On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912279104061440 |
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| author | Sinchuk, Sergei |
| author_facet | Sinchuk, Sergei |
| contents | In this paper we study the $\mathbb{A}^1$-invariance of the unstable functor $\mathrm{K}_2(Φ, R)$ in the case when $Φ$ is an irreducible root system of type $\mathsf{ADE}$ containing $\mathsf{A}_4$ and not of type $\mathsf{E}_8$. We show that in the geometric case, i. e. when $R$ is a regular ring containing a field $k$ one has $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$, which allows one to interpret the unstable $\mathrm{K}_2$ groups as $\mathbb{A}^1$-fundamental groups of Chevalley--Demazure group schemes in the $\mathbb{A}^1$-homotopy category. We also prove a variant of "early stability" theorem which allows one to find a generating set of $\mathrm{K}_2(Φ, A[X_1, \ldots X_n])$ in the case when $A$ is a Dedekind domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type Sinchuk, Sergei Group Theory K-Theory and Homology 19C20 In this paper we study the $\mathbb{A}^1$-invariance of the unstable functor $\mathrm{K}_2(Φ, R)$ in the case when $Φ$ is an irreducible root system of type $\mathsf{ADE}$ containing $\mathsf{A}_4$ and not of type $\mathsf{E}_8$. We show that in the geometric case, i. e. when $R$ is a regular ring containing a field $k$ one has $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$, which allows one to interpret the unstable $\mathrm{K}_2$ groups as $\mathbb{A}^1$-fundamental groups of Chevalley--Demazure group schemes in the $\mathbb{A}^1$-homotopy category. We also prove a variant of "early stability" theorem which allows one to find a generating set of $\mathrm{K}_2(Φ, A[X_1, \ldots X_n])$ in the case when $A$ is a Dedekind domain. |
| title | On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type |
| topic | Group Theory K-Theory and Homology 19C20 |
| url | https://arxiv.org/abs/2412.06944 |