Central extensions of three dimensional Artin-Schelter regular algebras
Artin-Schreier regular (AS-regular, for short) algebras of dimension \(r\) are certain graded algebras having very good homological properties and the same Hilbert series as a polynomial ring in \(r\) variables. The classification of these algebras for \(r=3\) was undertaken in \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)], \textit{M. Artin, J. Tate} and \textit{M. van den Bergh} [The Grothendieck Festschrift Vol. 1, Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)]. For general \(r\), there is an important family of examples -- the so called Sklyanin algebras [see e. g. \textit{J. Tate} and \textit{M. van den Bergh}, Invent. Math. 124, 619-647 (1996; Zbl 0876.17011)]. In the present paper, another class of 4-dimensional AS-regular algebras is investigated: central extensions of 3-dimensional ones. For this, the authors first consider quadratic algebras \(D\) with a normal element \(z\) of degree 1 such that \(D/\langle z \rangle\) is Koszul, and obtain precise conditions for \(D\) to be also Koszul. Then they classify central extensions of 3-dimensional AS-regular algebras which are also AS-regular algebras, and determine their point and line modules.
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