Formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization

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Suppose G is an abelian group and A is an abelian group. Then, the second cohomology group for trivial group action H^2(G;A) occurs in the following natural short exact sequence (this sequence arises as the n = 2 case of the dual universal coefficients theorem for group cohomology):

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0

Since G is also an abelian group, we have G^{\operatorname{ab}} = G and the Schur multiplier H_2(G;\mathbb{Z}) becomes the exterior square \bigwedge^2G (see Schur multiplier of abelian group is its exterior square), and in this case, the short exact sequence becomes:

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

The sequence splits (though not necessarily naturally), so we get:

H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \oplus \operatorname{Hom}(\bigwedge^2G,A)

This can be interpreted as follows:

  • The group \operatorname{Ext}^1_{\mathbb{Z}}(G,A) describes the abelian groups arising as extensions with G on top of A.
  • The group \operatorname{Hom}(\bigwedge^2G,A) can be viewed as group of all alternating \mathbb{Z}-bilinear maps from G to A.

Interpretation of left map of the sequence

The mapping sends each abelian group extension with G on top of A to the same extension now viewed as a cohomology class. The image of the mapping is the subgroup H^2_{sym}(G,A) comprising the cohomology classes that contain symmetric 2-cocycles.

Interpretation of right map of the sequence

  • In cocycle terms: The mapping from H^2(G;A) to \operatorname{Hom}(\bigwedge^2G,A) sends a given cohomology class to the skew of any 2-cocycle representing it (the skew of f:G \times G \to A is (x,y) \mapsto f(x,y) - f(y,x)).
  • In group extension terms: The mapping sends an extension group E to the commutator map of E, which can be viewed as an alternating bilinear map G \times G \to A.
  • In isoclinism terms: The group \operatorname{Hom}(\bigwedge^2G,A) can be viewed as the second cohomology group up to isoclinism for the trivial group action of G on A. The mapping that we have can thus be viewed as taking an element of H^2(G;A) to its equivalence class under isoclinism.

Comment on direct sum

As an internal direct sum, the summand for \operatorname{Ext}^1_{\mathbb{Z}}(G,A) is H^2_{sym}(G;A). Because of the non-naturality of splitting, the other summand cannot usually be identified explicitly as a particular subgroup of H^2(G;A).

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