In
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s, and since their introduction by , they have become important computational tools, particularly in algebraic topology,
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and homological algebra.
Discovery and motivation
Motivated by problems in algebraic topology, Jean Leray introduced the notion of a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper ser ...
and found himself faced with the problem of computing
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
. To compute sheaf cohomology, Leray introduced a computational technique now known as the
Leray spectral sequence In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
Definition
Let f:X\to Y be a cont ...
. 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
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
, 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
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
s and from algebraic situations involving
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s. While their theoretical importance has decreased since the introduction of
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
, 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 group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s or
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
. 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
Cohomological spectral sequence
Fix an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
, such as a category of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, and a nonnegative integer
. A cohomological spectral sequence is a sequence
of objects
and endomorphisms
, such that for every
#
,
#
, the
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
of
with respect to
.
Usually the isomorphisms are suppressed and we write
instead. An object
is called ''sheet'' (as in a sheet of
paper
Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, rags, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre evenly distributed ...
), or sometimes a ''page'' or a ''term''; an endomorphism
is called ''boundary map'' or ''differential''. Sometimes
is called the ''derived object'' of
.
Bigraded spectral sequence
In reality spectral sequences mostly occur in the category of doubly graded
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' (or doubly graded
sheaves of modules over a sheaf of rings), i.e. every sheet is a bigraded R-module
So in this case a cohomological spectral sequence is a sequence
of bigraded R-modules
and for every module the direct sum of endomorphisms
of bidegree
, such that for every
it holds that:
#
,
#
.
The notation used here is called ''complementary degree''. Some authors write
instead, where
is the ''total degree''. 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
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his ''Tôhoku'' paper, is a spectral sequence that computes the derived functors of the composition of two funct ...
, ''r''
0 = 2. Usually ''r''
0 is zero, one, or two. In the ungraded situation described above, ''r''
0 is irrelevant.
Homological spectral sequence
Mostly the objects we are talking about are
chain complexes
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
, that occur with descending (like above) or ascending order. In the latter case, by replacing
with
and
with
(bidegree
), one receives the definition of a homological spectral sequence analogously to the cohomological case.
Spectral sequence from a chain complex
The most elementary example in the ungraded situation is a
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
''C
•''. An object ''C
•'' in an abelian category of chain complexes naturally 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
to equal
, 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
•''
* ''E
r'' = ''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
.
Visualization
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''. An object
can be seen as the
checkered page of a book. On these sheets, we will take ''p'' to be the horizontal direction and ''q'' to be the vertical direction. At each lattice point we have the object
. Now turning to the next page means taking homology, that is the
page is a subquotient of the
page. The total degree ''n'' = ''p'' + ''q'' 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. The differentials change their direction with each turn with respect to r.
![Spectral Sequence Visualization](https://upload.wikimedia.org/wikipedia/commons/e/ed/Spectral_Sequence_Visualization.jpg)
The red arrows demonstrate the case of a first quadrant sequence (see example
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
), where only the objects of the first quadrant are non-zero. While turning pages, either the domain or the codomain of all the differentials become zero.
Properties
Categorical properties
The set of cohomological spectral sequences form a category: a morphism of spectral sequences
is by definition a collection of maps
which are compatible with the differentials, i.e.
, and with the given isomorphisms between the cohomology of the ''r''th step and the ''(r+1)''th sheets of ''E'' and ''E' '', respectively:
. In the bigraded case, they should also respect the graduation:
Multiplicative structure
A
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
gives a
ring structure to a cohomology group, turning it into a
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
. Thus, it is natural to consider a spectral sequence with a ring structure as well. Let
be a spectral sequence of cohomological type. We say it has multiplicative structure if (i)
are (doubly graded)
differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded alg ...
s and (ii) the multiplication on
is induced by that on
via passage to cohomology.
A typical example is the cohomological
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
for a fibration
, when the coefficient group is a ring ''R''. It has the multiplicative structure induced by the cup products of fibre and base on the
-page. However, in general the limiting term
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.
