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My question is about the compatibility and consistency between two definitions of cohomology in two books. I asked this question about 10 days ago on MathSE and I set a bounty on it, but I didn't ...

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...

I am reading Class field theory - Bonn Lectures by Neukirch. Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules ...