Global dimension four extensions of Artin-Schelter regular algebras

This dissertation classifies central and normal extensions from global dimension three Artin-Schelter regular algebras to global dimension four Artin-Schelter regular algebras. Let A be an AS regular algebra of global dimension three, and let D be an extension of A by a normal graded element z, i.e. D/z=A . The algebra A falls under a classification due to Artin, Schelter, Tate and Van den Bergh [1, 2, 3], and is either quadratic or cubic. The quadratic algebras A are Koszul, and this fact was used by Le Bruyn, Smith and Van den Bergh in [8] to classify the 4-dimensional AS regular algebras D when A is quadratic and degz=1 . Alternative methods are needed when A is cubic or degz>1 . I prove in all such cases that the regularity of D and the regularity of z are equivalent to the regularity of z in low degree (e.g. 2 or 3) and this is equivalent to easily verifiable matrix conditions on the relations for D.