Lemma 12.18.2. Let $(A, E, \alpha , f, g)$ be an exact couple in an abelian category $\mathcal{A}$. Set

1. $d = g \circ f : E \to E$ so that $d \circ d = 0$,

2. $E' = \mathop{\mathrm{Ker}}(d)/\mathop{\mathrm{Im}}(d)$,

3. $A' = \mathop{\mathrm{Im}}(\alpha )$,

4. $\alpha ' : A' \to A'$ induced by $\alpha$,

5. $f' : E' \to A'$ induced by $f$,

6. $g' : A' \to E'$ induced by “$g \circ \alpha ^{-1}$”.

Then we have

1. $\mathop{\mathrm{Ker}}(d) = f^{-1}(\mathop{\mathrm{Ker}}(g)) = f^{-1}(\mathop{\mathrm{Im}}(\alpha ))$,

2. $\mathop{\mathrm{Im}}(d) = g(\mathop{\mathrm{Im}}(f)) = g(\mathop{\mathrm{Ker}}(\alpha ))$,

3. $(A', E', \alpha ', f', g')$ is an exact couple.

Proof. Omitted. $\square$

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