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!



Which limits does group cohomology commute with?
Ken Brown shows in


* *Homological criteria for finiteness*, Comment. Math. Helv. 50 (1975), 129–135, doi:[10.1007/BF02565740](https://doi.org/10.1007/BF02565740), ([free author version](https://pi.math.cornell.edu/%7Ekbrown/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.


