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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
======================================================================
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.

The most elementary example is a chain complex 'C�'.  An object 'C�'
in an abelian category of chain complexes comes with a differential
'd'.  Let 'r'0 = 0, and let 'E'0 be 'C�'.  This forces 'E'1 to be the
complex 'H'('C�'): At the 'i''th location this is the 'i''th homology
group of 'C�'.  The only natural differential on this new complex is
the zero map, so we let 'd'1 = 0.  This forces E_2 to equal E_1, and
again our only natural differential is the zero map.  Putting the zero
differential on all the rest of our sheets gives a spectral sequence
whose terms are:

* 'E'0 = 'C�'
* 'Er' = 'H'('C�') for all 'r' � 1.

The terms of this spectral sequence stabilize at the first sheet
because its only nontrivial differential was on the zeroth sheet.
Consequently, we can get no more information at later steps.  Usually,
to get useful information from later sheets, we need extra structure
on the E_r.

In the ungraded situation described above, 'r'0 is irrelevant, but in
practice most spectral sequences occur in the category of doubly
graded modules over a ring 'R' (or doubly graded sheaves of modules
over a sheaf of rings).  In this case, each sheet is a doubly graded
module, so it decomposes as a direct sum of terms with one term for
each possible bidegree.  The boundary map is defined as the direct sum
of boundary maps on each of the terms of the sheet.  Their degree
depends on 'r' and is fixed by convention.  For a homological spectral
sequence, the terms are written E^r_{p,q} and the differentials have
bidegree (� 'r','r' � 1).  For a cohomological spectral sequence, the
terms are written E^{p,q}_r and the differentials have bidegree ('r',
1 � 'r').  (These choices of bidegree occur naturally in practice; see
the example of a double complex below.)  Depending upon the spectral
sequence, the boundary map on the first sheet can have a degree which
corresponds to 'r' = 0, 'r' = 1, or 'r' = 2.  For example, for the
spectral sequence of a filtered complex, described below, 'r'0 = 0,
but for the Grothendieck spectral sequence, 'r'0 = 2.  Usually 'r'0 is
zero, one, or two.

A morphism of spectral sequences 'E' � 'E' ' is by definition a
collection of maps 'fr' : 'Er' � 'E'r' which are compatible with the
differentials and with the given isomorphisms between cohomology of
the 'r'-th step and the ('r' + 1)-st sheets of 'E' and 'E' ',
respectively.

Let 'E''r' be a spectral sequence, starting with say 'r' = 1. Then
there is a sequence of subobjects
:0 = B_0 \subset B_1 \subset B_{2} \subset \dots \subset B_r \subset
\dots \subset Z_r \subset \dots \subset Z_2 \subset Z_1 \subset Z_0 =
E_1
such that E_r \simeq Z_{r-1}/B_{r-1}; indeed, recursively we let Z_0 =
E_1, B_0 = 0 and let Z_r, B_r be so that  Z_r/B_{r-1}, B_r/B_{r-1} are
the kernel and the image of E_r \overset{d_r}\to E_r.

We then  let Z_{\infty} = \cap_r Z_r, B_{\infty} = \cup_r B_r and
:E_{\infty} = Z_{\infty}/B_{\infty};
it is called the limiting term. (Of course, such E_{\infty} need not
exist in the category, but this is usually a non-issue since for
example in the category of modules such limits exist or since in
practice a spectral sequence one works with tends to degenerate; there
are only finitely many inclusions in the sequence above.)


                           Visualization
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The E2 sheet of a cohomological spectral sequence
A doubly graded spectral sequence has a tremendous amount of data to
keep track of, but there is a common visualization technique which
makes the structure of the spectral sequence clearer.  We have three
indices, 'r', 'p', and 'q'.  For each 'r', imagine that we have a
sheet of graph paper.  On this sheet, we will take 'p' to be the
horizontal direction and 'q' to be the vertical direction.  At each
lattice point we have the object E_r^{p,q}.

It is very common for 'n' = 'p' + 'q' to be another natural index in
the spectral sequence. 'n' runs diagonally, northwest to southeast,
across each sheet.  In the homological case, the differentials have
bidegree (�'r', 'r' � 1), so they decrease 'n' by one.  In the
cohomological case, 'n' is increased by one.  When 'r' is zero, the
differential moves objects one space down or up.  This is similar to
the differential on a chain complex.  When 'r' is one, the
differential moves objects one space to the left or right.  When 'r'
is two, the differential moves objects just like a knight's move in
chess.  For higher 'r', the differential acts like a generalized
knight's move.


                        Worked-out examples
======================================================================
When learning spectral sequences for the first time, it is often
helpful to work with simple computational examples. For more formal
and complete discussions, see the sections below. For the examples in
this section, it suffices to use this definition: one says a spectral
sequence converges to 'H' with an increasing filtration 'F' if
E^{\infty}_{p, q} = F_p H_{p+q}/F_{p-1} H_{p+q}. The examples below
illustrate how one relates such filtrations with the 'E'2-term in the
forms of exact sequences; many exact sequences in applications (e.g.,
Gysin sequence) arise in this fashion.


