On the lifting of the Nagata automorphism

Abstract

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K<x,y,z>$. The proof is based on the following two new results which have their own interests: degree estimate of $Q{\ast }_{F}F<x_1,...,x_n>$ and tameness of the automorphism group ${\text{Aut}}_Q(Q{\ast }_{F}F<x,y>)$.

15 pages

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