Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 6 / a

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- Solution a

Solution

A complex in an abelian category is a sequence

(A_{n}, d_{n})

with

d_{n} d_{n + 1} = 0

. A sequence is exact when the image of every incoming map equals the kernel of the outgoing map.

The Five lemma says that in a morphism between exact five-term sequences, suitable epimorphism assumptions on the left and monomorphism assumptions on the right, together with isomorphisms in the four surrounding positions, force the middle map to be an isomorphism.

The Snake lemma associates to a morphism of short exact sequences the exact sequence

ker f^{'} \to ker f \to ker f^{''} \partial coker f^{'} \to coker f \to coker f^{''} .

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