2014/01/21 by Huishi Li, Li, Huishi · 1 citation
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Polynomial and algebraic computation #math.RA #msc:16W70 #msc:16Z05
paper · pdf · doi:10.48550/arxiv.1401.5464
37 pages. arXiv admin note: text overlap with arXiv:1401.5206
arxiv created 2014/01/21 · arxiv updated 2014/01/23
Let A=K[a1,…,an] be a weighted ℕ-filtered solvable polynomial algebra with filtration FA=\ FpA\p∈ℕ, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. \it 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.