On prime ideals of noetherian skew power series ri

Authors: Edward S Letzter

Publish Date: 2012/05/12

Volume: 192, Issue: 1, Pages: 67-81

PDF Link

Abstract

We study prime ideals in skew power series rings T= Ry τ δ for suitably conditioned complete right Noetherian rings R automorphisms τ of R and τderivations δ of R Such rings were introduced by Venjakob motivated by issues in noncommutative Iwasawa theory Our main results concern “Cutting Down” and “Lying Over” In particular assuming that τ extends to a compatible automorphsim of T we prove If I is an ideal of R then there exists a τprime ideal P of T contracting to I if and only if I is a τδprime ideal of R Consequently under the more specialized assumption that δ = τ − id a basic feature of the Iwasawatheoretic context we can conclude If I is an ideal of R then there exists a prime ideal P of T contracting to I if and only if I is a τprime ideal of R Our approach depends essentially on two key ingredients First the algebras considered are Zariskian in the sense of Li and Van Oystaeyen and so the ideals are all topologically closed Second topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings

Keywords: