Spectral sequence

In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of 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 spreadsheet metaphor. The initial E₁ being the first column of data, the E₂ column is derived from it by a definite process; and so on. The 'end result' of the calculation would be a final column, which then repeated itself eternally, because the differentials from then on were all 0.

The column talk here is fairly appropriate, in that in practice the E_i tend to carry some grading data. This enters the definition in a covert way, but has to be made explicit as one sees the process by which d_n gives rise to d_{n + 1}.

Even more adequate is a description in which the 'column' is actually a two-dimensional array; this is the standard situation in practice. Notationally the E_n therefore carry two further index numbers

E_n^p,q.

This is the natural 'data structure' for the occurrence of spectral sequences, for the reason that the d_n then show up as knight's moves in the chess sense, on the (p,q) 'board', in the E₂ term of a spectral sequence (which very often is the initial term given), and in higher terms as elongated knight's moves. That is, the spectral sequence as process is analogous to a spreadsheet with double-indexed columns, or if one wishes book 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.)

Filtrations

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

<math>A_{-2} = A_{-1} = A_0 \supset A_1 \supset A_2 \supset \ldots <math>

of a module induces a short exact sequence

<math>0 \to A \hookrightarrow A \to B \to 0,<math>

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

<math>\ldots \to A_1 \to A_1 \to B_1 \to A_1 \to \ldots<math>

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

<math>\ldots \to A_2 \to A_2 \to B_2 \to A_2 \to \ldots<math>

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.

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