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Journal of the London Mathematical Society Advance Access published online on March 27, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp005
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© 2009 London Mathematical Society

Gerasimov's theorem and N-Koszul algebras

Roland Berger

Faculté des Sciences et Techniques
Laboratoire de Mathématiques de l’Université de Saint-Etienne
23, Rue Docteur Paul Michelon
42023 Saint-Etienne cedex 2
France

This article is devoted to graded algebras A having a single homogeneous relation. We give a criterion for A to be N-Koszul, where N is the degree of the relation. This criterion uses a theorem of Gerasimov. As a consequence of the criterion, some new examples of N-Koszul algebras are presented. We give an alternative proof of Gerasimov's theorem for N=2, which is related to Dubois-Violette's theorem concerning a matrix description of the Koszul and AS-Gorenstein algebras of global dimension 2. We determine which of the Poincaré–Birkhoff–Witt deformations of a symplectic form are Calabi–Yau.

2000 Mathematics Subject Classification 16S37, 16W50, 06D99 (primary), 16S38, 05A19, 16S80 (secondary).

Received February 1, 2008; revised December 11, 2008;
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