Cohomology of groups
Historically, the earliest theory of a cohomology of algebras.
With every pair , where
is a group and
a left
-module (that is, a module over the integral group ring
), there is associated a sequence of Abelian groups
, called the cohomology groups of
with coefficients in
. The number
, which runs over the non-negative integers, is called the dimension of
. The cohomology groups of groups are important invariants containing information both on the group
and on the module
.
By definition, is
, where
is the submodule of
-invariant elements in
. The groups
,
, are defined as the values of the
-th derived functor of the functor
. Let
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be some projective resolution of the trivial -module
in the category of
-modules, that is, an exact sequence in which every
is a projective
-module. Then
is the
-th cohomology group of the complex
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where is induced by
, that is,
.
The homology groups of a group are defined using the dual construction, in which is replaced everywhere by
.
The set of functors ,
is a cohomological functor (see Homology functor; Cohomology functor) on the category of left
-modules.
A module of the form , where
is an Abelian group and
acts on
according to the formula
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is said to be co-induced. If is injective or co-induced, then
for
. Every module
is isomorphic to a submodule of a co-induced module
. The exact homology sequence for the sequence
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then defines isomorphisms ,
, and an exact sequence
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Therefore, the computation of the -dimensional cohomology group of
reduces to calculating the
-dimensional cohomology group of
. This device is called dimension shifting.
Dimension shifting enables one to give an axiomatic definition of cohomology groups, namely, they can be defined as a sequence of functors from the category of
-modules into the category of Abelian groups forming a cohomological functor and satisfying the condition that
,
, for every co-induced module
.
The groups can also be defined as equivalence classes of exact sequences of
-modules of the form
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with respect to a suitably defined equivalence relation (see [1], Chapt. 3, 4).
To compute the cohomology groups, the standard resolution of the trivial -module
is generally used, in which
and, for
,
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where the symbol over
means that the term
is omitted. The cochains in
are the functions
for which
. Changing variables according to the rules
,
,
, one can go over to inhomogeneous cochains
. The coboundary operation then acts as follows:
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For example, a one-dimensional cocycle is a function for which
for all
, and a coboundary is a function of the form
for some
. A one-dimensional cocycle is also said to be a crossed homomorphism and a one-dimensional coboundary a trivial crossed homomorphism. When
acts trivially on
, crossed homomorphisms are just ordinary homomorphisms and all the trivial crossed homomorphisms are 0, that is,
in this case.
The elements of can be interpreted as the
-conjugacy classes of sections
in the exact sequence
, where
is the semi-direct product of
and
. The elements of
can be interpreted as classes of extensions of
by
. Finally,
can be interpreted as obstructions to extensions of non-Abelian groups
with centre
by
(see [1]). For
, there are no analogous interpretations known (1978) for the groups
.
If is a subgroup of
, then restriction of cocycles from
to
defines functorial restriction homomorphisms for all
:
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For ,
is just the imbedding
. If
is some quotient group of
, then lifting cocycles from
to
induces the functorial inflation homomorphism
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Let be a homomorphism. Then every
-module
can be regarded as a
-module by setting
for
. Combining the mappings
and
gives mappings
. In this sense
is a contravariant functor of
. If
is a group of automorphisms of
, then
can be given the structure of a
-module. For example, if
is a normal subgroup of
, the groups
can be equipped with a natural
-module structure. This is possible thanks to the fact that inner automorphisms of
induce the identity mapping on the
. In particular, for a normal subgroup
in
,
.
Let be a subgroup of finite index in the group
. Using the norm map
, one can use dimension shifting to define the functorial co-restriction mappings for all
:
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These satisfy .
If is a normal subgroup of
then there exists the Lyndon spectral sequence with second term
converging to the cohomology
(see [1], Chapt. 11). In small dimensions it leads to the exact sequence
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where is the transgression mapping.
For a finite group , the norm map
induces the mapping
, where
and
is the ideal of
generated by the elements of the form
,
. The mapping
can be used to unite the exact cohomology and homology sequences. More exactly, one can define modified cohomology groups (also called Tate cohomology groups)
for all
. Here
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For these cohomology groups there exists an exact cohomology sequence that is infinite in both directions. A -module
is said to be cohomologically trivial if
for all
and all subgroups
. A module
is cohomologically trivial if and only if there is an
such that
and
for every subgroup
. Every module
is a submodule or a quotient module of a cohomologically trivial module, and this allows one to use dimension shifting both to raise and to lower the dimension. In particular, dimension shifting enables one to define
and
(but not
) for all integral
. For a finitely-generated
-module
the groups
are finite.
The groups are annihilated on multiplication by the order of
, and the mapping
, induced by restrictions, is a monomorphism, where now
is a Sylow
-subgroup (cf. Sylow subgroup) of
. A number of problems concerning the cohomology of finite groups can be reduced in this way to the consideration of the cohomology of
-groups. The cohomology of cyclic groups has period 2, that is,
for all
.
For arbitrary integers and
there is defined a mapping
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(called -product, cup-product), where the tensor product of
and
is viewed as a
-module. In the special case where
is a ring and the operations in
are automorphisms, the
-product turns
into a graded ring. The duality theorem for
-products asserts that, for every divisible Abelian group
and every
-module
, the
-product
![]() |
defines a group isomorphism between and
(see [2]). The
-product is also defined for infinite groups
provided that
.
Many problems lead to the necessity of considering the cohomology of a topological group acting continuously on a topological module
. In particular, if
is a profinite group (the case nearest to that of finite groups) and
is a discrete Abelian group that is a continuous
-module, one can consider the cohomology groups of
with coefficients in
, computed in terms of continuous cochains [5]. These groups can also be defined as the limit
with respect to the inflation mapping, where
runs over all open normal subgroups of
. This cohomology has all the usual properties of the cohomology of finite groups. If
is a pro-
-group, the dimension over
of the first and second cohomology groups with coefficients in
are interpreted as the minimum number of generators and relations (between these generators) of
, respectively.
See [6] for different variants of continuous cohomology, and also for certain other types of cohomology groups. See Non-Abelian cohomology for cohomology with a non-Abelian coefficient group.
References
[1] | S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009 |
[2] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305 |
[3] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) MR0215665 Zbl 0153.07403 |
[4] | J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303 |
[5] | H. Koch, "Galoissche Theorie der ![]() |
[6] | Itogi Nauk. Mat. Algebra. 1964 (1966) pp. 202–235 |
Comments
The norm map is defined as follows. Let
be a set of representatives of
in
. Then
in
. For a definition of the transgression relation in general spectral sequences cf. Spectral sequence; for the particular case of group cohomology, where this gives a relation, sometimes called connection, between
and
for all
, cf. also [a1], Chapt. 11, Par. 9.
References
[a1] | K.S. Brown, "Cohomology of groups" , Springer (1982) MR0672956 Zbl 0584.20036 |
Cohomology of groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohomology_of_groups&oldid=46392