  We introduce a new method to construct 4-dimensional Artin-Schelter regular
algebras as normal extensions of (not necessarily noetherian) 3-dimensional
ones. The method produces large classes of new 4-dimensional Artin-Schelter
regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a
flat family of central extensions of it that are 4-Calabi-Yau, and all
4-Calabi-Yau central extensions having the same generating set as the original
3-Calabi-Yau algebra arise in this way. Each normal extension has the same
generators as the original 3-dimensional algebra, and its relations consist of
all but one of the relations for the original algebra and an equal number of
new relations determined by "the missing one" and a tuple of scalars satisfying
some numerical conditions. We determine the Nakayama automorphisms of the
4-dimensional algebras obtained by our method and as a consequence show that
their homological determinant is 1. This supports the conjecture by Mori-Smith
that the homological determinant of the Nakayama automorphism is 1 for all
Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved
this is true in the noetherian case.
