Let $B$ be a $T_{3\frac12}$ space, $C(B)$ be the ring of continuous real valued functions on $B$. Show that if $M$ is a finitely generated projective module over $C(B)$, then there is a vector bundle $E$ over $B$ so that $M= \Gamma(E)$.
Progress so far: $M$ finitely generated projective $\implies\;\exists N\in C(B)-\operatorname{Mod}\; M\oplus N = C(B)^n$ for some $n.$ But $C(B)^n = \Gamma(B\times \mathbb{R}^n)$ the rank $n$ trivial bundle. If we can separate the trivial bundle as $B\times\mathbb{R}^n = E\oplus F\; \Gamma(E) = M\;\Gamma(F) = N$ we'd be done. So we need to quotient the trivial bundle in some way so that the space of sections. My first guess would be $E = \{\mathbb{R}s(b) | s\in M, b\in B\}, F = \{\mathbb{R}s(b) | s\in N, b\in B\}$,because it seems like the most natural thing, but I can't figure out a way to show that this works.