On computation of minimal free resolutions over solvable polynomial algebras

Abstract

Let $A=K[a_1,\ldots ,a_n]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Grobner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1--26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A=\bigoplus_{p\in\mathbb{N}}A_p$ is an $\mathbb{N}$-graded algebra with the degree-0 homogeneous part $A_0=K$; (2) $A$ is an $\mathbb{N}$-filtered algebra with the filtration $\{F_pA\}_{p\in\mathbb{N}}$ determined by a positive-degree function on $A$.