1-cocycle for a group action
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This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group
A 1-cocycle of in , also called a crossed homomorphism from to , is a function satisfying:
Here the group operation in is expressed multiplicatively, and the group operation in is expressed additively.
If we suppress and simply denote the action by , the condition is:
Definition as part of the general definition of cocycle
A 1-cocycle for a group action is a special case of a cocycle for a group action in the case . This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of to as -modules.
For a group acting on an abelian group , the set of 2-cocycles for the action of on forms a group under pointwise addition.
As a group of homomorphisms
For any group , we can construct a -module such that for any abelian group , the group of 1-cocycles can be identified with the group of -module maps from to .
Particular cases and variations
|Case or variation||Condition for a function to be a 1-coycle||Further information|
|Action is trivial||, so is a homomorphism of groups from to . In fact, it descends to a homomorphism from the abelianization to|
|is an abelian group, hence is written additively|
As stability automorphisms for an extension
Suppose we are given a group acting on an abelian group , and a group which is the semidirect product of by with respect to this action. Now consider the group of those automorphisms of which fix pointwise and which are identity on (viewed as a quotient).
The claim is that this group is isomorphic to the group of 1-cocycles. The isomorphism is as follows. The automorphism takes every coset of to itself, and acts as identity on . Hence it must simply translate each coset of by an element of . Consider the function by the map , in other words, this describes the amount by which each coset gets translated. The fact that is an automorphism forces to be a 1-cocycle.
Interestingly, it is also true that the 1-coboundaries correspond to those stability automorphisms that arise as inner automorphisms by elements of . Refer 1-coboundary for a group action.
The quotient of these groups (the first cohomology group) thus measures the group of stability automorphisms of upto inner automorphisms of .