2 columns and 2 rows
======================
Let E^r_{p, q} be a spectral sequence such that E^2_{p, q} = 0 for all
'p' other than 0, 1. The differentials on the second page have degree
(-2, 1) and therefore they are all zero; i.e., the spectral sequence
degenerates: E^{\infty} = E^2. Say, it converges to 'H' with a
filtration
:0 = F_{-1} H_n \subset F_0 H_n \subset \dots \subset F_n H_n = H_n
such that E^{\infty}_{p, q} = F_p H_{p+q}/F_{p-1} H_{p+q}. Then F_0
H_n = E^2_{0, n}, F_1 H_n / F_0 H_n = E^2_{1, n -1}, F_2 H_n / F_1 H_n
= 0, F_3 H_n / F_2 H_n = 0, etc. Thus, there is the exact sequence:
:0 \to E^2_{0, n} \to H_n \to E^2_{1, n - 1} \to 0.

Next, let E^r_{p, q} be a spectral sequence whose second page consists
only of two lines 'q' = 0, 1. This need not degenerate at the second
page but it still degenerates at the third page as the differentials
there have degree (-3, 2). Note E^3_{p, 0} = \operatorname{ker} (d:
E^2_{p, 0} \to E^2_{p - 2, 1}), as the denominator is zero. Similarly,
E^3_{p, 1} = \operatorname{coker}(d: E^2_{p+2, 0} \to E^2_{p, 1}).
Thus,
:0 \to E^{\infty}_{p, 0} \to E^2_{p, 0} \overset{d}\to E^2_{p-2, 1}
\to E^{\infty}_{p-2, 1} \to 0.
Now, say, the spectral sequence converges to 'H' with a filtration 'F'
as in the previous example. Since F_{p-2} H_{p} / F_{p-3} H_{p} =
E^{\infty}_{p-2, 2} = 0, F_{p-3} H_p / F_{p-4} H_p = 0, etc., we have:
0 \to E^{\infty}_{p - 1, 1} \to H_p \to E^{\infty}_{p, 0} \to 0.
Putting everything together, one gets:
:\cdots \to H_{p+1} \to E^2_{p + 1, 0} \overset{d}\to E^2_{p - 1, 1}
\to H_p \to E^2_{p, 0} \overset{d}\to E^2_{p - 2, 1} \to H_{p-1} \to
\dots.


Wang sequence
===============
The computation in the previous section generalizes in a
straightforward way. Consider a fibration over a sphere:
:F \overset{i}\to E \overset{p}\to S^n
with 'n' at least 2. There is the Serre spectral sequence:
:E^2_{p, q} = H_p(S^n; H_q(F)) \Rightarrow H_{p+q}(E);
that is to say, E^{\infty}_{p, q} = F_p H_{p+q}(E)/F_{p-1} H_{p+q}(E)
with some filtration 'F'.

Since H_p(S^n) is nonzero only when 'p' is zero or 'n' and equal to Z
in that case, we see E^2_{p, q} consists of only two lines 'p' = 0,
'n' and moreover E^2_{p, q} = H_q(F) for 'p' = 0, 'n', by the
universal coefficient theorem. Clearly, E^{\infty} = E^{n+1}, E^{n} =
E^2 and by computing E^{n+1} we see:
:0 \to E^{\infty}_{n, q-n} \to E^n_{n, q-n} \overset{d}\to E^n_{0,
q-1} \to E^{\infty}_{0, q-1} \to 0.
Now, writing H = H(E), since F_1 H_q/F_0 H_q = E^{\infty}_{1, q - 1} =
0, etc., we have: E^{\infty}_{n, q-n} = F_n H_q / F_0 H_q and thus,
since F_n H_q = H_q,
:0 \to E^{\infty}_{0, q} \to H_q \to E^{\infty}_{n, q - n} \to 0.
Putting all calculations together, one gets:
:\dots \to H_q(F) \overset{i_*}\to H_q(E) \to H_{q-n}(F)
\overset{d}\to H_{q-1}(F) \overset{i_*}\to H_{q-1}(E) \to H_{q-n
-1}(F) \to \dots

(The Gysin sequence is obtained in a similar way.)


Low-degree terms
==================
With an obvious notational change, the type of the computations in the
previous examples can also be carried out for cohomological spectral
sequence. Let E_r^{p, q} be a first-quadrant spectral sequence
converging to 'H' with the decreasing filtration
:0 = F^{n+1} H^n \subset F^n H^n \subset \dots \subset F^0 H^n = H^n
so that E_{\infty}^{p,q} = F^p H^{p+q}/F^{p+1} H^{p+q}.
Since E_2^{p, q} is zero if 'p' or 'q' is negative, we have:
:0 \to E^{0, 1}_{\infty} \to E^{0, 1}_2 \overset{d}\to E^{2, 0}_2 \to
E^{2, 0}_{\infty} \to 0.
Since E_{\infty}^{1, 0} = E_2^{1, 0} for the same reason and since F^2
H^1 = 0,
:0 \to E_2^{1, 0} \to H^1 \to E^{0, 1}_{\infty} \to 0.
Since F^3 H^2 = 0, E^{2, 0}_{\infty} \subset H^2. Stacking the
sequences together, we get the so-called five-term exact sequence:
:0 \to E^{1, 0}_2 \to H^1 \to E^{0, 1}_2 \overset{d}\to E^{2, 0}_2 \to
H^2.


