I am reading the corresponding section in Cassels-Frohlich. This is the axiomatic definition of cup products given there: (I'm copying from Alison Miller's answer here)
The cup product is a family of maps from $H^p(G, A) \otimes H^p(G, B)\to H^p(G, A \otimes B)$ for all $A$, $B$ and all non-negative integers $p$, $q$ (for Tate cohomology, put hats on all the $H$'s and allow $p$, $q$ to be arbitrary integers).
(i) These homomorphisms are functorial in $A$ and $B$.
(ii) For $p = q = 0$, they are (induced by) the natural product $A^G \otimes B^G \to (A \otimes B)^G$.
(iii and iv) They are compatible with the delta map in the long exact sequences associated to short exact sequences of $G$-modules in the following way: $\delta (a'' \smile b) = (\delta a'') \smile b$ and $a'' \smile (\delta b) = (-1)^{(\deg a'')} \delta ( a'' \smile b)$.
I have two questions:
I don't have any intuition about how cup product should behave when we change coefficient - in particular I have no intuition what conditions (iii) and (iv) are supposed to mean. Can anyone give a brief explanation/motivation in terms of say, sheaf cohomology, or point me to a reference where I can gain better intuition?
I don't understand how cup product interplays with (co)restriction. Cassels-Frohlich listed some properties, but he proves it only for two zero-th (Tate) cohomology groups, and says the magic word "dimension shifting". This is not very convincing to me - but again, it's probably because I don't understand (iii) and (iv), which I believe is the crux of dimension shifting here.
Thanks!
