Matrix computations on projective modules using noncommutative Gr\"obner bases

TL;DR

This paper develops algorithms based on noncommutative Gr"obner bases to compute projective dimensions, check stable freeness, and construct bases for modules over bijective skew PBW extensions, advancing computational methods in noncommutative algebra.

Contribution

It introduces new algorithms utilizing noncommutative Gr"obner bases for module analysis over bijective skew PBW extensions, including projective dimension and stable freeness computations.

Findings

Algorithms for calculating projective dimension

Methods to check stable freeness of modules

Procedures to construct minimal presentations and bases

Abstract

Constructive proofs of fact that a stably free left $S$-module $M$ with rank$(M)\geq$sr$(S)$ is free, where sr$(S)$ denotes the stable rank of an arbitrary ring $S$, were developed in some articles. Additionally, in such papers, are presented algorithmic proofs for calculating projective dimension, and to check whether a left $S$-module $M$ is stably free. Given a left $A$-module $M$, with $A$ a bijective skew $PBW$ extension, we will use these results and Gr\"obner bases theory, to establish algorithms that allow us to calculate effectively the projective dimension for this module, to check whether is stably free, to construct minimal presentations, and to obtain bases for free modules.