Monomial methods in iterated local skew power series rings

Abstract

Let $A={\mathbb{F}}_p$ or ${\mathbb{Z}}_p$, and let $R=A[[x_1]][[x_2;{\sigma }_2,{\delta }_2]]\dots [[x_n;{\sigma }_n,{\delta }_n]]$, an iterated local skew power series ring over $A$. Under mild conditions, we show that (multiplicative) monomial orders exist, and develop the theory of Gröbner bases for $R$. We show that all rank-2 local skew power series rings over ${\mathbb{F}}_p$ satisfy polynormality, and give an example of a rank-2 local skew power series ring over ${\mathbb{Z}}_p$ which is a unique factorisation domain in the sense of Chatters-Jordan.

31 pages with 3 figures. Draft: comments welcome