Convergent presentations and polygraphic resolutions of associative algebras

TL;DR

This paper introduces polygraphic resolutions for associative algebras using higher-dimensional rewriting, improving homological methods and linking to the Koszul property.

Contribution

It defines polygraphs as higher-dimensional rewriting systems generalizing noncommutative Gr"obner bases and develops a categorical framework for resolutions.

Findings

Polygraphic resolutions can be computed from convergent presentations.

Polygraphs allow more flexible termination orders than monomial orders.

The approach links higher-dimensional rewriting to algebraic properties like Koszulness.

Abstract

Several constructive homological methods based on noncommutative Gr\"obner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Gr\"obner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Gr\"obner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property.