Noncommutative scheme theory and the Serre–Artin–Zhang–Verevkin theorem for semi-graded rings

Abstract

In this paper, we present a noncommutative scheme theory for the semi-graded rings generated in degree one defined by Lezama and Latorre [Internat. J. Algebra Comput. 27 (2017), 361–389] following the ideas about schematicness introduced by Van Oystaeyen and Willaert [J. Pure Appl. Algebra 104 (1995), 109–122] for $\mathbb{N}$-graded algebras. With this theory, we prove the Serre–Artin–Zhang–Verevkin theorem for several families of non-$\mathbb{N}$-graded algebras and finitely non-$\mathbb{N}$-graded algebras appearing in ring theory and noncommutative algebraic geometry. Our treatment contributes to the research on this theorem presented by Lezama from a different point of view.