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For a discrete group G, if $M$ is a direct/inverse limit of $M_i$, is $H^i(G, M)$ the direct/inverse limit of the $H^i(G, M_i)$? Of course, cohomology commutes with finite direct sums, but how about general limits/colimits over finite categories? Thank you!

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Related: mathoverflow.net/questions/120808 –  Martin Brandenburg Feb 18 '13 at 21:02
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up vote 3 down vote accepted

Ken Brown shows in http://www.math.cornell.edu/~kbrown/scan/1975.0050.pdf that group cohomology for a group G commutes with direct limits iff G is of type $FP_\infty$. That is the trivial module $\mathbb Z$ has a projective $\mathbb ZG$ resolution which is finitely generated in each degree.

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I believe all limits are preserved because Hom preserved the and group cohomology is a derived functor of Hom. –  Benjamin Steinberg Feb 18 '13 at 19:39
    
Don't you have problems since inverse limits are not exact? –  Lennart Meier Feb 18 '13 at 20:05
    
By "all limits" in the comment you mean "all inverse limits" ? –  Demin Hu Feb 18 '13 at 20:12
    
By limit I mean in the category sense as opposed to colimits like the misnamed direct limit. –  Benjamin Steinberg Feb 18 '13 at 20:17
    
My first comment about inverse limits is probably wrong because inverse limits are not exact –  Benjamin Steinberg Feb 18 '13 at 20:30
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