Summary: Lifting Standard Bases in Filtered Structures
W.W. Adams \Lambda P. Loustaunau y
In this paper we further study the theory of standard bases in filtered structures as introduced
by Robbiano. In particular, we study the situation in which a commutative ring A is filtered
by commutative ordered monoids \Delta and \Gamma in such a way that the \Gamma filtration refines the \Delta
filtration. We use these results to determine conditions under which a \Delta standard basis can
be lifted to a \Gamma standard basis. Finally, we interpret these results in three situations where
the refinements are given by changing the status of the variables in a polynomial ring, refining
a partial order on the monomials in a polynomial ring, and refining a filtration by an Iadic
filtration for an ideal I in the ring A.
The concept of a Gršobner basis in polynomial rings as defined originally by Buchberger (see  and
) has been extended to more general settings by many authors (see , , and ) where such
bases are called standard bases. The notions of Gršobner and standard bases are dual to each other.
Indeed, while Gršobner bases are defined with respect to orders that are wellorderings, standard
bases are defined for orders that are negative wellorderings (see Examples 2.2 (3)). Standard bases
play a role in the study of local algebra similar to the role of Gršobner bases in polynomial rings
(see for example ).
Much work has been done for the study and computation of Gršobner bases. However, standard