Gerasimov's theorem and N-Koszul algebras

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 Poincare-Birkhoff-Witt deformations of a symplectic form are Calabi-Yau.