H$^1 (G,\mathbb{Z}) \cong$ Der$(G, \mathbb{Z})$/PDer $(G, \mathbb{Z})\cong$ Hom$(\mathbb{Z}, \mathbb{Z}) \neq {0}$.

Question

$G \cong \mathbb{Z}$ is an infinite cyclic group.

Then H$^1 (G,\mathbb{Z}) \cong$ Der$(G, \mathbb{Z})$/PDer $(G, \mathbb{Z})\cong$ Hom$(\mathbb{Z}, \mathbb{Z}) \neq {0}$.

Is there something wrong with the last isomorphism. Should it be
H$^0 (G,\mathbb{Z})\cong$ Hom$(\mathbb{Z}, \mathbb{Z})$?

This is from P592 of Rotman's Homological algebra book. Any help would be appreciated!

Answer 1

In general $H^1(G,A)$ is the group of crossed homomorphisms $G\to A$ factored out by the principal crossed homomorphisms.

If $G$ acts trivially on $A$ then crossed homomorphisms are just homomorphisms and the principal crossed homomorphisms are all zero, so $H^1(G,A)=\text{Hom}(G,A)$.