Minimal filtered free resolutions for analytic D-modules

We define the notion of a minimal filtered free resolution for a filtered module over the ring image, a homogenization of the ring image of analytic differential operators. This provides us with analytic invariants attached to a (bi)filtered image-module. We also give an effective argument using a generalization of the division theorem in image due to Assi et al. (J. Pure. Appl. Algebra 164 (2001) 3–21), by which we obtain an upper bound for the length of minimal filtered resolutions.