spectral sequence

A spectral sequence is a way to compute a hard homology in stages, by successive approximation. Imagine developing a photograph: a blurry first image sharpens through a series of passes, each pass correcting the previous one, until it stabilizes into the true picture. A spectral sequence is exactly such a process for homology — a sequence of pages, each obtained as the homology of the last, the whole sequence converging to the answer you actually want.

Formally, a spectral sequence (of cohomological type) is a collection of bigraded modules E_r^{p,q} for r ≥ r_0, each equipped with a differential d_r : E_r^{p,q} -> E_r^{p+r, q-r+1} satisfying d_r ∘ d_r = 0, together with isomorphisms identifying E_{r+1} with the homology of (E_r, d_r). The index r is the page number; passing to the next page means taking homology with respect to the current differential. Under good conditions (e.g. first-quadrant) the differentials eventually vanish, the pages stabilize to a limit E_∞, and one says the spectral sequence converges to a graded module H, meaning E_∞^{p,q} is the associated graded of a filtration on H^{p+q}.

The two great sources are filtered complexes and double complexes; both produce spectral sequences whose E_2 page is computable and whose limit is the homology sought. Classic instances include the Leray spectral sequence relating cohomology of a space to that of a base and fiber, the Lyndon-Hochschild-Serre spectral sequence for group cohomology of an extension, and the Grothendieck spectral sequence for the derived functor of a composite. The honest caveat: a spectral sequence computes only the associated graded of the answer, so extension problems must still be resolved by hand to recover H exactly.