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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:

  1. 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?

  2. 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!

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It might be helpful to you to read part of Benson's book Representations and Cohomology I which shows how to construct group cohomology from the ground up and talks about induction/restriction, (cup) products, dimension shifting etc. –  mt_ Mar 26 '12 at 7:32
    
@mt_ Thanks for the reference! I just took a quick look at it - it seems to have an expanded discussion but based on homological algebra. I am willing to take these stuff (existence of cup products etc) granted for now, as long as I have the right intuition. So I would like to have something more concrete for me to play with, rather than something formal - an example of sheaf cohomology or even singular cohomology that lets me know what happens in change of coefficients would be great. Thanks again! –  user27126 Mar 26 '12 at 8:05

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