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 ) = kernel (mathematics) d n / image (mathematics) d n

is the homology (mathematics) 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 notebookspreadsheet metaphor. The initial E 1 being the first sheet of data, the E 2 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 previous 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 filtration (abstract algebra) of the initial module (mathematics) E 0. A filtration

:A_{-2} = A_{-1} = A_0 supset A_1 supset A_2 supset ldots

of a module induces a short exact sequence

:0 o A hookrightarrow A o B o 0,

where B , the quotient j of A by its image under the inclusion i , has the differential induced by that of A . Set A 1 = H ( A ) and B 1 = H ( B ); a long exact sequence

:ldots o A_1 o A_1 o B_1 o A_1 o ldots

is then provided by the snake lemma. If we call the displayed maps i 1, j 1, and k 1, and let A 2 = i1A 1 and B 2 = ker j1k1 / im j1k1 , it can be shown (and perhaps will be in a later version of this article) that

:ldots o A_2 o A_2 o B_2 o A_2 o ldots

is another exact sequence. Setting i 2 = i 1, j 2 = [ j 1 i 1−1], and k 2 =

::: ,

and designating A 3 = iA 2, B 3 = ker j 2 k 2 / im j 2 k 2, we arrive at a third exact sequence. If we continue in this pattern, ( B n , j n k 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=