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

Formula
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 $$H^2(G;A)$$.

Related facts

 * 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