Central Extensions of Stephenson’s Algebras

This paper completes the classification of central extensions of three dimensional Artin-Schelter regular algebras to four dimensional 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 central graded element z, i.e. D/〈z〉 = A. If A is generated by elements of degree one, those algebras D which are again AS regular have been classified in [1] and [2]. If A is not generated by elements of degree one, then A falls under a classification due to Stephenson [3, 4]. We classify the AS regular central extensions of Stephenson’s algebras by proving that the regularity of D and z is equivalent to the regularity of z in low degree and this is equivalent to easily verifiable conditions on the defining relations for D.