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It is, it seems, rather well-known that there's no universal coefficient theorem for cohomology with local coefficients. However in Spanier's book Algebraic topology, there is an exercise that asks the reader to do something that at least goes by that name! Specifically (from page 283),

4 If $\Gamma$ is a local system of $R$-modules on $X$ and $G$ is an $R$-module, there is a local system $\mathrm{Hom}(\Gamma,G)$ of $R$-modules on $X$ which assigns to $x\in X$ the module $\mathrm{Hom}(\Gamma(x),G)$. Prove that $$ \Delta^*(X,A;\mathrm{Hom}(\Gamma,G)) \approx \mathrm{Hom}(\Delta(X,A;\Gamma),G) $$ Deduce a universal-coefficient formula for cohomology with local coefficients.

It does not appear that he is assuming $R$ is a PID or anything special. A 2018 paper (Local Coefficients Revisited) says "there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited".

Though there's no actual statement in Spanier, as is apparent from the quote above!

What should the statement be, and why is it of limited application? (I'm not particularly interested in the proof at present, just what the theorem is meant to say!)

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    $\begingroup$ Doesn't Spanier assume $R$ is a PID more or less throughout all the chapters he does homological algebra in? (The fact that $R$ is hereditary is quite crucial for the UCT to be true, I think?) $\endgroup$
    – Pedro
    Commented May 11, 2021 at 16:42
  • $\begingroup$ @PedroTamaroff not sure! I didn't dig back to the beginning of the chapter to see any standing assumptions. $\endgroup$ Commented May 12, 2021 at 0:03

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I think the identity Spanier proposes is true, and should not be difficult to prove. The question is what piece of homological algebra one should then apply to it.

If $R$ is left hereditary (eg a PID) and either $G$ is an injective $R$-module (unlikely) or else $\Delta(X, A;\Gamma)$ is a complex of projective $R$-modules (which holds iff the $\Gamma(x)$ are projective $R$-modules), then page 114 of Cartan-Eilenberg gives a standard-looking UCT, of the form $$0 \to Ext^1_R(H_{i-1}(X,A;\Gamma), G) \to H^i(X,A; Hom_R(\Gamma, G)) \to Hom_R(H_i(X,A;\Gamma), G) \to 0.$$

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    $\begingroup$ Hi, welcome to the site! The statement of theorem in p. 114 of C&E you're referring to is, I suspect, Thm 3.3a 4(a) applied to $A=\Delta(X,A;\Gamma)$ and $G=C$? If so, I could edit it in for those that may not have the book immediately available to them. If you'd like to edit the answer yourself, that's no problem at all. Regards, $\endgroup$
    – Pedro
    Commented May 11, 2021 at 18:27
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    $\begingroup$ Thanks, Oscar, that helps a lot. The use case was $R=\mathbb{Z}$, $\Gamma(x)\simeq \mathbb{Z}^n$, so I think I'm probably in luck, modulo the dualising issue... $\endgroup$ Commented May 12, 2021 at 0:06
  • $\begingroup$ @PedroTamaroff I concur that sequence (4a) in Theorem 3.3a is what Oscar seems to be relying on, getting (3'a) from his "either...or..." assumption. $\endgroup$ Commented May 12, 2021 at 6:10
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After discussion with the OP, here's a little note that I wrote offering a modern point of view on this theorem - viewing it as a special case of the algebraic UCT, using the point of view that local coefficient systems are functors $X\to D(R)$, and that their co/homology can be interpreted as the homology of a limit/colimit of that functor.

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  • $\begingroup$ I want to thank you publicly for this. It was a very fun discussion, and I'm glad you got a thorough, general and modern formulation to share with the world! $\endgroup$ Commented Jun 1, 2021 at 23:41

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