Bockstein Spectral Sequence (Why is this a direct sum)

Question

(from McCleary's User guide pg 459).

I am curious why is $pH_n(X)+\ker p^r$ a direct sum? (Or is it?)

I tried to reason that $\ker p^r$ is equal to $\text{Im}\ \partial:H_{n+1}(X;\mathbb{F}_p)\to H_n(X)$, so those elements in $\ker p^r$ "come from" $H_{n+1}(X;\mathbb{F_p})$ where they are already reduced mod p, so the only way an element $u$ in $\ker p^r$ can be in $pH_n(X)$ (a multiple of $p$) is when $u$ is the zero element.

Is that correct reasoning?

Thanks.

Answer 1

It is not a direct sum. Consider for example when $r=1$ and we have $H_n(X)=\mathbb{Z}_{p^2}$. Then $p\cdot H_n(X)=\ker(H_n(X)\xrightarrow{\times p}H_n(X))$.