The passage from the integral to the rational group ring in algebraic $K$-theory
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915558823297024 |
|---|---|
| author | Lehner, Georg |
| author_facet | Lehner, Georg |
| contents | An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to the rational group ring is trivial for any group $G$. We will show that this is false, with a counterexample given by the group $QD_{32} *_{Q_{16}} QD_{32}$. We will also show how to compute the image of the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representation theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_01413 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The passage from the integral to the rational group ring in algebraic $K$-theory Lehner, Georg K-Theory and Homology 19A31 (Primary), 20C07, 19D35, 55P42 (Secondary) An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to the rational group ring is trivial for any group $G$. We will show that this is false, with a counterexample given by the group $QD_{32} *_{Q_{16}} QD_{32}$. We will also show how to compute the image of the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representation theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture. |
| title | The passage from the integral to the rational group ring in algebraic $K$-theory |
| topic | K-Theory and Homology 19A31 (Primary), 20C07, 19D35, 55P42 (Secondary) |
| url | https://arxiv.org/abs/2110.01413 |