Free Groups

 
Theorem 13.21. Let $G$ be a free group on a set $X$. Then $J$ is a free $ℤ G$-module with basis $\{x-1: x ∈ X\}$. Proof. Induct on the length of word. See Weibel Page 169, Proposition 6.2.6. Corollary 13.22. If $G$ is a free group on $X$, then $ℤ$ has free resolution \[ 0 → J → ℤ G → ℤ → 0 \] Therefore, $H_n(G, A)=H^n(G, A)=0$ for $n ≠ 0,1$, and $H_0(G, ℤ) ≅ H^0(G, ℤ) ≅ ℤ$, while \[ H_1(G, ℤ) ≅ \bigoplus_{x ∈ X} ℤ,   H^1(G, ℤ) ≅ \prod_{x ∈ X} ℤ \] Proof. $H_*(G ; A)$ is the homology of $0 →J ⊗_{ℤ G} A → A → 0$, and $H^*(G ; A)$ is the cohomology of $0 → A → \operatorname{Hom}_G(J, A) → 0$. For $A=ℤ$, the differentials are zero.