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 [29] following the ideas about schematicness introduced by Van Oystaeyen and Willaert [50] for N-graded algebras. With this theory, we prove the Serre-Artin-Zhang-Verevkin theorem [5,19,20,42,57,58] for several families of non-N-graded algebras and finitely non-N-graded algebras appearing in ring theory and noncommutative algebraic geometry. Our treatment contributes to the research on this theorem presented by Lezama [27,29] from a different point of view.