i aslo have a quesiton that what's your meaning of "$\bigoplus_{v \in S} U^+_v(X)$ is shifted with respect to $C^\bullet_{cont}(G_{K,S},X)$"? I can understand that why finally our map should have two parts with opposite sign, because locally condition's aim is to restrict the image, making it within a set of $\bigoplus_{v \in S} C^{\bullet}_{cont}(G_v, X)$, which is given by $U_v^+(X)$.

Thanks for your explanation. So I should regard the two morphism in the first form as the morphism from the direct sum to the $\bigoplus_{v \in S} C^{\bullet}_{cont}(G_v,X)$?

Are there some application of group rings for infinite groups on number theory? So I ask the question. I' m reading the paper you linked.

Hi Tim, sorry for my late response. My meaning of the question is when we research algebraic number theory, we also use the completer group ring which is the inverse limit of finite group rings produced by the same coefficients ring and quotient groups of the group by all open normal subgroups. However, when I study Selmer complex book wrriten by Jan Nekovar, I find that on the book there occur many symbols for group ring, like R[G], for profinite group. So it makes me confused and want to make the meaning of symbols clearly. Besides, it inspires me to ask the question.

@KConrad Thanks for your comment. If there is no special interpretation on context, where I can just find R-modules M and N, should I assume the tensor without subscript means tensor for R-modules?

@Denis T Thanks very much for pointing my mistake. I will amend it later.