Formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization
Suppose is an abelian group and is an abelian group. Then, the second cohomology group for trivial group action occurs in the following natural short exact sequence (this sequence arises as the case of the dual universal coefficients theorem for group cohomology):
Since is also an abelian group, we have and the Schur multiplier becomes the exterior square (see Schur multiplier of abelian group is its exterior square), and in this case, the short exact sequence becomes:
The sequence splits (though not necessarily naturally), so we get:
This can be interpreted as follows:
- The group describes the abelian groups arising as extensions with on top of .
- The group can be viewed as group of all alternating -bilinear maps from to .
Interpretation of left map of the sequence
The mapping sends each abelian group extension with on top of to the same extension now viewed as a cohomology class. The image of the mapping is the subgroup comprising the cohomology classes that contain symmetric 2-cocycles.
Interpretation of right map of the sequence
- In cocycle terms: The mapping from to sends a given cohomology class to the skew of any 2-cocycle representing it (the skew of is ).
- In group extension terms: The mapping sends an extension group to the commutator map of , which can be viewed as an alternating bilinear map .
- In isoclinism terms: The group can be viewed as the second cohomology group up to isoclinism for the trivial group action of on . The mapping that we have can thus be viewed as taking an element of to its equivalence class under isoclinism.
Comment on direct sum
As an internal direct sum, the summand for is . Because of the non-naturality of splitting, the other summand cannot usually be identified explicitly as a particular subgroup of .
- Dual universal coefficients theorem for group cohomology
- Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization
- Schur multiplier of abelian group is its exterior square
- Interpretation of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem