nLab derived module

Given a group homomorphism, $\varphi:H\to G$, the derived module of $\varphi$ is a $\mathbb{Z}[G]$-module, $D_\varphi$, together with a (universal) $\varphi$-derivation,

$$\partial_\varphi :H\to D_\varphi,$$

such that, given any $\varphi$-derivation, $f_1:H\to M$, for $M$ a $G$-module, there is a unique $G$-module morphism, $\overline{f_1}: D_\varphi\to M$ such that $f_1= \overline{f_1}\circ \partial_\varphi$.

Examples

• If $\varphi$ is the identity morphism on $G$ then the augmentation ideal, $I(G)$, together with

$$d_G:G\to I(G)$$

sending $g$ to $g-1$ is the derived module of $id_G$ aka the derived module of $G$.

References

For the original version of derived module, see

• R. H. Crowell, The derived module of a homomorphism, Advances in Math., 5, (1971), 210–238.

For applications in arithmetic topology

• Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)