The tame automorphism group of a polynomial ring

Abstract

Throughout this chapter R denotes a commutative ring, however in almost all results R will be a domain, and R[X]:= R[X1,…, Xn] the polynomial ring in n variables over R. We will consider the following subgroups of AutR R[X]: Aff(R, n):= the affine subgroup of AutR R[X] consisting of all R-automorphisms F such that deg Fi= 1 for all i. J(R,n):= the “de Jonquières” subgroup of AutR R[X] consisting of all R-automorphisms F of the form $$F = \left( {{a_1}{X_1} + {f_1}\left( {{X_2}, \ldots,{X_n}} \right),{a_2}{X_2} + {f_2}\left( {{X_3}, \ldots,{X_n}} \right), \ldots,{a_n}{X_n} + {f_n}} \right)$$where each aibelongs to R* and fi∈ R [Xi+1,… Xn] for all 1 ≤ i ≤ n - 1 and fn∈ R. E (R, n):= the subgroup of AutRR[X] generated by the elementary automorphisms, i.e. the automorphisms of the form$$F = \left( {{X_1}, \ldots,{X_{i - 1}},{X_i} + a,{X_{i + 1}}, \ldots,{X_n}} \right)$$for some $$a \in R\left[ {{X_1}, \ldots,{{\hat X}_i}, \ldots,{X_n}} \right]$$and 1 ≤ i ≤ n.