Second cohomology group
This article gives a basic definition in the following area: group cohomology
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Definition
Let be a group acting on an abelian group , via an action . Equivalently, is a module over the (possibly non-commutative) unital group ring of over the ring of integers.
Definition in cohomology terms
The second cohomology group (also denoted ) is an abelian group defined in the following equivalent ways.
When is understood from context, the subscript may be omitted in the notation for the cohomology group, as well as the notation for the groups of 2-cocycles and 2-coboundaries.
No. | Shorthand | Detailed description of , the second cohomology group |
---|---|---|
1 | Explicit, using the bar resolution | , is defined as the quotient where is the group of 2-cocycles for the action and is the group of 2-coboundaries. |
1' | Explicit, using the normalized bar resolution | Same as definition (1), but we use normalized cocycles and normalized coboundaries instead of arbitrary cocycles and coboundaries. |
2 | Complex based on arbitrary resolution | Let be a projective resolution for as a -module with the trivial action. Let be the complex . The cohomology group is defined as the second cohomology group for this complex. |
3 | Complex based on arbitrary injective resolution (works if category of -modules has enough injectives!) | Let be an injective resolution for as a -module with the specified action . Let be the complex where has the structure of a trivial action -module. The cohomology group is defined as the second cohomology group for this complex. |
4 | As an functor | where is a trivial -module and has the module structure specified by . |
5 | As a right derived functor | , i.e., it is the second right derived functor of the invariants functor for (denoted ) evaluated at . The invariants functor sends a -module to its submodule of elements fixed by all elements of . |
All these definitions have natural analogues for the cohomology group for all . For more, see cohomology group.
Definition in terms of group extensions
There is an alternative definition of that is specific to 2 and has no easy analogue for other . This is in terms of group extensions.
can also be identified with the set of congruence classes of group extensions with normal subgroup isomorphic to and quotient group isomorphic to where the induced action of the quotient is the specified action . By a group extension, we mean a group having as a normal subgroup and as a quotient group. Two extensions and are congruent if there is an isomorphism of to which is identity on and induces the identity map on as a quotient.
Equivalence of the definitions
Further information: Equivalence of definitions of second cohomology group
Particular cases
A very special case where a lot of additional things of interest happen is that where the action is trivial. See second cohomology group for trivial group action. In particular, in the case of a trivial action, the second cohomology group coincides with the second cohomology group of the classifying space of with coefficients in .
Functoriality and automorphisms
Covariance in the second group
Suppose is a group and are abelian groups. Suppose , , and are group homomorphisms such that for all .
In other words, is a homomorphism from the -module with action to the -module with action .
Then, we get an induced homomorphism between the second cohomology groups:
This association is functorial, i.e., it gives a (covariant) functor from the category of -modules (i.e., abelian groups with acting on them) to the category of abelian groups.
Contravariance in the first group
Further information: restriction functor on cohomology, inflation functor on cohomology
Suppose are groups and is an abelian group. Suppose , , and are homomorphisms such that , i.e., the -action and -action on are compatible. Then, we get an induced homomorphism between the second cohomology groups:
Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor. The functor in general is termed the restriction functor.
Automorphism group actions
- Due to the covariance in the second argument, there is a natural action on of the group , i.e., the subgroup of the automorphism group of comprising those automorphisms that commute with the action of .
- Due to the contravariance in the first argument, there is a natural action on of the subgroup of that send every coset of the subgroup to itself (or equivalently, induce the identity map on . Here, is a normal subgroup of defined as the kernel of .
Examples
Here, we use the notation with a group acting on an abelian group via a group action .
Extreme examples
- If is a trivial group, then the second cohomology group is also a trivial group.
- If is a trivial group, then the second cohomology group is also a trivial group.