Belov-Kanel, Alexei; Yu, Jie-Tai Stable tameness of automorphisms of \(F\langle x,y,z\rangle\) fixing \(z\). (English) Zbl 1268.16025 Sel. Math., New Ser. 18, No. 4, 799-802 (2012). 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)]. Reviewer: Vesselin Drensky (Sofia) Cited in 2 ReviewsCited in 3 Documents 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) Keywords:stably tame automorphisms; coordinates; polynomial algebras; free associative algebras; stable tameness; lifting problem Citations:Zbl 1246.14075; Zbl 1232.13005; Zbl 0692.13004; Zbl 1086.16019 PDFBibTeX XMLCite \textit{A. Belov-Kanel} and \textit{J.-T. Yu}, Sel. Math., New Ser. 18, No. 4, 799--802 (2012; Zbl 1268.16025) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.