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=                         Spectral sequence                          =
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                             Introduction
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In homological algebra and algebraic topology, a spectral sequence is
a means of computing homology groups by taking successive
approximations.  Spectral sequences are a generalization of exact
sequences, and since their introduction by , they have become
important computational tools, particularly in algebraic topology,
algebraic geometry and homological algebra.


                       Discovery and motivation
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Motivated by problems in algebraic topology, Jean Leray introduced the
notion of a sheaf and found himself faced with the problem of
computing sheaf cohomology.  To compute sheaf cohomology, Leray
introduced a computational technique now known as the Leray spectral
sequence.  This gave a relation between cohomology groups of a sheaf
and cohomology groups of the pushforward of the sheaf.  The relation
involved an infinite process.  Leray found that the cohomology groups
of the pushforward formed a natural chain complex, so that he could
take the cohomology of the cohomology.  This was still not the
cohomology of the original sheaf, but it was one step closer in a
sense.  The cohomology of the cohomology again formed a chain complex,
and its cohomology formed a chain complex, and so on.  The limit of
this infinite process was essentially the same as the cohomology
groups of the original sheaf.

It was soon realized that Leray's computational technique was an
example of a more general phenomenon.  Spectral sequences were found
in diverse situations, and they gave intricate relationships among
homology and cohomology groups coming from geometric situations such
as fibrations and from algebraic situations involving derived
functors.  While their theoretical importance has decreased since the
introduction of derived categories, they are still the most effective
computational tool available.  This is true even when many of the
terms of the spectral sequence are incalculable.

Unfortunately, because of the large amount of information carried in
spectral sequences, they are difficult to grasp.  This information is
usually contained in a rank three lattice of abelian groups or
modules.  The easiest cases to deal with are those in which the
spectral sequence eventually collapses, meaning that going out further
in the sequence produces no new information.  Even when this does not
happen, it is often possible to get useful information from a spectral
sequence by various tricks.


                          Formal definition
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Fix an abelian category, such as a category of modules over a ring.  A
spectral sequence is a choice of a nonnegative integer r_0 and a
collection of three sequences:
# For all integers r\ge r_0, an object E_r, called a 'sheet' (as in a
sheet of paper), or sometimes a 'page' or a 'term';
# Endomorphisms d_r\colon E_r \to E_r satisfying d_r \circ d_r = 0,
called 'boundary maps' or 'differentials';
# Isomorphisms of E_{r+1} with H(E_r), the homology of E_r with
respect to d_r.

Usually the isomorphisms between E_{r+1} and H(E_r) are suppressed,
and we write equalities instead.  Sometimes E_{r+1} is called the
derived object of E_r.