Bourbaki

Exercise 2, § 1

Algebra, X, p. 168

Let u:M→Nu:M\to N, v:N→Pv:N\to P be two homomorphisms of AA-modules. Now prove that there exists an exact sequence:

0→Ker⁡(u)→Ker⁡(v∘u)→Ker⁡(v)→Coker⁡(u)→Coker⁡(v∘u)→Coker⁡v→0. 0\to\operatorname{Ker}(u)\to\operatorname{Ker}(v\circ u)\to\operatorname{Ker}(v)\to\operatorname{Coker}(u)\to\operatorname{Coker}(v\circ u)\to\operatorname{Coker}v\to0.
Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.