Higher Koszul complexes

摘要

In this paper we generalize the Koszul complexes and Koszul algebras, and introduce the higher Koszul (t-Koszul) complexes and higher Koszul algebras, where t ≥ 2 is an integer. We prove that an algebra is t-Koszul if and only if its t-Koszul complex is augmented, i.e. the higher degree (≥ 1) homologies vanish. For arbitrary t-Koszul algebra Λ, we also give a description of the structure of the cohomology algebra Ext_Λ~· (Λ_0, Λ_0) by using the t-Koszul complexes, where the Λ_0 is the direct sum of the simples.