Exercise A0.III.7.1

Proposition

Suppose we have an exact complex

$$\cdots \longrightarrow \; 0\longrightarrow M \longrightarrow0\longrightarrow \cdots.$$

Then $M$ is trivial.

Proof

From 6.18 we know that $M$ is generated by the disjoint union of generators of $0$ and $0.$ But $0 \cong R^{\oplus 0}$ and hense is a free object on $0$ generators. So $M$ is generated by an empty set.

$\blacksquare$