In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (E_n,d_n) such that

E_n+1 = H(E_n) = ker d_n / im d_n

is the homology of E_n.

There are several ways in practice that such a 'linked' sequence can arise in homological algebra. Historically (since about 1950) spectral sequence arguments have been an important research tool, particularly in homotopy theory; even though, as explained by one of the leading experts, such a discussion may become 'too messy to publish in that form'. That is, they are intricate, certainly as compared to exact sequence arguments that are in effect a simple special case.

Overall explanation

One way to visualise what is occurring in a spectral sequence is by means of a notebook–spreadsheet metaphor. The initial E₁ being the first sheet of data, the E₂ sheet is derived from it by a definite process; and so on. The 'end result' of the calculation would be a final sheet.

The spreadsheet talk here is fairly appropriate, because in practice the E_i tend to carry some grading data, often a double grading. Each sheet is then ruled into cells, indexed by row and column, with an abelian group in each cell. Each sheet also has mappings called differentials, acting from each cell on the sheet to some other cell in a way referred to pictorially as knight's moves. The 'definite process' mentioned above is then a way to calculate each cell in the next sheet using the previuos sheet's data and differentials. The process often stabilizes at the final sheet, and then repeats itself eternally because all the differentials from that sheet onwards are identically zero.

Notationally the E_n would therefore carry two index numbers

E_n^p,q

with differentials d_n^p,q acting from each E_n^p,q to some E_n^p+a,q+b, with a and b depending only on n. That is, the spectral sequence as process is analogous to a book with pages ruled out into grids, one for each E_n. (As David Mumford writes, it becomes easier to work it out on one's own, rather than try to follow someone else's notations.)

The spectral sequence is often used to derive some data about the final sheet knowing the data from initial sheets, or vice versa. Take, for instance, the Leray-Serre spectral sequence of a fibration in algebraic topology. For many fibrations, on the second sheet the first column is cohomology of the fiber, and the first row is cohomology of the base space, while the final sheet is determined in a certain way by cohomology of the total space of the fibration. One might, for example, use this spectral sequence to calculate cohomology of the group SU(3) from the fibration SU(3) → S⁵. This fibration has total space SU(3), base space S⁵, and fiber SU(2) which is the same as the 3-sphere S³. So it is possible to calculate the cohomology of SU(3) knowing the cohomology of spheres.

Filtrations

Spectral sequences arise frequently from filtrations of the initial module E₀. A filtration

of a module induces a short exact sequence

where B, the quotient j of A by its image under the inclusion i, has the differential induced by that of A. Set A₁ = H(A) and B₁ = H(B); a long exact sequence

is then provided by the snake lemma. If we call the displayed maps i₁, j₁, and k₁, and let A₂ = i₁A₁ and B₂ = ker j₁k₁ / im j₁k₁, it can be shown (and perhaps will be in a later version of this article) that

is another exact sequence. Setting i₂ = i₁, j₂ = [j₁i₁⁻¹], and k₂ =

 ? ,

and designating A₃ = iA₂, B₃ = ker j₂k₂ / im j₂k₂, we arrive at a third exact sequence. If we continue in this pattern, (B_n, j_nk_n) is a spectral sequence.

Examples

Some notable spectral sequences are:

• Leray-Serre spectral sequence of a fibration
• Hochschild-Serre spectral sequence in group cohomology
• Adams spectral sequence in stable homotopy theory
• Atiyah-Hirzebruch spectral sequence of an extraordinary cohomology theory
• Adams-Novikov spectral sequence for an extraordinary cohomology theory
• Grothendieck spectral sequence for composing derived functors
• Chromatic spectral sequence for the stable homotopy groups of spheres
• Eilenberg-Moore spectral sequence
• Bockstein spectral sequence

Exact couples

An axiomatic approach that produces spectral sequences is that of exact couples, defined by W. S. Massey in work from around 1952. This is much more elegant, but ultimately depends on the same computational mechanism.

Reference

• A User's Guide to Spectral Sequences by John McCleary
• Cohomology Operations and Applications in Homotopy Theory, by Robert Mosher and Martin Tangora,