Let $R$ be a ring, and let $C_\bullet = (C_n, d_n)_{n \in \mathbb{Z}}$ be a chain complex of $R$-modules, where each differential is an $R$-module homomorphism
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\begin{align*}
d_n: C_n \to C_{n-1}
\end{align*}
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and satisfies $d_n \circ d_{n+1} = 0$ for every $n \in \mathbb{Z}$. For each $n \in \mathbb{Z}$, define
Then $C_\bullet$ is exact, meaning $\operatorname{im} d_{n+1} = \ker d_n$ for every $n \in \mathbb{Z}$, if and only if $H_n(C_\bullet) = 0$ for every $n \in \mathbb{Z}$.