A Categorical Characterization of Quantum Projective \(\mathbb {Z}\)-spaces

Abstract

In this paper we study a generalization of the notion of AS-regularity for connected \({\mathbb Z}\)-algebras defined in Mori and Nyman (J Pure Appl Algebra, 225(9), 106676, 2021). Our main result is a characterization of those categories equivalent to noncommutative projective schemes associated to right coherent regular \({\mathbb Z}\)-algebras, which we call quantum projective \({\mathbb Z}\)-spaces in this paper. As an application, we show that smooth quadric hypersurfaces and the standard noncommutative smooth quadric surfaces studied in Smith and Van den Bergh (J Noncommut Geom 7(3), 817–856, 2013) , Mori and Ueyama (J Noncommut Geom, 15(2), 489–529, 2021) have right noetherian AS-regular \({\mathbb Z}\)-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative \({\mathbb P}^1\times {\mathbb P}^1\) [in the sense of Van den Bergh (Int Math Res Not 17:3983–4026, 2011)].