Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.