The cancellation property for projective modules over integral group rings
Abstract
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 $Q$. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition and use it to establish the cancellation property for a large class of finite groups. 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 the cancellation property for certain groups.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.08692
- arXiv:
- arXiv:2406.08692
- Bibcode:
- 2024arXiv240608692N
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Algebraic Topology;
- Mathematics - K-Theory and Homology;
- Mathematics - Number Theory;
- 20C05;
- 20C10;
- 19B28
- E-Print:
- 39 pages