Spectral sequence

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

Filtrations

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

Related Topics:
Filtrations - Module

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

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of a module induces a short exact sequence

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:0 o A hookrightarrow A o B o 0,

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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 o A_1 o A_1 o B_1 o A_1 o ldots

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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 o A_2 o A_2 o B_2 o A_2 o ldots

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is another exact sequence. Setting i2 = i1, j2 = , and k2 =

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::: ? ,

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

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