First cohomology group
This article gives a basic definition in the following area: group cohomology
View other basic definitions in group cohomology |View terms related to group cohomology |View facts related to group cohomology
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 first cohomology group 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 1-cocycles and 1-coboundaries.
No. | Shorthand | Detailed description of ![]() |
---|---|---|
1 | Explicit, using the bar resolution | ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() ![]() |
3 | As an ![]() |
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4 | As a right derived functor | ![]() ![]() ![]() ![]() ![]() ![]() |
All these definitions have natural analogues for the cohomology group
for all
. For more, see cohomology group.
Definition in terms of stability automorphisms of extensions
Suppose is a group that has an abelian normal subgroup identified with
and such that the quotient group
is identified with
(we abuse notation here and treat
and
as the actual subgroup and quotient group respectively). Further, assume that the induced action of the quotient on the subgroup is the same as the group action
.
Then, is a group quotient:
(Group of stability automorphisms of the chain )/(Subgroup comprising the stability automorphisms of the chain that are induced from conjugation by elements of
)
More precisely, the group of all stability automorphisms can be naturally identified with the 1-cocycle group and the group of stability automorphisms arising via conjugation by an element of
can be naturally identified with the 1-coboundary group
.
Particular cases
If the action of on
is the trivial group action (i.e., every element of
fixes every element of
), then the first cohomology group
can be naturally identified with the set
endowed with a group structure under pointwise addition. Specifically, the group of 1-cocycles is identified with the group of homomorphisms under pointwise addition and the group of 1-coboundaries is the trivial group.Further information: First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms