# Intuition for cohomology of groups [duplicate]

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I just finish reading a pdf about group cohomology that explain a lot about it. It goes from the definition to Galois's cohomology. But there is one think I would like to know and I think I haven't enough read about cohomology to know : What is the intuition behind all of this theory ? If someone could explain me or just give me some references I would appreciate. Thank you very much !

## marked as duplicate by Watson, Pedro Tamaroff♦Jun 2 '18 at 14:36

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• This is not intuition, but possible motivation : if you are interested in computing the rational points of an elliptic curve $E$, say $y^2 = x^3 - k$, then you notice that $E(\Bbb Q)$ is the set of invariants of $E(\overline{\Bbb Q})$ under the action of the absolute Galois group. Taking these invariants provides a left-exact functor [unfortunately it is not right exact, i.e. invariants don't commute with quotients], whose right derived functors are precisely giving group cohomology (here Galois cohomology). – Watson Jun 2 '18 at 14:30
• – Watson Jun 2 '18 at 14:33
• Actually I don't know what is an Elliptic curve neither than an absolute Galois group, is this hard to understand ? – Pierre21 Jun 2 '18 at 14:44
• Dear @Pierre21, the equation $y^2 = x^3 - k$ (for a fixed non-zero integer $k$) is the equation of an elliptic curve, and $E(\Bbb Q)$ is the set of pairs $(x,y)$ satisfying this equation (there is just a "subtelty" with the "point at infinity"). This is not hard at all! (What is harder is to have a group structure on $E(\Bbb Q)$, though). – Watson Jun 2 '18 at 14:46
• Hmm okay thank you ! But like is this that type of oservation that lead the mathematician to construct all this structure ? – Pierre21 Jun 2 '18 at 14:49