P⊕Λ is free, but P is not free for n≠1,3,7. If n is even, P is indecomposable. In Example 2, he discusses the number of generators of projective modules of rank k by considering a line bundle over projective space. In Example 3, he states the following lemma. If Λ is a regular domain, Λ is a UFD if and only if every rank 1 projective module is free. (A rank 1 projective module corresponds to a line bundle.) He then proves the theorem: Λn is UFD (n>1) and Λ1 is not UFD. CΛn is UFD (n>1,n≠2) and CΛ2 is not UFD, where CΛn denotes a complex affine ring [Nagata, J. Math. Soc. Japan 9 (1957), 143–145; MR0084490 (18,869a)]. For the proof of the fact that Λ1 and CΛ2 are not UFD and of the following theorem concerning the ground field extension, he uses various vector bundles. Theorem: There is a f.g. proj. module A over Λ4 such that A has no free complement but CA is free over CΛ4.