The Serre-Swan theorem in topology says that if $X$ is compact Hausdorff and $C(X)$ the ring of continuous functions on $X$, then the category of finitely generated projective $C(X)$-modules is equivalent to the category of vector bundles over $X$. Is there an analogous theorem for the dual notion of injective modules?
Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?
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general-topology
vector-bundles
topological-k-theory
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2It's a nice question. I'll be flabbergasted if the answer is "yes", but I'm not sure exactly how to turn that into a mathematical pronouncement. By the way, I don't see any differential anything in the question: it seems to be pure topology and algebra. (On the other hand, there is a straightforward analogue of S-S where you work with smooth manifolds, smooth bundles and rings of smooth functions. IIRC, the differential geometric content in the proof of this is simply the existence of smooth partitions of unity.) – 2011-05-23
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0@Pete: Yea I don't know why I tagged it as differential geometry and topology. I've now edited the tags. – 2011-05-23
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3@Pete: indeed, it is hard to pin down exactly what "analogous to S-S" should mean to show that it cannot happen :) – 2011-05-23