On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups
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arXiv
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| Format: | Preprint |
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2017
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| author | Kanel-Belov, A. Yu, J. -T. Elishev, A. |
| author_facet | Kanel-Belov, A. Yu, J. -T. Elishev, A. |
| contents | We study authomorphisms of $Ind$-groups of polynomial automorphisms (wich are singular) via tame approximations. Such objects were pioneeered in research by B.I.Plotkin
We obtain a number of properties of $Aut(Aut(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all $Ind$-scheme automorphisms of $Aut(K[x_1,\dots,x_n])$ are inner for $n\ge 3$, and all $Ind$-scheme automorphisms of $Aut(K\langle x_1,\dots, x_n\rangle)$ are semi-inner.
As an application, we prove that $Aut(K[x_1,\dots,x_n])$ cannot be embedded into $Aut(K\langle x_1,\dots,x_n\rangle)$ by the natural abelianization. In other words, the {\it Automorphism Group Lifting Problem} has a negative solution. We explore close connection between the above results and the Jacobian conjecture type questions, formulate the Jacobian conjecture for fields of any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_01490 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups Kanel-Belov, A. Yu, J. -T. Elishev, A. Algebraic Geometry 14R10, 14R15, 14R20 We study authomorphisms of $Ind$-groups of polynomial automorphisms (wich are singular) via tame approximations. Such objects were pioneeered in research by B.I.Plotkin We obtain a number of properties of $Aut(Aut(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all $Ind$-scheme automorphisms of $Aut(K[x_1,\dots,x_n])$ are inner for $n\ge 3$, and all $Ind$-scheme automorphisms of $Aut(K\langle x_1,\dots, x_n\rangle)$ are semi-inner. As an application, we prove that $Aut(K[x_1,\dots,x_n])$ cannot be embedded into $Aut(K\langle x_1,\dots,x_n\rangle)$ by the natural abelianization. In other words, the {\it Automorphism Group Lifting Problem} has a negative solution. We explore close connection between the above results and the Jacobian conjecture type questions, formulate the Jacobian conjecture for fields of any characteristic. |
| title | On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups |
| topic | Algebraic Geometry 14R10, 14R15, 14R20 |
| url | https://arxiv.org/abs/1712.01490 |