Spectral sequence of an exact couple
Another technique for constructing spectral sequences is
William Massey
William Ferguson Massey (26 March 1856 – 10 May 1925), commonly known as Bill Massey, was a politician who served as the 19th prime minister of New Zealand from May 1912 to May 1925. He was the founding leader of the Reform Party, New Zea ...
's method of exact couples. Exact couples are particularly common in algebraic topology. 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'', ''C''), together with three homomorphisms between these objects: ''f'' : ''A'' → ''A'', ''g'' : ''A'' → ''C'' and ''h'' : ''C'' → ''A'' subject to certain exactness conditions:
*
Image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
''f'' =
Kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
''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 ''E
1'' term of the spectral sequence. We can iterate this procedure to get exact couples (''A''
(''n''), ''C''
(''n''), ''f''
(''n''), ''g''
(''n''), ''h''
(''n'')).
In order to construct a spectral sequence, let ''E
n'' be ''C''
(''n'') and ''d
n'' be ''g''
(''n'') o ''h''
(''n'').
Spectral sequences constructed with this method
*
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
- used to compute (co)homology of a fibration
*
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, i ...
- used to compute (co)homology of extraordinary cohomology theories, such as
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
*
Bockstein spectral sequence In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after Meyer Bockstein.
Definition
Let ''C'' be a chain complex of ...
.
* Spectral sequences of filtered complexes
The spectral sequence of a filtered complex
A very common type of spectral sequence comes from a
filtered
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
cochain complex, as it naturally induces a bigraded object. Consider a cochain complex
together with a descending filtration,
. We require that the boundary map is compatible with the filtration, i.e.
, and that the filtration is ''exhaustive'', that is, the union of the set of all
is the entire chain complex
. Then there exists a spectral sequence with
and
. Later, we will also assume that the filtration is ''Hausdorff'' or ''separated'', that is, the intersection of the set of all
is zero.
The filtration is useful because it gives a measure of nearness to zero: As ''p'' increases,
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 .
Construction
has only a single grading and a filtration, so we first construct a doubly graded object for the first page of the spectral sequence. 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
step:
:
:
:
:
Since we assumed that the boundary map was compatible with the filtration,
is a doubly graded object and there is a natural doubly graded boundary map
on
. To get
, we take the homology of
.
:
:
:
:
Notice that
and
can be written as the images in
of
:
:
and that we then have
:
are exactly the elements which the differential pushes up one level in the filtration, and
are exactly the image of the elements which the differential pushes up zero levels in the filtration. This suggests that we should choose
to be the elements which the differential pushes up ''r'' levels in the filtration and
to be image of the elements which the differential pushes up ''r-1'' levels in the filtration. In other words, the spectral sequence should satisfy
:
:
:
and we should have the relationship
:
For this to make sense, we must find a differential
on each
and verify that it leads to homology isomorphic to
. The differential
:
is defined by restricting the original differential
defined on
to the subobject
. It is straightforward to check that the homology of
with respect to this differential is
, 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.
Spectral sequences constructed with this method
*
Hodge–de Rham spectral sequence
* Spectral sequence of a double complex
* Can be used to construct Mixed Hodge structures
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 ''C
i,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
and
. We will do this by filtering our double complex in two different ways. Here are our filtrations:
:
:
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
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 ''C
i,j'' give two filtrations on the total complex:
:
:
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:
:
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
:
Consequently, the differential on ''E
0'' 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
:
To find ''E
2'', we need to determine
:
Because ''E''
1 was exactly the homology with respect to ''d
II'', ''d
II'' is zero on ''E''
1. Consequently, we get
:
Using the other filtration gives us a different spectral sequence with a similar ''E''
2 term:
:
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
Interpretation as a filtration of cycles and boundaries
Let ''E''
''r'' be a spectral sequence, starting with say ''r'' = 1. Then there is a sequence of subobjects
:
such that
; indeed, recursively we let
and let
be so that
are the kernel and the image of
We then let
and
:
;
it is called the limiting term. (Of course, such
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.)
Terms of convergence
We say a spectral sequence converges weakly if there is a graded object
with a filtration
for every
, and for every
there exists an isomorphism
. It converges to
if the filtration
is Hausdorff, i.e.
