Confluence Algebras and Acyclicity of the Koszul Complex

The N-Koszul algebras are N-homogeneous algebras satisfying a homological property. These algebras are characterised by their Koszul complex: an N-homogeneous algebra is N-Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove N-Koszulness: an algebra admitting a side-confluent presentation is N-Koszul if and only if the extra-condition holds. However, in general, these methods do not provide an explicit contracting homotopy for the Koszul complex. In this article we present a way to construct such a contracting homotopy. The property of side-confluence enables us to define specific representations of confluence algebras. These representations provide a candidate for the contracting homotopy. When the extra-condition holds, it turns out that this candidate works. We make explicit our construction on several examples.