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!

$\begingroup$ Related: mathoverflow.net/questions/120808 $\endgroup$ – Martin Brandenburg Feb 18 '13 at 21:02
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.

1$\begingroup$ I believe all limits are preserved because Hom preserved the and group cohomology is a derived functor of Hom. $\endgroup$ – Benjamin Steinberg Feb 18 '13 at 19:39

$\begingroup$ Don't you have problems since inverse limits are not exact? $\endgroup$ – Lennart Meier Feb 18 '13 at 20:05

$\begingroup$ By "all limits" in the comment you mean "all inverse limits" ? $\endgroup$ – Demin Hu Feb 18 '13 at 20:12

$\begingroup$ By limit I mean in the category sense as opposed to colimits like the misnamed direct limit. $\endgroup$ – Benjamin Steinberg Feb 18 '13 at 20:17

$\begingroup$ My first comment about inverse limits is probably wrong because inverse limits are not exact $\endgroup$ – Benjamin Steinberg Feb 18 '13 at 20:30