. We write
:
to mean that whenever ''p'' + ''q'' = ''n'',
converges to
.
We say that a spectral sequence
abuts to
if for every
there is
such that for all
,
. Then
is the limiting term. The spectral sequence is regular or degenerates at
if the differentials
are zero for all
. If in particular there is
, such that the
sheet is concentrated on a single row or a single column, then we say it collapses. In symbols, we write:
:
The ''p'' indicates the filtration index. It is very common to write the
term on the left-hand side of the abutment, because this is the most useful term of most spectral sequences. The spectral sequence of an unfiltered chain complex degenerates at the first sheet (see first example): since nothing happens after the zeroth sheet, the limiting sheet
is the same as
.
The
five-term exact sequence
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence.
More precisely, let
:E_2^ \Rightarrow H^n(A)
be a first quadrant spectral sequence, meaning tha ...
of a spectral sequence relates certain low-degree terms and ''E''
∞ terms.
Examples of degeneration
The spectral sequence of a filtered complex, continued
Notice that we have a chain of inclusions:
:
We can ask what happens if we define
:
:
:
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:
:
:
To see what this implies for
recall that we assumed that the filtration was separated. This implies that as ''r'' increases, the kernels shrink, until we are left with
. For
, recall that we assumed that the filtration was exhaustive. This implies that as ''r'' increases, the images grow until we reach
. We conclude
:
,
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:
:
Long exact sequences
Using the spectral sequence of a filtered complex, we can derive the existence of
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s. 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 ''H
n''(''A
•'') → ''H
n''(''B
•'') → ''H
n''(''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
•'':
:
:
:
This gives:
:
:
The differential has bidegree (1, 0), so ''d
0,q'' : ''H
q''(''C
•'') → ''H''
''q''+1(''A
•''). These are the connecting homomorphisms from the
snake lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...
, and together with the maps ''A
•'' → ''B
•'' → ''C
•'', they give a sequence:
:
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:
:
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:
:
:
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:
:
:
In general, ''the two gradings on H
p+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, TOR or ToR may refer to:
Places
* Tor, Pallars, a village in Spain
* Tor, former name of Sloviansk, Ukraine, a city
* Mount Tor, Tasmania, Australia, an extinct volcano
* Tor Bay, Devon, England
* Tor River, Western New Guinea, Indonesia
Sc ...
. Tor is defined using a projective resolution of its first argument. However, it turns out that
. While this can be verified without a spectral sequence, it is very easy with spectral sequences.
Choose projective resolutions
and
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
and whose differentials are
and
. (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:
:
:
Since the two complexes are resolutions, their homology vanishes outside of degree zero. In degree zero, we are left with
:
:
In particular, the
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:
:
:
Finally, when ''p'' and ''q'' are equal, the two right-hand sides are equal, and the commutativity of Tor follows.
Worked-out examples
First-quadrant sheet
Consider a spectral sequence where
vanishes for all
less than some
and for all
less than some
. If
and
can be chosen to be zero, this is called a first-quadrant spectral sequence.
The sequence abuts because
holds for all
if
and
. To see this, note that either the domain or the codomain of the differential is zero for the considered cases. In visual terms, the sheets stabilize in a growing rectangle (see picture above). 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
vanishes for all
greater than some
and for all
greater than some
.
2 non-zero adjacent columns
Let
be a homological spectral sequence such that
for all ''p'' other than 0, 1. Visually, this is the spectral sequence with
-page
:
The differentials on the second page have degree (-2, 1), so they are of the form
:
These maps are all zero since they are
:
,
hence the spectral sequence degenerates:
. Say, it converges to
with a filtration
:
such that
. Then
,
,
,
, etc. Thus, there is the exact sequence:
:
.
Next, let
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
, as the denominator is zero. Similarly,
. Thus,
:
.
Now, say, the spectral sequence converges to ''H'' with a filtration ''F'' as in the previous example. Since
,
, etc., we have:
. Putting everything together, one gets:
:
Wang sequence
The computation in the previous section generalizes in a straightforward way. Consider a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
over a sphere:
:
with ''n'' at least 2. There is the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
:
:
;
that is to say,
with some filtration
.
