nLab derived module

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

φ:HD φ,\partial_\varphi :H\to D_\varphi,

such that, given any φ\varphi-derivation, f 1:HMf_1:H\to M, for MM a GG-module, there is a unique GG-module morphism, f 1¯:D φM\overline{f_1}: D_\varphi\to M such that f 1=f 1¯ φf_1= \overline{f_1}\circ \partial_\varphi.


  • If φ\varphi is the identity morphism on GG then the augmentation ideal, I(G)I(G), together with
d G:GI(G)d_G:G\to I(G)

sending gg to g1g-1 is the derived module of id Gid_G aka the derived module of GG.


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

Last revised on August 27, 2018 at 14:03:59. See the history of this page for a list of all contributions to it.