                    Edge maps and transgressions
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Let E^r_{p, q} be a spectral sequence. If E^r_{p, q} = 0 for every 'q'
< 0, then it must be that: for 'r' � 2,
:E^{r+1}_{p, 0} = \operatorname{ker}(d: E^r_{p, 0} \to E^r_{p-r, r-1})
as the denominator is zero. Hence, there is a sequence of
monomorphisms:
:E^{r}_{p, 0} \to E^{r-1}_{p, 0} \to \dots \to E^3_{p, 0} \to E^2_{p,
0}.
They are called the edge maps. Similarly, if E^r_{p, q} = 0 for every
'p' < 0, then there is a sequence of epimorphisms (also called the
edge maps):
:E^2_{0, q} \to E^3_{0, q} \to \dots \to E^{r-1}_{0, q} \to E^r_{0,
q}.
The transgression is a partially-defined map (more precisely, a map
from a subobject to a quotient)
:\tau: E^2_{p, 0} \to E^2_{0, p - 1}
given as a composition E^2_{p, 0} \to E^p_{p, 0} \overset{d}\to
E^p_{0, p-1} \to E^2_{0, p - 1}, the first and last maps being the
inverses of the edge maps.

For a spectral sequence E_r^{p, q} of cohomological type, the
analogous statements hold. If E_r^{p, q} = 0 for every 'q' < 0,
then there is a sequence of epimorphisms
:E_{2}^{p, 0} \to E_{3}^{p, 0} \to \dots \to E_{r-1}^{p, 0} \to
E_r^{p, 0}.
And if E_r^{p, q} = 0 for every 'p' < 0, then there is a sequence
of monomorphisms:
:E_{r}^{0, q} \to E_{r-1}^{0, q} \to \dots \to E_{3}^{0, q} \to
E_2^{0, q}.
The transgression is a not necessarily well-defined map:
:\tau: E_2^{0, q-1} \to E_2^{q, 0}
induced by d: E_q^{0, q-1} \to E_q^{q, 0}.


                      Multiplicative structure
======================================================================
A cup product gives a ring structure to a cohomology group, turning it
into a cohomology ring. Thus, it is natural to consider a spectral
sequence with a ring structure as well. Let E^{p, q}_r be a spectral
sequence of cohomological type.  We say it has multiplicative
structure if (i) E_r are (doubly graded) differential graded algebras
and (ii) the multiplication on E_{r+1} is induced by that on E_r via
passage to cohomology.

A typical example is the cohomological Serre spectral sequence for a
fibration F \to E \to B, when the coefficient group is a "ring" 'R'.
It has the multiplicative structure induced by the cup products of
fibre and base on the E_{2}-page.  However, in general the limiting
term E_{\infty} is not isomorphic as a graded algebra to H('E'; 'R').
The multiplicative structure can be very useful for calculating
differentials on the sequence.


                Constructions of spectral sequences
======================================================================
Spectral sequences can be constructed by various ways. In algebraic
topology, an exact couple is perhaps the most common tool for the
construction. In algebraic geometry, spectral sequences are usually
constructed from filtrations of cochain complexes.


Exact couples
===============
right
The most powerful technique for the construction of spectral sequences
is William Massey's method of exact couples.  Exact couples are
particularly common in algebraic topology, where there are many
spectral sequences for which no other construction is known.  In fact,
all known spectral sequences can be constructed using exact couples.
Despite this they are unpopular in abstract algebra, where most
spectral sequences come from filtered complexes. To define exact
couples, we begin again with an abelian category.  As before, in
practice this is usually the category of doubly graded modules over a
ring.  An exact couple is a pair of objects 'A' and 'C', together with
three homomorphisms between these objects: 'f' : 'A' � 'A', 'g' : 'A'
� 'C' and 'h' : 'C' � 'A' subject to certain exactness conditions:

*Image 'f' = Kernel 'g'
*Image 'g' = Kernel 'h'
*Image 'h' = Kernel 'f'

We will abbreviate this data by ('A', 'C', 'f', 'g', 'h').  Exact
couples are usually depicted as triangles.  We will see that 'C'
corresponds to the 'E'0 term of the spectral sequence and that 'A' is
some auxiliary data.

