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Let $f: A \to B$ and $g: B \to C$ be two arrows in an abelian category $\mathsf{A}$. Prove that they induce an exact sequence:

$$0 \to kerf \to kergf \to kerg \to cokerf \to cokergf \to cokerg \to 0$$

This is exercise 8.4.6 in Category Theory for Working Mathematicians.

Here is my attempt:

This looks much like a corrolary of the snake lemma. I tried to ensemble them into a diagram (q.uiver link) but this does not seem to work as the middle rows are far from exact.

In this diagram (again q.uiver link) the 'connecting' morphism $kerg \to cokerf$ is obvious. However, there seems not to exist an easy method to check exactness other than laboriously chasing elements.

SalutaFungo
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  • Hint: Try fiddling with the entries in your diagram such that the middle rows are exact. You are on the right track, wanting to use the snake lemma. – Julius J. Feb 21 '24 at 14:36
  • @JuliusJ. Thats exactly what I have tried but it seemed impossible. To fit $a \to a \to b$ into a short exact sequence without any restrictions whatsoever... I cannot figure out how that would work. – SalutaFungo Feb 23 '24 at 06:45
  • I found the solution presented as an exercise in Algebre Commutative by Bourbaki. I will write an answer shortly- – SalutaFungo Mar 28 '24 at 05:41

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I was right with the idea to ensemble all $f,g,gf$ into a snake lemma diagram and here is its construction (q.uiver link). This diagram is precisely the mapping cone of this simpler diagram, which should be of interest.

SalutaFungo
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