Computation of Minimal Filtered Free Resolutions over $\mathbb{N}$-Filtered Solvable Polynomial Algebras

Abstract

Let $A=K[a_1,\dots ,a_n]$ be a weighted $\mathbb{N}$-filtered solvable polynomial algebra with filtration $FA=\{F_{p}A{\}}_{p\in \mathbb{N}}$, 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.

37 pages. arXiv admin note: text overlap with arXiv:1401.5206

References in corpus (2)

• A Constructive Characterization of Solvable Polynomial Algebras
• On (De)homogenized Gröbner Bases

Cited by in corpus (2)

• Computation of Minimal Graded Free Resolutions over $\mathbb{N}$-Graded Solvable Polynomial Algebras
• Notes on Gröbner Bases and Free Resolutions of Modules over Solvable Polynomial Algebras