Spectral sequence

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.

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

$A_{-2}=A_{-1}=A_{0}\supset A_{1}\supset A_{2}\supset \ldots$

of a module induces a short exact sequence

$0\to A\hookrightarrow A\to B\to 0,$

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

$\ldots \to A_{1}\to A_{1}\to B_{1}\to A_{1}\to \ldots$

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

$\ldots \to A_{2}\to A_{2}\to B_{2}\to A_{2}\to \ldots$

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