A categorical characterization of quantum projective $\mathbb Z$-spaces

In this paper, we study a generalization of the notion of AS-regularity for connected $\mathbb{Z}$-algebras. 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 have right noetherian AS-regular $\mathbb{Z}$-algebras as homogeneous coordinate algebras. In particular, the latter are thus noncommutative $\mathbb{P}^1\times \mathbb{P}^1$.