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?
