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Stable tameness of automorphisms of \(F\langle x,y,z\rangle\) fixing \(z\). (English) Zbl 1268.16025

Let \(F\) be an arbitrary field and let \(F\langle x_1,\dots,x_n\rangle\) be the free associative algebra with free generators \(x_1,\dots,x_n\). An \(F\)-automorphism \((f_1,\dots,f_n)\) of \(F\langle x_1,\dots,x_n\rangle\) is tame if it is a composition of automorphisms each of them fixing \(n-1\) of the variables. The automorphism is stably tame if \((f_1,\dots,f_n,x_{n+1},\dots,x_{n+m})\) is a tame automorphism of \(F\langle x_1,\dots,x_n,x_{n+1},\dots,x_{n+m}\rangle\) for some \(m\).
The paper under review is devoted to the following well known and long standing problem: Is every automorphism of \(F\langle x_1,\dots,x_n\rangle\) stably tame? Recently J. Berson, A. van den Essen, and D. Wright [Adv. Math. 230, No. 4-6, 2176-2197 (2012; Zbl 1246.14075)] have established that all automorphisms fixing the variable \(z\) of the polynomial algebra \(F[x,y,z]\) are stably tame. In the present paper the authors prove an important noncommutative analogue of this result. They establish that every automorphism fixing \(z\) of the free associative algebra \(F\langle x,y,z\rangle\) is stably tame and becomes tame if adding one new variable. As a direct consequence, if \(f\in F\langle x,y,z\rangle\) is a \(z\)-coordinate (i.e., there exists a \(g\in F\langle x,y,z\rangle\) such that \((f,g,z)\) is an automorphism), then this coordinate is also stably tame. The proof of the main result is quite different from the proof of the result of Berson, van den Essen, and Wright. It is based on methods of the recent paper by the authors [Sel. Math., New Ser. 17, No. 4, 935-945 (2011; Zbl 1232.13005)] combined with ideas from M. K. Smith [J. Pure Appl. Algebra 58, No. 2, 209-212 (1989; Zbl 0692.13004] and V. Drensky and J.-T. Yu [J. Algebra 291, No. 1, 250-258 (2005; Zbl 1086.16019)].

MSC:

16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16W20 Automorphisms and endomorphisms
13B10 Morphisms of commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
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References:

[1] Belov-Kanel, A., Yu, J.-T.: On the lifting of the Nagata automorphism, Appeared in Selecta Math (New Series), Online FirstTM (2011), http://www.springerlink.com/content/6770653845642152/fulltext.pdf · Zbl 1232.13005
[2] Drensky V., Yu J.-T.: Automorphisms fixing a variable of $${K\(\backslash\)langle x,y,z\(\backslash\)rangle}$$ . J. Algebra 291, 250–258 (2005) · Zbl 1086.16019 · doi:10.1016/j.jalgebra.2005.04.009
[3] Drensky V., Yu J.-T.: The strong Anick conjecture is true. J. Eur. Math. Soc. (JEMS) 9, 659–679 (2007) · Zbl 1161.16019 · doi:10.4171/JEMS/92
[4] Drensky V., Yu J.-T.: The strong Anick conjecture. Proc. Natl. Acad. Sci. USA (PNAS) 103, 4836–4840 (2006) · Zbl 1161.16020 · doi:10.1073/pnas.0509951103
[5] Smith M.: Stably tame automorphisms. J. Pure Appl. Algebra 58, 209–212 (1989) · Zbl 0692.13004 · doi:10.1016/0022-4049(89)90158-8
[6] Berson, J., van den Essen, A., Wright, D.: Stable tameness of two-dimensional polynomial automorphisms over a regular ring. arXiv:0707.3151 · Zbl 1246.14075
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