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In [1], the author proves that given a group $G$ and a directed system $(M_\lambda)_\lambda$ of $G$-modules, the induced maps $$\varinjlim H^k(G,M_\lambda) \to H^k(G,\varinjlim M_\lambda)$$ are isomorphisms for all $k$ if and only if $G$ is of type $FP_\infty$.

This seems like a useful result whenever one considers directed systems of $G$-modules, but I haven't been able to find any concrete examples.

My questions are basically:

  • Are there any applications or examples where one wants to compute group cohomology with respect to a direct system of coefficients?
  • Are there any concrete application of Brown's theorem above, i.e. given some fixed group $G$ of type $FP_\infty$, one wants to compute $H^k(G,\varinjlim M_\lambda)$ and uses this "commutativity"?

[1] Brown, Kenneth S., Homological criteria for finiteness, Comment. Math. Helv. 50, 129-135 (1975). ZBL0302.18010.

Mark Backhaus
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    My accepted answer to this question https://mathoverflow.net/questions/272715/cohomological-dimension-of-g-times-g/272755#272755 uses this property of $FP_{\infty}$ groups. The main applications I use tend to be of the flavor that you can reduce things to finitely generated modules because every module is a direct limit of finitely generated modules, but I don't do group cohomology for a living. – Benjamin Steinberg Jul 07 '21 at 12:26

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