Reference for the Swan-Serre theorem as a monoidal equivalence Let $X$ be a compact Hausdorff The well-known Swan--Serre theorem gives an equivalence between the continuous vector bundles over a compact Hausdorff space $X$, and finitely-generated projective $C(X)$-modules. Both the category of vector bundles and the category of projective modules have evident monoidal structures. With respect to these structures, is the Swan--Serre equivalence a monoidal equivalence? I would guess that this is the case but I cannot find a reference.
 A: Recall that the equivalence in question is given by taking global sections of a vector bundle, $V\mapsto \Gamma(X,V)$.
Now if you have a section $s$ of $V$ and a section $t$ of $W$, this gives you a section $s\otimes t$ of $V\otimes W$ so this gives you a natural morphism $\Gamma(X,V)\otimes_{C(X)}\Gamma(X,W)\to \Gamma(X,V\otimes W)$
(note that if $f$ is a function on $X$, then $((f\cdot s)\otimes t)_x = f(x)s_x \otimes t_x = s_x \otimes f(x)t_x = (s\otimes (f\cdot t))_x = (f\cdot (s\otimes t))_x$, so this does descend to the $C(X)$-tensor product, and is indeed $C(X)$-linear)
Furthermore, this morphism is an isomorphism when $V=W$ is the trivial line bundle over $X$, and both sides are additive functors in each variable, so it is an isomorphism whenever $V$ and $W$ are summands of trivial bundles, i.e. for all $V,W$ because any vector bundle is a summand of a trivial vector bundle (equivalently any finitely generated projective $C(X)$-module is a summand of a finitely generated free $C(X)$-module).
So the answer is yes, $\Gamma(X,-)$ has a natural symmetric monoidal structure.
I don't know the references though, so if this is what you're looking for then this doesn't quite answer.
