Inverse limits of polynomial rings
The author deals with the following question of M.~Miyanishi. Let \(\{A_i\}\) be an inverse system of reduced affine algebras over a field. Assume that \(A_i\to A_{i-1}\) is onto for all \(i>0\). Similarly consider polynomial rings in one variable \(\{A_i[T_i]\}\) with surjective maps \(A_i[T_i]\to A_{i-1}[T_{i-1}]\) compatible with the previous inverse system. Then Miyanishi asked whether \(\lim_{\leftarrow}A_i[T_i]\) is isomorphic to \(A[\mathbb{Y}]\) for a variable \(\mathbb{Y}\) and \(A=\lim_{\leftarrow} A_i\). The author proves that the answer is in the affirmative locally and globally under a suitable Mittag-Leffler hypothesis on the inverse system of units of \(A_i\)'s. The paper ends with several illuminating examples due to David Wright showing how one can go wrong. The study of these inverse systems is primarily motivated by the well-known Jacobian conjecture.
scientific article; zbMATH DE number 2134871