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The $N$-Koszul algebras are $N$-homogeneous algebras which satisfy anhomological property. These algebras are characterised by their Koszul complex:an $N$-homogeneous algebra is $N$-Koszul if and only if its Koszul complex isacyclic. 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, thesemethods do not provide an explicit contracting homotopy for the Koszul complex.In this article we present a way to construct such a contracting homotopy. Theproperty of side-confluence enables us to define specific representations ofconfluence algebras. These representations provide a candidate for thecontracting homotopy. When the extra-condition holds, it turns out that thiscandidate works. We explicit our construction on several examples.

Confluence algebras and acyclicity of the Koszul complex