LIFTING STANDARD BASES IN FILTERED STRUCTURES

In this paper we further study the theory of standard bases in filtere d structures as introduced by Robbiano. In particular, we study the si tuation in which a commutative ring A is filtered by commutative order ed monoids Delta and Gamma in such a way that the Gamma filtration ref ines the a filtration. We use these results to determine conditions un der which a Delta standard basis can be lifted to a Gamma standard bas is. Finally, we interpret these results in three situations where the refinements are given by changing the status of the variables in a pol ynomial ring, refining a partial order on the monomials in a polynomia l ring, and refining a filtration by an I-adic filtration for an ideal I in the ring A.