Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach
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| Format: | Preprint |
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2020
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| _version_ | 1866912278604939264 |
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| author | Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor |
| author_facet | Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor |
| contents | We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(Φ,\, R)$ for any root system $Φ$ of rank $\geq 3$. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism $\mathrm{St}(Φ, R) \to \mathrm{G}_\mathrm{sc}(Φ, R)$, which has not been known previouly for exceptional $Φ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_03999 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach Lavrenov, Andrei Sinchuk, Sergey Voronetsky, Egor Group Theory 19C09, 20G35, 20H05 We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(Φ,\, R)$ for any root system $Φ$ of rank $\geq 3$. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism $\mathrm{St}(Φ, R) \to \mathrm{G}_\mathrm{sc}(Φ, R)$, which has not been known previouly for exceptional $Φ$. |
| title | Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach |
| topic | Group Theory 19C09, 20G35, 20H05 |
| url | https://arxiv.org/abs/2009.03999 |