Computing minimal free resolutions of right modules over noncommutative algebras

TL;DR

This paper introduces a general algorithmic method for computing minimal free resolutions of finitely presented graded right modules over noncommutative algebras, extending syzygy computations from commutative to noncommutative settings.

Contribution

It presents a novel approach to compute minimal free resolutions over noncommutative algebras using syzygy computations, including bounds for Betti numbers in monomial cases.

Findings

Algorithmic method for resolutions over noncommutative algebras

Bound for Betti number degrees in monomial modules

Complete resolution computation when finite

Abstract

In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of the algebra then one obtains its graded homology. The approach is based on the possibility to obtain the resolution via the computation of syzygies for modules over commutative algebras. The method behaves algorithmically if one bounds the degree of the required elements in the resolution. Of course, this implies a complete computation when the resolution is a finite one. Finally, for a monomial right module over a monomial algebra we provide a bound for the degrees of the non-zero Betti numbers of any single homological degree in terms of the maximal degree of the monomial relations of the module and the algebra.