Just a random thought here: Can cohomology theories (e.g. sheaf cohomology) on the Stone space $S_n(T)$ (the space of complete n-types) of a first-order theory $T$ tell us anything interesting (e.g. the classification of theories)? Is there any result in model theory that is obtained (probably most easily) by this kind of application of cohomology theories? Thanks!
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2$\begingroup$ I am not aware of any such result. But don't cohomology theories have problems with zero dimensional spaces? The spaces of types are zero-dimensional. I don't know about sheaf cohomology, though. $\endgroup$– Stefan GeschkeCommented Jul 22, 2010 at 8:55
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$\begingroup$ @Stefan: indeed, you were right. The (sheaf-)cohomological dimension of the type spaces is always zero. So sheaf cohomology can't really give anything more interesting than the global section functor. $\endgroup$– Jizhan HongCommented Apr 7, 2011 at 15:17
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$\begingroup$ Maybe you should put it as an answer so that I can accept it? $\endgroup$– Jizhan HongCommented Apr 7, 2011 at 15:22
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2 Answers
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I can not really inform you about this since I don't know, but I can point you to some notes of Angus Macintyre, http://modular.math.washington.edu/swc/notes/files/03MacintyreNotes.pdf
Here are some excerpts:
"For me personally, the main surprise arising from the discovery of ACFA was how much there was to be done in terms of a model-theoretic reaction to the development of etale cohomology and its relatives."
"Again, in a different direction, one begins to see cohomological ideas coming up all over applied model theory, for example in o-minimality."
I hope that you find this useful.
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You "only" need to change the topology of the stone space to make it interesting. In o-minimality, the "spectral" topology is often used: see e.g. many papers by Edmundo. A similar approach can be used in other topological structures, as long as the structure is definably connected; for structures that are (totally) definably disconnected (like the p-adics) you would need to come out with something different.
