Gelfand-Kirillov dimension of multi-filtered algebras

We consider associative algebras filtered by the additive monoid N-p. We pr ove that, under quite general conditions, the study of Gelfand-Kirillov dim ension of modules over multi-filtered algebra R can be reduced to the assoc iated N-p-graded algebra G(R). As a consequence, we show the exactness of t he Gelfand-Kirillov dimension when the multi-filtration is finite-dimension al and G(R) is a finitely generated noetherian algebra. Our methods apply t o examples like iterated Ore extensions with arbitrary derivations and "hom othetic" automorphisms (e.g. quantum matrices, quantum Weyl algebras) and t he quantum enveloping algebra of sl(v + 1).