Locally isotropic Steinberg groups I. Centrality of the $\mathrm K_2$-functor
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915984798908416 |
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| author | Voronetsky, Egor |
| author_facet | Voronetsky, Egor |
| contents | We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of presheaves. We also show that it is a crossed module over $G$ in a unique way, in particular, that the $\mathrm K_2$-functor is central. If $G$ is globally isotropic in a suitable sense, then the Steinberg group functor exists as an ordinary group-valued functor and all such abstract Steinberg groups are crossed modules over the groups of points of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally isotropic Steinberg groups I. Centrality of the $\mathrm K_2$-functor Voronetsky, Egor Representation Theory Group Theory K-Theory and Homology 19C09 We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of presheaves. We also show that it is a crossed module over $G$ in a unique way, in particular, that the $\mathrm K_2$-functor is central. If $G$ is globally isotropic in a suitable sense, then the Steinberg group functor exists as an ordinary group-valued functor and all such abstract Steinberg groups are crossed modules over the groups of points of $G$. |
| title | Locally isotropic Steinberg groups I. Centrality of the $\mathrm K_2$-functor |
| topic | Representation Theory Group Theory K-Theory and Homology 19C09 |
| url | https://arxiv.org/abs/2410.14039 |