Group cohomology of free groups

We discuss here the group homology and cohomology for the free group $$F_n$$ on a freely generating set of size $$n$$.

Over the integers
$$H_q(F_n;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & q = 0 \\ \mathbb{Z}^n, & q = 1 \\ 0, & q > 1 \\\end{array}\right.$$

Over an abelian group
The homology groups over an abelian group $$M$$ for the trivial group action are as follows:

$$H_q(F_n;M) = \left\lbrace \begin{array}{rl} M, & q = 0 \\ M^n, & q = 1 \\ 0, & q > 1 \\\end{array}\right.$$

Over the integers
$$H^q(F_n;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & q = 0 \\ \mathbb{Z}^n, & q = 1 \\ 0, & q > 1 \\\end{array}\right.$$

Over an abelian group
The cohomology groups over an abelian group $$M$$ for the trivial group action are as follows:

$$H^q(F_n;M) = \left\lbrace \begin{array}{rl} M, & q = 0 \\ M^n, & q = 1 \\ 0, & q > 1 \\\end{array}\right.$$