To pass to the next sheet of the spectral sequence, we will form the
derived couple.  We set:

*'d' = 'g' o 'h'
*'A'' = 'f'('A')
*'C'' = Ker 'd' / Im 'd'
*'f'' = 'f'|'A'', the restriction of 'f' to 'A''
*'h'' : 'C'' � 'A'' is induced by 'h'.  It is straightforward to see
that 'h' induces such a map.
*'g'' : 'A'' � 'C'' is defined on elements as follows: For each 'a' in
'A'', write 'a' as 'f'('b') for some 'b' in 'A'.  'g''('a') is defined
to be the image of 'g'('b') in 'C''.  In general, 'g'' can be
constructed using one of the embedding theorems for abelian
categories.

From here it is straightforward to check that ('A'', 'C'', 'f'', 'g'',
'h'') is an exact couple.  'C'' corresponds to the 'E1' term of the
spectral sequence.  We can iterate this procedure to get exact couples
('A'('n'), 'C'('n'), 'f'('n'), 'g'('n'), 'h'('n')).  We let 'En' be
'C'('n') and 'dn' be 'g'('n') o 'h'('n').  This gives a spectral
sequence.

For a simple example, see the Bockstein spectral sequence.


The spectral sequence of a filtered complex
=============================================
A very common type of spectral sequence comes from a filtered cochain
complex.  This is a cochain complex 'C�' together with a set of
subcomplexes 'FpC�', where 'p' ranges across all integers.  (In
practice, 'p' is usually bounded on one side.)  We require that the
boundary map is compatible with the filtration; this means that
'd'('FpCn') � 'FpC''n'+1.  We assume that the filtration is
'descending', i.e., 'FpC�' � 'F''p'+1'C'�.  We will number the terms
of the cochain complex by 'n'.  Later, we will also assume that the
filtration is 'Hausdorff' or 'separated', that is, the intersection of
the set of all 'FpC�' is zero, and that the filtration is
'exhaustive', that is, the union of the set of all 'FpC�' is the
entire chain complex 'C'�.

The filtration is useful because it gives a measure of nearness to
zero: As 'p' increases, 'FpC�' gets closer and closer to zero.  We
will construct a spectral sequence from this filtration where
coboundaries and cocycles in later sheets get closer and closer to
coboundaries and cocycles in the original complex.  This spectral
sequence is doubly graded by the filtration degree 'p' and the
'complementary degree' . (The complementary degree is often a more
convenient index than the total degree 'n'. For example, this is true
of the spectral sequence of a double complex, explained below.)

We will construct this spectral sequence by hand.  'C'� has only a
single grading and a filtration, so we first construct a doubly graded
object from 'C'�.  To get the second grading, we will take the
associated graded object with respect to the filtration.  We will
write it in an unusual way which will be justified at the 'E'1 step:

:Z_{-1}^{p,q} = Z_0^{p,q} = F^p C^{p+q}
:B_0^{p,q} = 0
:E_0^{p,q} = \frac{Z_0^{p,q}}{B_0^{p,q} + Z_{-1}^{p+1,q-1}} =
\frac{F^p C^{p+q}}{F^{p+1} C^{p+q}}
:E_0 = \bigoplus_{p,q\in\mathbf{Z}} E_0^{p,q}

Since we assumed that the boundary map was compatible with the
filtration, 'E'0 is a doubly graded object and there is a natural
doubly graded boundary map 'd'0 on 'E'0.  To get 'E'1, we take the
homology of 'E'0.

:\bar{Z}_1^{p,q} = \ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1}
= \ker d_0^{p,q} : F^p C^{p+q}/F^{p+1} C^{p+q} \rightarrow F^p
C^{p+q+1}/F^{p+1} C^{p+q+1}
:\bar{B}_1^{p,q} = \mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow
E_0^{p,q} = \mbox{im } d_0^{p,q-1} : F^p C^{p+q-1}/F^{p+1} C^{p+q-1}
\rightarrow F^p C^{p+q}/F^{p+1} C^{p+q}
:E_1^{p,q} = \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}} = \frac{\ker
d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1}}{\mbox{im } d_0^{p,q-1}
: E_0^{p,q-1} \rightarrow E_0^{p,q}}
:E_1 = \bigoplus_{p,q\in\mathbf{Z}} E_1^{p,q} =
\bigoplus_{p,q\in\mathbf{Z}} \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}}

Notice that \bar{Z}_1^{p,q} and \bar{B}_1^{p,q} can be written as the
images in E_0^{p,q} of

:Z_1^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow
C^{p+q+1}/F^{p+1} C^{p+q+1}
:B_1^{p,q} = (\mbox{im } d_0^{p,q-1} : F^p C^{p+q-1} \rightarrow
C^{p+q}) \cap F^p C^{p+q}

and that we then have

:E_1^{p,q} = \frac{Z_1^{p,q}}{B_1^{p,q} + Z_0^{p+1,q-1}}.

