Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a, b, c, and d with a > 1, c ≥ 0, b ≥ a + c, and d ≥ b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)-Koszul are provided.