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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$\begingroup$ Related: mathoverflow.net/questions/120808 $\endgroup$– Martin BrandenburgFeb 18, 2013 at 21:02
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1 Answer
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Ken Brown shows in
- Homological criteria for finiteness, Comment. Math. Helv. 50 (1975), 129–135, doi:10.1007/BF02565740, (free author version)
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.
answered Feb 18, 2013 at 19:34
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1$\begingroup$ I believe all limits are preserved because Hom preserved the and group cohomology is a derived functor of Hom. $\endgroup$ Feb 18, 2013 at 19:39
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$\begingroup$ Don't you have problems since inverse limits are not exact? $\endgroup$ Feb 18, 2013 at 20:05
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$\begingroup$ By "all limits" in the comment you mean "all inverse limits" ? $\endgroup$– Demin HuFeb 18, 2013 at 20:12
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$\begingroup$ By limit I mean in the category sense as opposed to colimits like the misnamed direct limit. $\endgroup$ Feb 18, 2013 at 20:17
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$\begingroup$ My first comment about inverse limits is probably wrong because inverse limits are not exact $\endgroup$ Feb 18, 2013 at 20:30