Exercise A0.III.7.04

Proposition

There exist short exact sequences

$$0 \longrightarrow \mathbb{Z}^{\oplus\mathbb{N}} \longrightarrow \mathbb{Z}^{\oplus\mathbb{N}} \longrightarrow \mathbb{Z} \longrightarrow 0$$

and

$$0 \longrightarrow \mathbb{Z}^{\oplus\mathbb{N}} \longrightarrow \mathbb{Z}^{\oplus\mathbb{N}} \longrightarrow \mathbb{Z}^{\oplus\mathbb{N}} \longrightarrow 0.$$

Proof

For the first sequence, consider the map $f : \mathbb{Z}^{\oplus\mathbb{N}} \to \mathbb{Z}^{\oplus\mathbb{N}}$ sending $g_n \mapsto g_{n+1}$ where $g_n$ is the $n$-th generator of $\mathbb{Z}^{\oplus\mathbb{N}}.$ This map is injective, since it simply reindexes identical free generators. Consider the map $g : \mathbb{Z}^{\oplus\mathbb{N}} \to \mathbb{Z}$ which sends $g_1$ to the generator of $\mathbb{Z},$ and all other generators to $0.$ It is simply the $0$-th canonical projection of $\mathbb{Z}^{\oplus\mathbb{N}},$ and hence is surjective.

This sequence is exact, since the image of $f$ is $\mathbb{Z}^{\oplus\mathbb{N}},$ and the kernel of $g$ is the same $\mathbb{Z}^{\oplus\mathbb{N}}$ by construction. The image of $g$ is $\mathbb{Z},$ since it is surjective, and the kernel of $f$ is $0$, since it is injective.

For the second sequence we first reindex all generators by $g_n \to g_{2n}$ and then discard all even-indexed generators.

$\blacksquare$