Group cohomology of finite homocyclic groups

We are interested in describing the homology groups and cohomology groups for a homocyclic group of the form $$(\mathbb{Z}/m\mathbb{Z})^n$$. This can be viewed as the additive group of a free $$m$$-dimensional module over the ring $$\mathbb{Z}/m\mathbb{Z}$$. It is isomorphic to the external direct product of $$n$$ copies of the finite cyclic group of order $$m$$.

Related discussions

 * Group cohomology of elementary abelian groups is very similar, and is the special situation where the underlying cyclic group is a group of prime order.

For this article, we will use the following notation:


 * $$m$$ is the exponent of the group, i.e., the order of the underlying cyclic group.
 * $$n$$ is the minimum size of generating set for the group, i.e., its rank as a free module over the ring $$\mathbb{Z}/m\mathbb{Z}$$.
 * $$q$$ is the degree in which we are looking at the homology or cohomology, i.e., we are looking at $$H_q$$ or $$H^q$$.

Over the integers
All the formulas obtained here are obtained by combining information about the group cohomology of finite cyclic groups with the Kunneth formula for group homology, as well as basic facts about computation of tensor products and $$\operatorname{Ext}$$ for finitely generated abelian groups. Details of the derivations are pending.

Rank as polynomial in homology degree for fixed rank of homocyclic group
The zeroth homology group is always $$\mathbb{Z}$$. All higher homology groups are themselves homocyclic groups with exponent $$m$$, i.e., they are free modules over the ring $$\mathbb{Z}/m\mathbb{Z}$$. For fixed $$n$$, there are two polynomials in $$q$$, both of degree $$n - 1$$ (one for even $$q$$, one for odd $$q$$) such that the rank of $$H_q((\mathbb{Z}/m\mathbb{Z})^n;\mathbb{Z})$$ is that polynomial in $$q$$. The polynomials are given below. In all cases, the following are true:


 * Both polynomials have degree $$n - 1$$.
 * For $$n > 1$$, the leading coefficient of both polynomials is $$1/(2(n - 1)!)$$ (is it? Just guesswork right now).
 * The polynomials differ only in their constant terms, with the polynomial for even $$q$$ having zero constant term.

Rank as polynomial in rank for fixed degree of homocyclic group
The zeroth homology group is always $$\mathbb{Z}$$. All higher homology groups are themselves homocyclic groups with exponent $$m$$, i.e., they are free modules over the ring $$\mathbb{Z}/m\mathbb{Z}$$. For fixed $$q$$, we can find a polynomial in $$n$$ such that the rank of $$H_q((\mathbb{Z}/m\mathbb{Z})^n;\mathbb{Z})$$ is that polynomial in $$q$$. The polynomials are given below:

Combined information on ranks
Below are the ranks of the homology groups $$H_q((\mathbb{Z}/m\mathbb{Z})^n;\mathbb{Z})$$ in terms of both $$n$$ and $$q$$.