Relative Reduction and Buchberger's Algorithm in Filtered Free Modules

In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference– differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchbergertype algorithm for constructing relative Gröbner bases of filtered free modules.