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

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

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

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

then defines isomorphisms , , and an exact sequence

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

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 ,

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:

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 :

For , is just the imbedding . If is some quotient group of , then lifting cocycles from to induces the functorial inflation homomorphism

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 :

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

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

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

(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 -Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970) [6] Itogi Nauk. Mat. Algebra. 1964 (1966) pp. 202–235