I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.

The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence $$0 \to P \to M \to N \to 0,$$

we only have the following partial sequence for the $0$-th cohomology groups: $$0 \to P^G \to M^G \to N^G$$

and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?

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    $\begingroup$ How you already studied, say, simplicial homology and cohomology (or similar)? That would be a really good motivation... $\endgroup$ – Mark S. Sep 3 '18 at 15:59
  • $\begingroup$ @MarkS. I haven't, this is my first contact with the topic, I only know a bit about algebraic topology and homotopy groups. Any better reference to grasp the topic would be also warmly welcome! $\endgroup$ – TheStudent Sep 4 '18 at 2:35
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    $\begingroup$ I can't find a source that explains the basic ideas very directly, but Chapter 5 of seas.upenn.edu/~jean/sheaves-cohomology.pdf seems to be a decent. Simplicial homology seems to have a lot more free sources online, and Chapter 2 of Hatcher may suffice. $\endgroup$ – Mark S. Sep 4 '18 at 10:57

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