Noetherian Connected Graded Algebras of Global Dimension 3

Let A be a noetherian connected graded algebra of global dimension 3. We show that A is regular in the sense of Artin and Schelter if one of the following conditions holds: (1) A is generated by two elements; (2) the graded simple module has a standard resolution; (3) the degrees of minimal relations are the same (this includes the quadratic case). Some general properties of A are studied without assuming regularity.