If you trawl the internet for more exotic Universal Coefficient Theorems, then you’ll come across comments to the effect that there isn’t a UCT for cohomology with local coefficients in general (for instance, here, for group cohomology, which is ordinary cohomology of a , or the Groupprops wiki page, which only gives the trivial coefficients version). I had reason to want a UCT for group cohomology with coefficients a nontrivial module, to try to clear some hypotheses using formal properties of group homology (it’s better behaviour with respect to filtered colimits, in particular). The only reference I could find for a suitable UCT was an *exercise* in Spanier’s venerable book:

I found a reference in a 2018 paper, that said “there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited”. Up until this point, I had not actually seen the statement written out anywhere! In particular, it’s not clear what Spanier’s assumptions are (it might be he is assuming is a PID throughout this section, but I couldn’t see it on a quick search), and in particular, something must break for this to be of “limited” application.

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