The cancellation property for projective modules over integral group rings
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916477505896448 |
|---|---|
| author | Nicholson, John |
| author_facet | Nicholson, John |
| contents | We obtain a partial classification of the finite groups $G$ for which the integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which $P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for projective $\mathbb{Z} G$-modules $P$ and $Q$. In particular, we determine when projective cancellation holds for a finite group with no exceptional binary polyhedral quotients. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition. We then use a group theoretic argument to precisely determine the class of groups not covered by this result. The final classification is then obtained by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure of projective cancellation for certain groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The cancellation property for projective modules over integral group rings Nicholson, John Group Theory Algebraic Topology K-Theory and Homology Number Theory 20C05, 20C10, 19B28 We obtain a partial classification of the finite groups $G$ for which the integral group ring $\mathbb{Z} G$ has projective cancellation, i.e. for which $P \oplus \mathbb{Z} G \cong Q \oplus \mathbb{Z} G$ implies $P \cong Q$ for projective $\mathbb{Z} G$-modules $P$ and $Q$. In particular, we determine when projective cancellation holds for a finite group with no exceptional binary polyhedral quotients. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition. We then use a group theoretic argument to precisely determine the class of groups not covered by this result. The final classification is then obtained by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure of projective cancellation for certain groups. |
| title | The cancellation property for projective modules over integral group rings |
| topic | Group Theory Algebraic Topology K-Theory and Homology Number Theory 20C05, 20C10, 19B28 |
| url | https://arxiv.org/abs/2406.08692 |