Global Dimension 4 Extensions of Artin–Schelter Regular Algebras

Abstract

This paper classifies central and normal extensions from global dimension 3 Artin–Schelter regular algebras to global dimension 4 Artin–Schelter regular algebras. Let A be an AS regular algebra of global dimension 3, 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 [Artin and Schelter, Adv. Math.66 (1987), pp. 171–216; Artin et al., in “The Grothendieck Festschrift,” Vol. 1, pp. 33–85, Birkhäuser, Basel, 1990; Artin et al., Invent Math.106 (1991), pp. 335–388] 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 [Le Bruyn et al., Math. Z.222 (1996), 171–212] to classify the four-dimensional AS regular algebras D when A is quadratic and deg(z)=1. Alternative methods are needed when A is cubic or deg(z)>1. We prove in all such cases that the regularity of D and z is 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.