Top-degree rational cohomology in the symplectic group of a number ring
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911333784485888 |
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| author | Brück, Benjamin Himes, Zachary |
| author_facet | Brück, Benjamin Himes, Zachary |
| contents | Let $K$ be a number field with ring of integers $R = \mathcal{O}_K$. We show that if $R$ is not a principal ideal domain, then the symplectic group $\operatorname{Sp}_{2n}(R)$ has non-trivial rational cohomology in its virtual cohomological dimension. This demonstrates a sharp contrast to the situation where $R$ is Euclidean. To prove our result, we study the symplectic Steinberg module, i.e. the top-dimensional homology group of the spherical building associated to $\operatorname{Sp}_{2n}(K)$. We show that this module is not generated by integral apartment classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05456 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Top-degree rational cohomology in the symplectic group of a number ring Brück, Benjamin Himes, Zachary Number Theory Algebraic Topology Group Theory 11F75, 20E42, 55U10 Let $K$ be a number field with ring of integers $R = \mathcal{O}_K$. We show that if $R$ is not a principal ideal domain, then the symplectic group $\operatorname{Sp}_{2n}(R)$ has non-trivial rational cohomology in its virtual cohomological dimension. This demonstrates a sharp contrast to the situation where $R$ is Euclidean. To prove our result, we study the symplectic Steinberg module, i.e. the top-dimensional homology group of the spherical building associated to $\operatorname{Sp}_{2n}(K)$. We show that this module is not generated by integral apartment classes. |
| title | Top-degree rational cohomology in the symplectic group of a number ring |
| topic | Number Theory Algebraic Topology Group Theory 11F75, 20E42, 55U10 |
| url | https://arxiv.org/abs/2309.05456 |