Exercise A0.III.7.2

Proposition

Suppose we have an exact complex

$$\cdots \longrightarrow \; 0 \overset{f_1}\longrightarrow M \overset{f_2}\longrightarrow M' \overset{f_3}\longrightarrow 0 \longrightarrow \cdots.$$

Then $M$ is isomorphic to $M'$.

Proof

Note that $\im f_1 = \ker f_2 = 0$ so $f_2$ is injective. Also, $M' = \ker f_3 = \im f_2$ so $f_2$ is surjective. So $f_2$ is an isomorphism.

$\blacksquare$