Spectral sequence

Summary

A spectral sequence is a sequence of pages $E_r$ of objects indexed by two integers. Each page is the homology of the previous one for a differential $d_r$. It is a bookkeeping tool that computes the homology of a filtered complex or a double complex by successive approximations. Spectral sequences connect derived functors of composites, the homology of fibrations, and many other computations in algebra and topology.