Non-commutative Castelnuovo–Mumford regularity and AS-regular algebras

Let A be a connected graded k-algebra with a balanced dualizing complex. We prove that A is a Koszul AS-regular algebra if and only if that the Castelnuovo–Mumford regularity and the Ext-regularity coincide for all finitely generated A-modules. This can be viewed as a non-commutative version of [T. Römer, On the regularity over positively graded algebras, J. Algebra 319 (2008) 1–15, Theorem 1.3]. By using Castelnuovo–Mumford regularity, we prove that any Koszul standard AS-Gorenstein algebra is AS-regular. As a preparation to prove the main result, we also prove that the following statements are equivalent: (1) A is AS-Gorenstein; (2) A has finite left injective dimension; (3) the dualizing complex has finite left projective dimension. This generalizes [I. Mori, Homological properties of balanced Cohen–Macaulay algebras, Trans. Amer. Math. Soc. 355 (2002) 1025–1042, Corollary 5.9].