Z_1^{p,q} is exactly the stuff which the differential pushes up one
level in the filtration, and B_1^{p,q} is exactly the image of the
stuff which the differential pushes up zero levels in the filtration.
This suggests that we should choose Z_r^{p,q} to be the stuff which
the differential pushes up 'r' levels in the filtration and B_r^{p,q}
to be image of the stuff which the differential pushes up 'r-1' levels
in the filtration.  In other words, the spectral sequence should
satisfy

:Z_r^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow
C^{p+q+1}/F^{p+r} C^{p+q+1}
:B_r^{p,q} = (\mbox{im } d_0^{p-r+1,q+r-2} : F^{p-r+1} C^{p+q-1}
\rightarrow C^{p+q}) \cap F^p C^{p+q}
:E_r^{p,q} = \frac{Z_r^{p,q}}{B_r^{p,q} + Z_{r-1}^{p+1,q-1}}

and we should have the relationship

:B_r^{p,q} = d_0^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).

For this to make sense, we must find a differential 'dr' on each 'Er'
and verify that it leads to homology isomorphic to 'E''r'+1.  The
differential

:d_r^{p,q} : E_r^{p,q} \rightarrow E_r^{p+r,q-r+1}

is defined by restricting the original differential 'd' defined on
C^{p+q} to the subobject Z_r^{p,q}.

It is straightforward to check that the homology of 'Er' with respect
to this differential is 'E''r'+1, so this gives a spectral sequence.
Unfortunately, the differential is not very explicit.  Determining
differentials or finding ways to work around them is one of the main
challenges to successfully applying a spectral sequence.


The spectral sequence of a double complex
===========================================
Another common spectral sequence is the spectral sequence of a double
complex.  A 'double complex' is a collection of objects 'Ci,j' for all
integers 'i' and 'j' together with two differentials, 'd I' and 'd
II'.  'd I' is assumed to decrease 'i', and 'd II' is assumed to
decrease 'j'.  Furthermore, we assume that the differentials
'anticommute', so that 'd I d II' + 'd II d I' = 0.  Our goal is to
compare the iterated homologies H^I_i(H^{II}_j(C_{\bullet,\bullet}))
and H^{II}_j(H^I_i(C_{\bullet,\bullet})).  We will do this by
filtering our double complex in two different ways.  Here are our
filtrations:

:(C_{i,j}^I)_p = \begin{cases}
0 & \text{if } i < p \\
C_{i,j} & \text{if } i \ge p \end{cases}
:(C_{i,j}^{II})_p = \begin{cases}
0 & \text{if } j < p \\
C_{i,j} & \text{if } j \ge p \end{cases}

To get a spectral sequence, we will reduce to the previous example.
We define the 'total complex' 'T'('C'�,�) to be the complex whose
'n''th term is \bigoplus_{i+j=n} C_{i,j} and whose differential is 'd
I' + 'd II'.  This is a complex because 'd I' and 'd II' are
anticommuting differentials.  The two filtrations on 'Ci,j' give two
filtrations on the total complex:

:T_n(C_{\bullet,\bullet})^I_p = \bigoplus_{i+j=n \atop i > p-1}
C_{i,j}
:T_n(C_{\bullet,\bullet})^{II}_p = \bigoplus_{i+j=n \atop j > p-1}
C_{i,j}

To show that these spectral sequences give information about the
iterated homologies, we will work out the 'E'0, 'E'1, and 'E'2 terms
of the 'I' filtration on 'T'('C'�,�).  The 'E'0 term is clear:

:{}^IE^0_{p,q} =
T_n(C_{\bullet,\bullet})^I_p / T_n(C_{\bullet,\bullet})^I_{p+1} =
\bigoplus_{i+j=n \atop i > p-1} C_{i,j} \Big/
\bigoplus_{i+j=n \atop i > p} C_{i,j} =
C_{p,q},
where .

To find the 'E'1 term, we need to determine 'd I' + 'd II' on 'E'0.
Notice that the differential must have degree �1 with respect to 'n',
so we get a map

:d^I_{p,q} + d^{II}_{p,q} :
T_n(C_{\bullet,\bullet})^I_p / T_n(C_{\bullet,\bullet})^I_{p+1} =
C_{p,q} \rightarrow
T_{n-1}(C_{\bullet,\bullet})^I_p /
T_{n-1}(C_{\bullet,\bullet})^I_{p+1} =
C_{p,q-1}

Consequently, the differential on 'E0' is the map 'C''p','q' �
'C''p','q'�1 induced by 'd I' + 'd II'.  But  'd I' has the wrong
degree to induce such a map, so 'd I' must be zero on 'E'0.  That
means the differential is exactly 'd II', so we get

:{}^IE^1_{p,q} = H^{II}_q(C_{p,\bullet}).

To find 'E2', we need to determine

:d^I_{p,q} + d^{II}_{p,q} :
H^{II}_q(C_{p,\bullet}) \rightarrow
H^{II}_q(C_{p+1,\bullet})

Because 'E'1 was exactly the homology with respect to 'd II', 'd II'
is zero on 'E'1.  Consequently, we get

:{}^IE^2_{p,q} = H^I_p(H^{II}_q(C_{\bullet,\bullet})).

Using the other filtration gives us a different spectral sequence with
a similar 'E'2 term:

:{}^{II}E^2_{p,q} = H^{II}_q(H^{I}_p(C_{\bullet,\bullet})).

