Augmented chain complex
= Augmented chain complex
{title2=$\cdots\to C_1\to C_0\xrightarrow{\varepsilon}\mathbb Z\to0$}
An augmented <chain complex> includes an augmentation $\varepsilon:C_0\to\mathbb Z$ with $\varepsilon d_1=0$. For a nonempty <simplicial complex>, the augmentation sends every vertex to $1$ and the augmented homology in degree zero is its <reduced homology>.