On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866909540006494208 |
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| author | Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor |
| author_facet | Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor |
| contents | Let $k$ be an arbitrary field. In this paper we show that in the linear case ($Φ=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($Φ= \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(Φ, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable $\mathrm{K}_2$ groups can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ as fundamental groups of the simply-connected Chevalley--Demazure group schemes $\mathrm{G}(Φ,-)$. Our invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(Φ, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_11087 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor Group Theory K-Theory and Homology 19C09, 20G35, 20H05 Let $k$ be an arbitrary field. In this paper we show that in the linear case ($Φ=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($Φ= \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(Φ, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(Φ, R[t]) = \mathrm{K}_2(Φ, R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable $\mathrm{K}_2$ groups can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ as fundamental groups of the simply-connected Chevalley--Demazure group schemes $\mathrm{G}(Φ,-)$. Our invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(Φ, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$. |
| title | On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups |
| topic | Group Theory K-Theory and Homology 19C09, 20G35, 20H05 |
| url | https://arxiv.org/abs/2110.11087 |