On the lifting of the Nagata automorphism

Jie-Tai, Y.; Belov-Kanel, A.

Авторы: Jie-Tai Yu, Belov-Kanel A.
Журнал: Selecta Mathematica
Том: 17
Номер: 4
Год издания: 2011
Первая страница: 935
Последняя страница: 945
DOI: 10.1007/s00029-011-0061-3
Аннотация: et F be an arbitrary field and let F[x,y,z] and F⟨x,y,z⟩ be, respectively, the polynomial and the free associative algebras in three variables. Studying the automorphisms of F[x,y,z] and F⟨x,y,z⟩, a central role is played by the z-automorphisms, i.e. automorphisms fixing the variable z. In particular, ten years ago {\it I. P. Shestakov} and {\it U. U. Umirbaev} [Dokl. Math. 66, No. 2, 266--269 (2002); translation from Dokl. Akad. Nauk 386, No. 6, 745--748 (2002; Zbl 1170.14310); J. Am. Math. Soc. 17, No. 1, 181--196, 197--227 (2004; Zbl 1044.17014); J. Am. Math. Soc. 17, No. 1, 197--227 (2004; Zbl 1056.14085)] showed the wildness of a big class of z-automorphisms of F[x,y,z] including the famous Nagata automorphism (char(F)=0). More recently {\it U. U. Umirbaev} [J. Reine Angew. Math. 605, 165--178 (2007; Zbl 1126.16021)] established the wildness of some z-automorphisms of F⟨x,y,z⟩ including the Anick automorphism (which induces a tame automorphism of F[x,y,z], again char(F)=0). The paper under review deals with the lifting of z-automorphisms of F[x,y,z] to z-automorphisms of F⟨x,y,z⟩. It contains several important results. The main result states that a wild automorphism of the F[z]-algebra F[x,y,z] cannot be lifted to a z-automorphism of F⟨x,y,z⟩. Hence the Nagata automorphism cannot be lifted to a z-automorphism of F⟨x,y,z⟩. Working in the spirit of the result of {\it U. U. Umirbaev} and {\it J.-T. Yu} [Proc. Natl. Acad. Sci. USA 101, No. 13, 4352--4355 (2004; Zbl 1065.13011)], the authors prove that all wild z-coordinates of F[x,y,z] including the Nagata coordinates, cannot be lifted to z-coordinates of F⟨x,y,z⟩. The main idea of the proof is to work in the algebra Q∗FF⟨x,y⟩ for an extension Q of the field F and to describe the structure of the group of Q-automorphisms there. This is combined with a new estimate of the degree of the elements in the two-generated subalgebras of Q∗FF⟨x1,…,xn⟩. The main results are achieved applying the results on AutQQ∗FF⟨x,y⟩ for Q=F(z). In particular, the authors present a canonical decomposition of the z-automorphisms of F⟨x,y,z⟩ as a product of linear automorphisms in AutF[z]F⟨x,y,z⟩ and elementary automorphisms in AutF[z]F⟨x,y,z⟩ conjugated by linear automorphisms in AutF(z)F⟨x,y,z⟩.Reviewer: Vesselin Drensky (Sofia)
Добавил в систему: Белов Алексей Яковлевич

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