What remains is to find a relationship between these two spectral
sequences.  It will turn out that as 'r' increases, the two sequences
will become similar enough to allow useful comparisons.


              Convergence, degeneration, and abutment
======================================================================
In the elementary example that we began with, the sheets of the
spectral sequence were constant once 'r' was at least 1.  In that
setup it makes sense to take the limit of the sequence of sheets:
Since nothing happens after the zeroth sheet, the limiting sheet 'E'�
is the same as 'E'1.

In more general situations, limiting sheets often exist and are always
interesting.  They are one of the most powerful aspects of spectral
sequences.  We say that a spectral sequence E_r^{p,q} converges to or
abuts to E_\infty^{p,q} if there is an 'r'('p', 'q') such that for all
'r' � 'r'('p', 'q'), the differentials d_r^{p-r,q+r-1} and d_r^{p,q}
are zero.  This forces E_r^{p,q} to be isomorphic to E_\infty^{p,q}
for large 'r'.  In symbols, we write:

:E_r^{p,q} \Rightarrow_p E_\infty^{p,q}

The 'p' indicates the filtration index.  It is very common to write
the E_2^{p,q} term on the left-hand side of the abutment, because this
is the most useful term of most spectral sequences.

In most spectral sequences, the E_\infty term is not naturally a
doubly graded object.  Instead, there are usually E_\infty^n terms
which come with a natural filtration F^\bullet E_\infty^n.  In these
cases, we set E_\infty^{p,q} = \mbox{gr}_p E_\infty^{p+q} =
F^pE_\infty^{p+q}/F^{p+1}E_\infty^{p+q}.  We define convergence in the
same way as before, but we write

:E_r^{p,q} \Rightarrow_p E_\infty^n

to mean that whenever 'p' + 'q' = 'n', E_r^{p,q} converges to
E_\infty^{p,q}.

The simplest situation in which we can determine convergence is when
the spectral sequences degenerates.  We say that the spectral
sequences degenerates at sheet r if, for any 's' � 'r', the
differential 'ds' is zero.  This implies that 'Er' �
 'E''r'+1 �

'E''r'+2 �
...  In particular, it implies that 'Er' is isomorphic to
'E�'.  This is what happened in our first, trivial example of an
unfiltered chain complex: The spectral sequence degenerated at the
first sheet.  In general, if a doubly graded spectral sequence is zero
outside of a horizontal or vertical strip, the spectral sequence will
degenerate, because later differentials will always go to or from an
object not in the strip.

The spectral sequence also converges if E_r^{p,q} vanishes for all 'p'
less than some 'p'0 and for all 'q' less than some 'q'0.  If 'p'0 and
'q'0 can be chosen to be zero, this is called a first-quadrant
spectral sequence.  This sequence converges because each object is a
fixed distance away from the edge of the non-zero region.
Consequently, for a fixed 'p' and 'q', the differential on later
sheets always maps E_r^{p,q} from or to the zero object; more
visually, the differential leaves the quadrant where the terms are
nonzero.  The spectral sequence need not degenerate, however, because
the differential maps might not all be zero at once.  Similarly, the
spectral sequence also converges if E_r^{p,q} vanishes for all 'p'
greater than some 'p'0 and for all 'q' greater than some 'q'0.

The five-term exact sequence of a spectral sequence relates certain
low-degree terms and 'E'� terms.

See also Boardman, [http://hopf.math.purdue.edu/Boardman/ccspseq.pdf
Conditionally Convergent Spectral Sequences].


The spectral sequence of a filtered complex, continued
========================================================
Notice that we have a chain of inclusions:

:Z_0^{p,q} \supe Z_1^{p,q} \supe Z_2^{p,q}\supe\cdots\supe B_2^{p,q}
\supe B_1^{p,q} \supe B_0^{p,q}

We can ask what happens if we define

:Z_\infty^{p,q} = \bigcap_{r=0}^\infty Z_r^{p,q},
:B_\infty^{p,q} = \bigcup_{r=0}^\infty B_r^{p,q},
:E_\infty^{p,q} =
\frac{Z_\infty^{p,q}}{B_\infty^{p,q}+Z_\infty^{p+1,q-1}}.

E_\infty^{p,q} is a natural candidate for the abutment of this
spectral sequence.  Convergence is not automatic, but happens in many
cases.  In particular, if the filtration is finite and consists of
exactly 'r' nontrivial steps, then the spectral sequence degenerates
after the 'r''th sheet.  Convergence also occurs if the complex and
the filtration are both bounded below or both bounded above.

