Chain complex

Summary

A chain complex is a sequence of objects, such as abelian groups or modules, joined by morphisms $d_n \colon C_n \to C_{n-1}$ with $d_{n-1} \circ d_n = 0$. The condition says that the image of each map lies in the kernel of the next, so the homology $\ker d_n / \im d_{n+1}$ is defined. Chain complexes are the central objects of homological algebra. They arise from topological spaces as in singular homology, and from modules through resolutions.