Since
is nonzero only when ''p'' is zero or ''n'' and equal to Z in that case, we see
consists of only two lines
, hence the
-page is given by
:
Moreover, since
:
for
by the
universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
, the
page looks like
:
Since the only non-zero differentials are on the
-page, given by
:
which is
:
the spectral sequence converges on
. By computing
we get an exact sequence
:
and written out using the homology groups, this is
:
To establish what the two
-terms are, write
, and since
, etc., we have:
and thus, since
,
:
This is the exact sequence
:
Putting all calculations together, one gets:
:
(The
Gysin sequence In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool f ...
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
be a first-quadrant spectral sequence converging to ''H'' with the decreasing filtration
:
so that
Since
is zero if ''p'' or ''q'' is negative, we have:
:
Since
for the same reason and since
:
.
Since
,
. Stacking the sequences together, we get the so-called
five-term exact sequence
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence.
More precisely, let
:E_2^ \Rightarrow H^n(A)
be a first quadrant spectral sequence, meaning tha ...
:
:
Edge maps and transgressions
Homological spectral sequences
Let
be a spectral sequence. If
for every ''q'' < 0, then it must be that: for ''r'' ≥ 2,
:
as the denominator is zero. Hence, there is a sequence of monomorphisms:
:
.
They are called the edge maps. Similarly, if
for every ''p'' < 0, then there is a sequence of epimorphisms (also called the edge maps):
:
.
The
transgression is a partially-defined map (more precisely, a
map from a subobject to a quotient)
:
given as a composition
, the first and last maps being the inverses of the edge maps.
Cohomological spectral sequences
For a spectral sequence
of cohomological type, the analogous statements hold. If
for every ''q'' < 0, then there is a sequence of epimorphisms
:
.
And if
for every ''p'' < 0, then there is a sequence of monomorphisms:
:
.
The transgression is a not necessarily well-defined map:
:
induced by
.
Application
Determining these maps are fundamental for computing many differentials in the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
. For instance the transgression map determines the differential
:
for the homological spectral spectral sequence, hence on the Serre spectral sequence for a fibration
gives the map
:
.
Further examples
Some notable spectral sequences are:
Topology and geometry
*
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, i ...
of an
extraordinary cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
*
Bar spectral sequence
Bar or BAR may refer to:
Food and drink
* Bar (establishment), selling alcoholic beverages
* Candy bar
* Chocolate bar
Science and technology
* Bar (river morphology), a deposit of sediment
* Bar (tropical cyclone), a layer of cloud
* Bar (un ...
for the homology of the classifying space of a group.
*
Bockstein spectral sequence In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after Meyer Bockstein.
Definition
Let ''C'' be a chain complex of ...
relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''.
*
Cartan–Leray spectral sequence converging to the homology of a quotient space.
*
Eilenberg–Moore spectral sequence In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the ho ...
for the
singular cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
of the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
of a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
*
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
of a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
Homotopy theory
*
Adams spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
in
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
*
Adams–Novikov spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
, a generalization to
extraordinary cohomology theories
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
.
*
Barratt spectral sequence converging to the homotopy of the initial space of a cofibration.
*
Bousfield–Kan spectral sequence converging to the homotopy colimit of a functor.
*
Chromatic spectral sequence In mathematics, the chromatic spectral sequence is a spectral sequence, introduced by , used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy ...
for calculating the initial terms of the
Adams–Novikov spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
.
*
Cobar spectral sequence
Cobar is a town in central western New South Wales, Australia whose economy is based mainly upon base metals and gold mining. The town is by road northwest of the state capital, Sydney. It is at the crossroads of the Kidman Way and Barrier Hi ...
*
EHP spectral sequence In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime ''p''. It is described in more detail in and . It is related to the EHP long exact sequence o ...
converging to
stable homotopy groups of spheres
*
Federer spectral sequence
Roger Federer (; born 8 August 1981) is a Swiss former professional tennis player. He was ranked world No. 1 by the Association of Tennis Professionals (ATP) for 310 weeks, including a record 237 consecutive weeks, and finished as the year-e ...
converging to homotopy groups of a function space.
