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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?

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It'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.) –  Pete L. Clark May 23 '11 at 2:23
    
@Pete: Yea I don't know why I tagged it as differential geometry and topology. I've now edited the tags. –  Eric O. Korman May 23 '11 at 2:50
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@Pete: indeed, it is hard to pin down exactly what "analogous to S-S" should mean to show that it cannot happen :) –  Mariano Suárez-Alvarez May 23 '11 at 2:54
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1 Answer

This is not really an answer. Rather, it deals with a different question, namely the analogue of your question in algebraic geometry. Perhaps it is still of some interest.


In algebraic geometry, one has an analogue of Serre--Swan, in which f.g. projective modules over a ring (commutative, with 1) $A$ correspond to finite rank locally free sheaves on Spec $A$.

If I recall correctly, under some assumptions, one can obtain injective $A$-modules as local cohomology sheaves of the structure sheaf supported at (not necessarily closed) points of Spec $A$.

E.g. if $A = k[x]$ (with $k$ an algebraically closed field), then the field of rational functions $k(x)$ is obtained as the local cohomology sheaf $\mathcal H^0_{\eta}(\mathcal O)$, where $\eta$ is the generic point of Spec $A$, while for a closed point $(x-a)$, the injective module $k[x,1/(x-a)]/k[x]$ is obtained as the local cohomology sheaf $\mathcal H^1_{(x-a)}(\mathcal O).$

This construction is due to Grothendieck, I think, and is discussed in Hartshorne's book Residues and duality, as a way of constructing canonical injective resolutions.

E.g. when $A = k[x]$, we have the injective resolution $$0 \to k[x] \rightarrow k(x) \to \bigoplus_{a \in k} k[x,1/(x-a)]/k[x] \to 0,$$ which can be rewritten in terms of sheaves on $X = $ Spec $A$ as $$0 \to \mathcal O_X \to \mathcal H^0_{\eta}(X,\mathcal O_X) \to \bigoplus_{a \in k} \mathcal H^1_{(x-a)}(X,\mathcal O_X) \to 0;$$ so the structure sheaf has an injective resolution whose $p$th term involves a sum over local cohomology sheaves supported at codimension $p$ points.

I don't have much of a feeling as to whether one can carry this over in any useful way to the topological setting, since the geometry of closed subsets in algebraic geometry is much more rigid than in general topology.

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Don't you get only indecomposable injectives like that? –  Mariano Suárez-Alvarez May 23 '11 at 3:25
    
@Mariano: Dear Mariano, Quite likely; it's been a while since I thought about this! I posted it since it's the only link between sheaf theory and injectives that I know. Best wishes, –  Matt E May 23 '11 at 3:27
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