Confluence Algebras and Acyclicity of the Koszul Complex

Cyrille Chenavier 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : 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.
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Submitted on : Monday, January 25, 2016 - 1:54:20 PM
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Cyrille Chenavier. Confluence Algebras and Acyclicity of the Koszul Complex. Algebras and Representation Theory, 2016, ⟨10.1007/s10468-016-9595-6⟩. ⟨hal-01141738v2⟩

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