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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.

Gerasimov's theorem and N ‐Koszul algebras