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Benjamin Steinberg
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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.

Benjamin Steinberg
  • 38.6k
  • 3
  • 104
  • 186