In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients.
These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of can be thought of as the singular cohomology of the circle S1. Likewise, the group cohomology of is the singular cohomology of infinite real projective space
A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
Motivation A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. We will write G multiplicatively and M additively.
Given such a G-module M, it is natural to consider the submodule of G-invariant elements:
Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology is to precisely measure this difference.
The group cohomology functors in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.
Definitions The collection of all -modules is a category (the morphisms are equivariant group homomorphisms, that is group homomorphisms with the property for all in and in ). Sending each module to the group of invariants yields a functor from the category of -modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by , "the -th cohomology group of with coefficients in ". Furthermore, the group may be identified with .
Cochain complexes The categorical definition using derived functors can be realized and computed using a standard cochain complex, which some authors also use as a definition.Page 62 of Milne 2008 or section VII.3 of Serre 1979 For let be the group of all functions from to M (here means ). This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms are defined by
One may check that so this defines a cochain complex whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex
Here the groups of n-cocycles, and n-coboundaries, respectively, are defined as
The functors Extn and formal definition of group cohomology Interpreting G-modules as modules over the group ring one can note that
i.e., the subgroup of G-invariant elements in M is identified with the group of homomorphisms from , which is treated as the trivial G-module (every element of G acts as the identity) to M.
Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism
These Ext groups can also be computed via a projective resolution of , the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective -resolution (e.g. a free -resolution) of the trivial -module :
e.g., one may always take the resolution of group rings, with morphisms
Recall that for -modules N and M, HomG(N, M) is an abelian group consisting of -homomorphisms from N to M. Since is a contravariant functor and reverses the arrows, applying to F termwise and dropping produces a cochain complex :
The cohomology groups of G with coefficients in the module M are defined as the cohomology of the above cochain complex:
This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of , that is, functions that obey
The coboundary operator is now naturally defined by, for example,
The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains , is given by reparameterizing so that
and so on. Thus
as in the preceding section.
Cohomology of the classifying space Another way to define group cohomology is to use topological cohomology theories (such as simplicial cohomology, singular cohomology or sheaf cohomology).
More precisely, the cohomology groups as defined above can also be expressed as
where is a classifying space for , that is a space whose fundamental group is and whose higher homotopy groups vanish (usually called an Eilenberg–MacLane space ). For example, classifying spaces for and are the circle , infinite real projective space and lens spaces, respectively. In general, can be constructed as the quotient , where is a contractible space on which acts freely. The space need not have an easily amenable geometric description, though this is the case for several important classes of groups, for instance Gromov-hyperbolic groups and arithmetic groups.
More generally, one can attach to any -module a local coefficient system on and the above isomorphism generalizes to an isomorphism, Chapter II.
Group homology Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m − m, g ∈ G, m ∈ M. Assign to M its coinvariant module, the quotient
this defines a right exact functor. Its left derived functors are by definition the group homology
The covariant functor which assigns MG to M is isomorphic to the functor which sends M to where is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors,
Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches: superscripts correspond to cohomology H* and invariants XG while subscripts correspond to homology H∗ and coinvariants XG := X/G.
Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolution F of the trivial -module as in the previous section. Apply the covariant functor to F termwise to get a chain complex :
Then Hn(G, M) are the homology groups of this chain complex, for n ≥ 0.
Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.
The group homology of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra .
Low-dimensional cohomology groups By definition, the zero-degree cohomology is , the module of invariants. The first and second degree cohomologies also have simple algebraic interpretations.
H 1 The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(g) = gm−m for some fixed m ∈ M. This follows from the definition of cochains above.
If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms G → M, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i.e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary.
On the other hand, consider the case of where denotes the non-trivial -structure on the additive group of integers, which sends a to -a for every ; and where we regard as the group . By considering all possible cases for the images of , it may be seen that crossed homomorphisms constitute all maps satisfying and for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy for some integer m: hence every crossed homomorphism sending -1 to an even integer is principal, and therefore:
with the group operation being pointwise addition: , noting that is the identity element.
H 2 If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H2(G,M) is in one-to-one correspondence with the set of central extensions of the group G by the additive group of M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H2(G,M) classifies the isomorphism classes of all group extensions of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with a G-module structure isomorphic to M.
In the example from the section on immediately above, as the only extension of by with the given nontrivial action is the infinite dihedral group, which is a split extension (semi-direct product) and thus trivial inside the group. This is in fact the significance in group-theoretical terms of the unique non-trivial element of .
An example of a second cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:
This classifies division algebras over k. See also .
Basic examples
Group cohomology of a finite cyclic group For the finite cyclic group of order with generator , the element in the associated group ring is a divisor of zero because its product with , given by
gives
This property can be used to construct the resolution of the trivial -module via the complex
giving the group cohomology computation for any -module . Note the augmentation map gives the trivial module its -structure by
This resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groups
showing that applying the functor to the complex above (with removed since this resolution is a quasi-isomorphism), gives the computation
for
For example, if , the trivial module, then , , and , hence
Explicit cocycles Cocycles for the group cohomology of a cyclic group can be given explicitly See Proposition 2.3. using the Bar resolution. We get a complete set of generators of -cocycles for odd as the maps
given by for odd, , a primitive -th root of unity, a field containing -th roots of unity, and