Vector bundles, finitely generated projective module?

Question

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module consisting of all cross-sections of $\xi$. I have two questions.

• If $\xi \oplus \eta$ is trivial, does it follow that $S(\xi)$ is a finitely generated projective module?
• Conversely, if $Q$ is a finitely generated projective module over $\mathbb{R}(B)$, does it follow that $Q \cong S(\xi)$ for some $\xi$?

Answer 1

1. Yes. By hypothesis, $\xi$ is a direct summand of a f.g. trivial vector bundle, so $S(\xi)$ is a direct summand of a f.g. free module.

2. Yes. A f.g. projective $R$-module is a direct summand of a f.g. free module, meaning it's obtained by taking fixed points of the action of some idempotent $e \in M_n(R)$ acting on some $R^n$. When $R = C(B, \mathbb{R})$ there is a corresponding idempotent acting on the trivial vector bundle $\mathbb{R}^n$, since the endomorphism ring of $\mathbb{R}$ (the vector bundle) is $R$. It remains to verify that taking fixed points of this idempotent produces a vector bundle. (Edit: Previously this section included the hypothesis that $B$ is locally connected, which is unnecessary.) The rank of the idempotent $e$, as a function on $B$, is $\text{tr}(e)$. This takes a discrete set of values, namely the integers from $0$ to $n$, and so its preimages break up $B$ into a finite disjoint union of clopen connected components. On each of these components $\text{tr}(e)$ is constant, and from here it should be straightforward to produce local trivializations.

Answer 2

Let me start by mentioning that for sure this is true when $B$ is compact (which for me includes Hausdorff): this is a celebrated theorem of Swan. I give a fairly detailed proof in $\S$6 of these notes.

For a more general base: it's been a little while since I've thought about this carefully, but it seems to me that the proof of this part of Swan's Theorem does not use any assertions about the base. In $\S$6.2 there is established an easy categorical equivalence between trivial vector bundles and finitely generated free modules over $\mathbb{R}(B)$. It is then explained that every finitely generated projective module is the image of an idempotent endomorphism $P$ of a finitely generated free module, so under the categorical equivalence we have an idempotent endomorphism $P$ of a trivial vector bundle $E$ on $X$ and we need to show that $P(E)$ is a vector bundle on $X$. This is Proposition 6.4. (The real meat of Swan's Theorem -- and here it seems that paracompactness is needed -- is to realize every vector bundle with bounded fiber dimensions as a direct summand of a trivial bundle.)

So...along with Qiaochu, I also say yes.

Added: I have just consulted the lovely recent text Ideals and Reality by Ischebeck and Rao. It has a superior (e.g., to mine!) treatment of Swan's Theorem. In fact it gives a more general result which both nails down an answer to the OP's second question (the first one is much easier) and addresses the issues raised in the comments.

For any topological space $X$, a vector bundle $E$ on $X$ is strongly of finite type if
(SFT1) There is a finite open cover $\mathcal{U} = \{U_i\}_{i=1}^n$ on which $E$ trivializes: i.e., for all $i$, the pullback of $E$ to $U_i$ is a trivial bundle, and
(SFT2) There is a (continuous) partition of unity subordinate to the cover $\mathcal{U}$.

Some remarks:
(i) A topological space $X$ is normal (every pair of disjoint closed subsets can be enlarged to a pair of disjoint open subsets; note that we do not require $X$ to be Hausdorff) iff every finite open covering admits a subordinate partition of unity. (By contrast, a Hausdorff space is paracompact iff every locally finite open covering admits a subordinate partition of unity.)
(ii) (SFT1) implies that the fibers have bounded dimension. If $X$ is a direct sum of infinitely many nonempty subspaces, then there are vector bundles without this property.

And here we go:

Generalized Swan's Theorem (Theorem 5.3.15 in Ischebeck & Rao): For any topological space $X$, the global section functor gives an equivalence from the category of strongly finite type vector bundles on $X$ to the category of finitely generated projective $\mathbb{R}(X)$-modules.