*
Homotopy fixed point spectral sequence
*
Hurewicz spectral sequence for calculating the homology of a space from its homotopy.
*
Miller spectral sequence
A miller is a person who operates a mill, a machine to grind a grain (for example corn or wheat) to make flour. Milling is among the oldest of human occupations. "Miller", "Milne" and other variants are common surnames, as are their equivalents ...
converging to the mod ''p'' stable homology of a space.
*
Milnor spectral sequence is another name for the
bar spectral sequence
Bar or BAR may refer to:
Food and drink
* Bar (establishment), selling alcoholic beverages
* Candy bar
* Chocolate bar
Science and technology
* Bar (river morphology), a deposit of sediment
* Bar (tropical cyclone), a layer of cloud
* Bar (un ...
.
*
Moore spectral sequence
Moore may refer to:
People
* Moore (surname)
** List of people with surname Moore
* Moore Crosthwaite (1907–1989), a British diplomat and ambassador
* Moore Disney (1765–1846), a senior officer in the British Army
* Moore Powell (died c. 1 ...
is another name for the bar spectral sequence.
*
Quillen spectral sequence In the area of mathematics known as K-theory, the Quillen spectral sequence, also called the Brown–Gersten–Quillen or BGQ spectral sequence (named after Kenneth Brown,
Stephen Gersten, and Daniel Quillen), is a spectral sequence conve ...
for calculating the homotopy of a simplicial group.
*
Rothenberg–Steenrod spectral sequence is another name for the bar spectral sequence.
*
van Kampen spectral sequence for calculating the homotopy of a wedge of spaces.
Algebra
*
Čech-to-derived functor spectral sequence from
Čech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
Motivation
Let ''X'' be a topolo ...
to
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
.
*
Change of rings spectral sequences for calculating Tor and Ext groups of modules.
*
Connes spectral sequence Connes is a surname. Notable people with the surname include:
*Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncomm ...
s converging to the cyclic homology of an algebra.
*
Gersten–Witt spectral sequence
*
Green's spectral sequence for
Koszul cohomology In mathematics, the Koszul cohomology groups K_(X,L) are groups associated to a projective variety ''X'' with a line bundle ''L''. They were introduced by , and named after Jean-Louis Koszul as they are closely related to the Koszul complex.
su ...
*
Grothendieck spectral sequence
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his ''Tôhoku'' paper, is a spectral sequence that computes the derived functors of the composition of two funct ...
for composing
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s
*
Hyperhomology spectral sequence for calculating hyperhomology.
*
Künneth spectral sequence for calculating the homology of a tensor product of differential algebras.
*
Leray spectral sequence In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
Definition
Let f:X\to Y be a cont ...
converging to the cohomology of a sheaf.
*
Local-to-global Ext spectral sequence
*
Lyndon–Hochschild–Serre spectral sequence
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup ''N' ...
in
group (co)homology
*
May spectral sequence for calculating the Tor or Ext groups of an algebra.
*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
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a ...
*
van Est spectral sequence converging to relative Lie algebra cohomology.
Complex and algebraic geometry
*
Arnold's spectral sequence In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to canonical form near critical points. It was ...
in
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
.
*
Bloch–Lichtenbaum spectral sequence converging to the algebraic K-theory of a field.
*
Frölicher spectral sequence In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler ma ...
starting from the
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault ...
and converging to the
algebraic de Rham cohomology
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
of a variety.
*
Hodge–de Rham spectral sequence converging to the
algebraic de Rham cohomology
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
of a variety.
*
Motivic-to-''K''-theory spectral sequence
Notes
References
Introductory
*
*
References
*
*
*
*
*
*
*
*
*
Further reading
*
External links
*
* {{cite web, url=https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.15/share/doc/Macaulay2/SpectralSequences/html/, title=SpectralSequences — a package for working with filtered complexes and spectral sequences, publisher=
Macaulay2
Macaulay2 is a free computer algebra system created by Daniel Grayson (from the University of Illinois at Urbana–Champaign) and Michael Stillman (from Cornell University) for computation in commutative algebra and algebraic geometry.
Overvi ...
*