To describe the abutment of our spectral sequence in more detail,
notice that we have the formulas:

:Z_\infty^{p,q} = \bigcap_{r=0}^\infty Z_r^{p,q} =
\bigcap_{r=0}^\infty \ker(F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+r}
C^{p+q+1})
:B_\infty^{p,q} = \bigcup_{r=0}^\infty B_r^{p,q} =
\bigcup_{r=0}^\infty (\mbox{im } d^{p,q-r} : F^{p-r} C^{p+q-1}
\rightarrow C^{p+q}) \cap F^p C^{p+q}

To see what this implies for Z_\infty^{p,q} recall that we assumed
that the filtration was separated.  This implies that as 'r'
increases, the kernels shrink, until we are left with Z_\infty^{p,q} =
\ker(F^p C^{p+q} \rightarrow C^{p+q+1}).  For B_\infty^{p,q}, recall
that we assumed that the filtration was exhaustive.  This implies that
as 'r' increases, the images grow until we reach B_\infty^{p,q} =
\text{im }(C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}.  We
conclude

:E_\infty^{p,q} = \mbox{gr}_p H^{p+q}(C^\bull),

that is, the abutment of the spectral sequence is the 'p''th graded
part of the 'p+q''th homology of 'C'.  If our spectral sequence
converges, then we conclude that:

:E_r^{p,q} \Rightarrow_p H^{p+q}(C^\bull)


Long exact sequences
======================
Using the spectral sequence of a filtered complex, we can derive the
existence of long exact sequences.  Choose a short exact sequence of
cochain complexes 0 � 'A�' � 'B�' � 'C�' � 0, and call the first map
'f�' : 'A�' � 'B�'.  We get natural maps of homology objects
'Hn'('A�') � 'Hn'('B�') � 'Hn'('C�'), and we know that this is exact
in the middle.  We will use the spectral sequence of a filtered
complex to find the connecting homomorphism and to prove that the
resulting sequence is exact.  To start, we filter 'B�':

:F^0 B^n = B^n
:F^1 B^n = A^n
:F^2 B^n = 0

This gives:

:E^{p,q}_0
= \frac{F^p B^{p+q}}{F^{p+1} B^{p+q}} = \begin{cases}
0 & \text{if } p < 0 \text{ or } p > 1 \\
C^q & \text{if } p = 0 \\
A^{q+1} & \text{if } p = 1 \end{cases}
:E^{p,q}_1
= \begin{cases}
0 & \text{if } p < 0 \text{ or } p > 1 \\
H^q(C^\bull) & \text{if } p = 0 \\
H^{q+1}(A^\bull) & \text{if } p = 1 \end{cases}

The differential has bidegree (1, 0), so 'd0,q' : 'Hq'('C�') �
'H''q'+1('A�').  These are the connecting homomorphisms from the snake
lemma, and together with the maps 'A�' � 'B�' � 'C�', they give a
sequence:

:\cdots\rightarrow H^q(B^\bull) \rightarrow H^q(C^\bull) \rightarrow
H^{q+1}(A^\bull) \rightarrow H^{q+1}(B^\bull) \rightarrow\cdots

It remains to show that this sequence is exact at the 'A' and 'C'
spots.  Notice that this spectral sequence degenerates at the 'E'2
term because the differentials have bidegree (2, �1).  Consequently,
the 'E'2 term is the same as the 'E'� term:

:E^{p,q}_2
\cong \text{gr}_p H^{p+q}(B^\bull)
= \begin{cases}
0 & \text{if } p < 0 \text{ or } p > 1 \\
H^q(B^\bull)/H^q(A^\bull) & \text{if } p = 0 \\
\text{im } H^{q+1}f^\bull : H^{q+1}(A^\bull) \rightarrow
H^{q+1}(B^\bull) &\text{if } p = 1 \end{cases}

But we also have a direct description of the 'E'2 term as the homology
of the 'E'1 term.  These two descriptions must be isomorphic:

: H^q(B^\bull)/H^q(A^\bull) \cong \ker d^1_{0,q} : H^q(C^\bull)
\rightarrow H^{q+1}(A^\bull)
: \text{im } H^{q+1}f^\bull : H^{q+1}(A^\bull) \rightarrow
H^{q+1}(B^\bull) \cong H^{q+1}(A^\bull) / (\mbox{im } d^1_{0,q} :
H^q(C^\bull) \rightarrow H^{q+1}(A^\bull))

The former gives exactness at the 'C' spot, and the latter gives
exactness at the 'A' spot.


The spectral sequence of a double complex, continued
======================================================
Using the abutment for a filtered complex, we find that:

:H^I_p(H^{II}_q(C_{\bull,\bull})) \Rightarrow_p
H^{p+q}(T(C_{\bull,\bull}))
:H^{II}_q(H^I_p(C_{\bull,\bull})) \Rightarrow_q
H^{p+q}(T(C_{\bull,\bull}))

In general, 'the two gradings on Hp+q(T(C�,�)) are distinct'.  Despite
this, it is still possible to gain useful information from these two
spectral sequences.


Commutativity of Tor
======================
Let 'R' be a ring, let 'M' be a right 'R'-module and 'N' a left
'R'-module.  Recall that the derived functors of the tensor product
are denoted Tor.  Tor is defined using a projective resolution of its
first argument.  However, it turns out that Tor'i'('M', 'N') =
Tor'i'('N', 'M').  While this can be verified without a spectral
sequence, it is very easy with spectral sequences.

