Linear polygraphs and Koszulity of algebras

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

1 PI.R2 - Design, study and implementation of languages for proofs and programs
2 PPS - Preuves, Programmes et Systèmes
3 LAGA - Laboratoire Analyse, Géométrie et Applications
4 ICJ - Institut Camille Jordan

We define higher-dimensional linear rewriting systems, called linear polygraphs, for presentations of associative algebras, generalizing the notion of noncommutative Gröbner bases. They are constructed on the notion of category enriched in higher-dimensional vector spaces. Linear polygraphs allow more possibilities of termination orders than those associated to Gröbner bases. We introduce polygraphic resolutions of algebras giving a description obtained by rewriting of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how to relate these resolutions with the Koszul property of algebras.

Keywords: koszulity of algebras; polygraphic resolutions; gröbner bases; higher-dimensional rewriting

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

Yves Guiraud, Eric Hoffbeck, Philippe Malbos. Linear polygraphs and Koszulity of algebras. 2014. ⟨hal-01006220v1⟩