AN EXPLICIT DIAGONAL RESOLUTION FOR A NON-ABELIAN METACYCLIC GROUP. Issue 2 (23rd March 2017) - Remez, J. J.
Abstract : We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups G(pq) however admit periodic resolutions. In the particular case of G(21) it is possible to achieve a fully diagonalized resolution . In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over [STIX]x1D6EC . Such a list aids the decomposition of the augmentation ideal (the first syzygy ) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.