  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
\cite[Theorem 1.3]{ro}. 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 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
\cite[Corollary 5.9]{mori}.
