Computation of Minimal Filtered Free Resolutions over N-Filtered Solvable Polynomial Algebras

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Let A=K[a₁,,aₙ] be a weighted N-filtered solvable polynomial algebra with filtration FA={ FₚA}p, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput., 9(1990), 1--26), and FA is constructed with respect to a positive-degree function d() on A. By introducing minimal F-bases and minimal standard bases respectively for left A-modules and their submodules with respect to good filtrations, minimal filtered free resolutions for finitely generated A-modules are introduced. It is shown that any two minimal F-bases, respectively any two minimal standard bases have the same number of elements and the same number of elements of the same filtered degree; that minimal filtered free resolutions are unique up to strict filtered isomorphism of chain complexes in the category of filtered A-modules; and that minimal finite filtered free resolutions can be algorithmically computed by employing Gröbner basis theory for modules over A with respect to any graded left monomial ordering on free left A-modules.