A <complex in an abelian category> is a sequence $(A_n,d_n)$ with $d_nd_{n+1}=0$. A sequence is <exact sequence in an abelian category>[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
$$
\ker f'\to\ker f\to\ker f''
\xrightarrow{\partial}
\operatorname{coker}f'\to\operatorname{coker}f\to\operatorname{coker}f''.
$$

Solution

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

A sequence is exact at an object when the image of the incoming morphism equals the kernel of the outgoing morphism. It is exact when this holds everywhere.


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 Incoming links: Exact sequence in an abelian category