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

TL;DR

This paper extends existing algorithms for computing minimal free resolutions from commutative polynomial algebras to weighted b-graded solvable polynomial algebras, enabling effective computation in non-commutative settings.

Contribution

It adapts and extends algorithms for minimal homogeneous generating sets to weighted b-graded solvable polynomial algebras, providing new computational methods.

Findings

Algorithms successfully adapted for non-commutative algebras.

Procedures for minimal finite graded free resolutions developed.

Enhanced computational tools for solvable polynomial algebras.

Abstract

It is shown that the methods and algorithms, developed in (A. Capani et al., Computing minimal finite free resolutions, {\it Journal of Pure and Applied Algebra}, (117& 118)(1997), 105 -- 117; M. Kreuzer and L. Robbiano, {\it Computational Commutative Algebra 2}, Springer, 2005.) for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a commutative polynomial algebra, can be adapted for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a weighted $\mathbb{N}$-graded solvable polynomial algebra, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gr\"obner bases in algebras of solvable type. {\it J. Symbolic Comput.}, 9(1990), 1--26). Consequently, algorithmic procedures for computing minimal finite graded free resolutions over weighted $\mathbb{N}$-graded solvable polynomial algebras are achieved.