Convergent presentations and polygraphic resolutions of associative algebras

Yves Guiraud (1, 2) , Eric Hoffbeck (3) , Philippe Malbos (4)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 PI.R2 - Design, study and implementation of languages for proofs and programs
3 LAGA - Laboratoire Analyse, Géométrie et Applications
4 AGL - Algèbre, géométrie, logique

Several constructive homological methods based on noncommutative Gröbner 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 Gröbner 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 Gröbner 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.

Keywords: Linear rewriting; Higher-dimensional associative algebras; Confluence and termination; Polygraphs; Free resolutions; Koszulness

Domains: Category Theory [math.CT]; K-Theory and Homology [math.KT]

Yves Guiraud, Eric Hoffbeck, Philippe Malbos. Convergent presentations and polygraphic resolutions of associative algebras. Mathematische Zeitschrift, 2019, 293 (1-2), pp.113-179. ⟨10.1007/s00209-018-2185-z⟩. ⟨hal-01006220v3⟩