Group cohomology of free abelian groups

This article describes the homology and cohomology groups of the free abelian group $$\mathbb{Z}^d$$ with $$d$$ generators.

Classifying space and corresponding chain complex
The free abelian group $$\mathbb{Z}^d$$ has classifying space equal to the $$d$$-fold torus, i.e., the space $$S^1 \times S^1 \times \dots \times S^1$$. A chain complex that can be used to compute the homology for the classifying space and hence also the group is below:

Over the integers
$$H_q(\mathbb{Z}^d;\mathbb{Z}) = (\mathbb{Z})^{\binom{d}{q}}$$

Here, $$\binom{d}{q}$$ denotes the binomial coefficient: the number of subsets of size $$q$$ in a set of size $$d$$. In particular, the homology group is $$0$$ for $$q > d$$.

Note that all homology groups themselves are free abelian groups. The homology groups for small values of $$d$$ and $$q$$ are given below. Note that missing cells correspond to zero homology groups:

Over an abelian group
The homology groups with coefficients in an arbitrary abelian group are given below:

$$H_q(\mathbb{Z}^d;M) = M^{\binom{d}{q}}$$