Exercise 1.1.1. Set $C_n = \Bbb{Z}/8$ for $n \geq 0$ and $C_n = 0$ for $n \lt 0$. Let $d_n : x \pmod{8} \to 4x \pmod{8}$ Compute the homology modules of the chain complex $C_{\cdot}$.
I got that $\ker d_n = \{0,2,4,6\}$ for $n \gt 0$, $ \ker d_n = \Bbb{Z}/8$ for $n = 0$ and $\ker d_n = 0$ for $n \lt 0$.
I also got that $\text{im } d_{n+1} = \{0, 4\}$ for $n \gt 0$, and $\text{im } d_{n+1} = 0$ for $n \leq 0$.
So the homology modules are $\ker d_n / \text{im } d_{n+1} = \{\bar{0}, \bar{2}\} \approx \Bbb{Z}/2$ for $n \gt 0$, $(\Bbb{Z}/8) / 0 \approx \Bbb{Z}/8$ for $n = 0$ and $0/0 \approx 0$ for $n \lt 0$. Did I do that right? Also do I now say "the homology modules are $\Bbb{Z}/2$, etc." or that they're isomorphic to these?