Standard bases in mixed power series and polynomial rings over rings

Abstract

In this paper we study standard bases for submodules of a mixed power series and polynomial ring R〚t1,…,tm〛[x1,…,xn]s respectively of their localisation with respect to a t_-local monomial ordering for a certain class of noetherian rings R, also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora and to generalise the Buchberger criterion. Everything else then translates naturally. Setting either m=0 or n=0 we get standard bases for polynomial rings respectively for power series rings over R as a special case.