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On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups

A. Kanel-Belov, J. -T. Yu, A. Elishev

Abstract

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.

On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups

Abstract

We study authomorphisms of -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 , where is the polynomial or free associative algebra over the base field . We prove that all -scheme automorphisms of are inner for , and all -scheme automorphisms of are semi-inner. As an application, we prove that cannot be embedded into 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.

Paper Structure

This paper contains 22 sections, 62 theorems, 179 equations.

Table of Contents

  1. Introduction and main results
  2. Automorphisms of $K[x_1,\dots,x_n]$ and $K\langle x_1,\dots,x_n\rangle$
  3. Main results
  4. Varieties of automorphisms
  5. Elementary and tame automorphisms
  6. $\operatorname{Ind}$-schemes and $\operatorname{Ind}$-groups
  7. The Jacobian conjecture in any characteristic, Kanel-Belov -- Kontsevich conjecture, and approximation
  8. Approximation problems and Kanel-Belov -- Kontsevich Conjecture
  9. The Jacobian conjecture in any characteristic
  10. Approximation for the automorphism group of affine spaces
  11. Lifting of automorphism groups
  12. Lifting of automorphisms from $\operatorname{Aut}(K[x_1,\dots,x_n])$ to $\operatorname{Aut}(K\langle x_1,\dots,x_n\rangle)$
  13. Automorphisms of the polynomial algebra and the approach of Bodnarchuk--Rips
  14. Reduction to the case when $\Psi$ is identical on $\operatorname{SL}_n$
  15. The lemma of Rips
  16. ...and 7 more sections

Key Result

Theorem 1.1

Any $\operatorname{Ind}$-scheme automorphism $\varphi$ of $\operatorname{NAut}(K[x_1,\dots,x_n])$ for $n\ge 3$ is inner, i.e. is a conjugation via some automorphism.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 57 more