Having already seen group cohomology, I was just introduced to the formula $U otimes Ind W = Ind(Res(U) otimes W)$ from representation theory. This seems oddly like the formula $mathrm{Cor}(u) cup v = mathrm{Cor}(u cup mathrm{Res}(v))$, which can be found as Proposition 1.39 in Chapter 2 of Milne's CFT Notes. Can one be proven from the other? In one case, $U, W$ are actual modules, whereas in the other case, $u,v$ are elements of modules. Maybe this means that certain $G$-modules might somehow classify representations, and the cup product would represent the tensor product of representations?
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