An example of direct sum but not split exact sequence

Recall the definition of split exact sequence

We say a short exact sequence split iff. it is isomorphic to the exact sequence

$$0\xrightarrow{} A\xrightarrow{f} A\oplus B \xrightarrow{g} B\xrightarrow{}0$$

where $f$ is the inclusion of the first component and $g$ is the projection onto the second
component.

注意到split exact seq 在定义里要求$f$ 和$g$ 分别是到$A\oplus B$的每一个component, 是否有例子使得一般的short exact seq s.t. $0\xrightarrow{} A\xrightarrow{f} A\oplus B \xrightarrow{g} B\xrightarrow{}0$ 但是不是exact seq?

答案是Yes.

Example of non split exact sequence

Consider the short exact sequence

$$0\xrightarrow{} \mathbb{Z}/2\mathbb{Z}\xrightarrow{f} \mathbb{Z}/4\mathbb{Z} \xrightarrow{g} \mathbb{Z}/2\mathbb{Z}\xrightarrow{}0$$

where $f$ is the inclusion of the subgroup generated by 2, so $f(1) = 2$, and $g$ is the quotient
onto that subgroup, meaning $g(1) = 1$. This is not a split short exact sequence(obviously). Consider $M\bigoplus\limits_{\mathbb{N}}(\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z})$.

Thus the exact seq

$$0\xrightarrow{} \mathbb{Z}/2\mathbb{Z}\xrightarrow{h} \mathbb{Z}/4\mathbb{Z}\oplus M \xrightarrow{i} \mathbb{Z}/2\mathbb{Z}\oplus M\xrightarrow{}0$$

with $h(a) = (f(a), 0)$ and $t(a, m) = (g(a), m)$ is exact. $\mathbb{Z}/4\mathbb{Z}\oplus M$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z}\oplus M)$. However, this is not a split exact seq since

$$0\xrightarrow{} \mathbb{Z}/2\mathbb{Z}\xrightarrow{h} \mathbb{Z}/4\mathbb{Z}\oplus M \xrightarrow{i} \mathbb{Z}/2\mathbb{Z}\oplus M\xrightarrow{}0$$

will give a exact seq $$0\xrightarrow{} \mathbb{Z}/2\mathbb{Z}\xrightarrow{f} \mathbb{Z}/4\mathbb{Z} \xrightarrow{g} \mathbb{Z}/2\mathbb{Z}\xrightarrow{}0$$ which is not split exact.