Choose projective resolutions 'P�' and 'Q'� of 'M' and 'N',
respectively.  Consider these as complexes which vanish in negative
degree having differentials 'd' and 'e', respectively.  We can
construct a double complex whose terms are 'Ci,j' = 'Pi' � 'Qj' and
whose differentials are 'd' � 1 and (�1)'i'(1 � 'e').  (The factor of
�1 is so that the differentials anticommute.)  Since projective
modules are flat, taking the tensor product with a projective module
commutes with taking homology, so we get:

:H^I_p(H^{II}_q(P_\bull \otimes Q_\bull)) = H^I_p(P_\bull \otimes
H^{II}_q(Q_\bull))
:H^{II}_q(H^I_p(P_\bull \otimes Q_\bull)) = H^{II}_q(H^I_p(P_\bull)
\otimes Q_\bull)

Since the two complexes are resolutions, their homology vanishes
outside of degree zero.  In degree zero, we are left with

:H^I_p(P_\bull \otimes N) = \mbox{Tor}_p(M,N)
:H^{II}_q(M \otimes Q_\bull) = \mbox{Tor}_q(N,M)

In particular, the E^2_{p,q} terms vanish except along the lines 'q' =
0 (for the 'I' spectral sequence) and 'p' = 0 (for the 'II' spectral
sequence).  This implies that the spectral sequence degenerates at the
second sheet, so the 'E'� terms are isomorphic to the 'E'2 terms:

:\mbox{Tor}_p(M,N) \cong E^\infty_p = H_p(T(C_{\bull,\bull}))
:\mbox{Tor}_q(N,M) \cong E^\infty_q = H_q(T(C_{\bull,\bull}))

Finally, when 'p' and 'q' are equal, the two right-hand sides are
equal, and the commutativity of Tor follows.


                          Further examples
======================================================================
Some notable spectral sequences are:
*Adams spectral sequence in stable homotopy theory
*Adams�Novikov spectral sequence, a generalization to extraordinary
cohomology theories.
*Arnold's spectral sequence in singularity theory.
*Atiyah�Hirzebruch spectral sequence of an extraordinary cohomology
theory
*Bar spectral sequence for the homology of the classifying space of a
group.
*Barratt spectral sequence converging to the homotopy of the initial
space of a cofibration.
*Bloch�Lichtenbaum spectral sequence converging to the algebraic
K-theory of a field.
*Bockstein spectral sequence relating the homology with mod 'p'
coefficients and the homology reduced mod 'p'.
*Bousfield�Kan spectral sequence converging to the homotopy colimit of
a functor.
*Cartan�Leray spectral sequence converging to the homology of a
quotient space.
*�ech-to-derived functor spectral sequence from �ech cohomology to
sheaf cohomology.
*Change of rings spectral sequences for calculating Tor and Ext groups
of modules.
*Chromatic spectral sequence for calculating the initial terms of the
Adams�Novikov spectral sequence.
*Cobar spectral sequence
*Connes spectral sequences converging to the cyclic homology of an
algebra.
*EHP spectral sequence converging to stable homotopy groups of spheres
*Eilenberg-Moore spectral sequence for the singular cohomology of the
pullback of a fibration
*Federer spectral sequence converging to homotopy groups of a function
space.
*Frölicher spectral sequence starting from the Dolbeault cohomology
and converging to the algebraic de Rham cohomology of a variety.
*Gersten-Witt spectral sequence
*Green's spectral sequence for Koszul cohomology
*Grothendieck spectral sequence for composing derived functors
*Hodge�de Rham spectral sequence converging to the algebraic de Rham
cohomology of a variety.
*Homotopy fixed point spectral sequence
*Hurewicz spectral sequence for calculating the homology of a space
from its homotopy.
*Hyperhomology spectral sequence for calculating hyperhomology.
*Künneth spectral sequence for calculating the homology of a tensor
product of differential algebras.
*Leray spectral sequence converging to the cohomology of a sheaf.
*Local-to-global Ext spectral sequence
*Lyndon�Hochschild�Serre spectral sequence in group (co)homology
*May spectral sequence for calculating the Tor or Ext groups of an
algebra.
*Miller spectral sequence converging to the mod 'p' stable homology of
a space.
*Milnor spectral sequence is another name for the bar spectral
sequence.
*Moore spectral sequence is another name for the bar spectral
sequence.
*Motivic-to-'K'-theory spectral sequence
*Quillen spectral sequence for calculating the homotopy of a
simplicial group.
*Rothenberg�Steenrod spectral sequence is another name for the bar
spectral sequence.
*Serre spectral sequence of a fibration
*Spectral sequence of a differential filtered group: described in this
article.
*Spectral sequence of a double complex: described in this article.
*Spectral sequence of an exact couple: described in this article.
*Universal coefficient spectral sequence
*van Est spectral sequence converging to relative Lie algebra
cohomology.
*van Kampen spectral sequence for calculating the homotopy of a wedge
of spaces.


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