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

TL;DR

This paper develops a noncommutative scheme theory for semi-graded rings and extends the Serre-Artin-Zhang-Verevkin theorem to non-$\mathbb{N}$-graded algebras, broadening its applicability in noncommutative algebraic geometry.

Contribution

It introduces a new noncommutative scheme framework for semi-graded rings and proves the theorem for classes of non-$\mathbb{N}$-graded algebras, expanding existing theory.

Findings

Extended Serre-Artin-Zhang-Verevkin theorem to non-$\mathbb{N}$-graded algebras

Developed a noncommutative scheme theory for semi-graded rings

Applied the theory to rings in noncommutative algebraic geometry

Abstract

In this paper, we present a noncommutative scheme theory for the semi-graded rings generated in degree one defined by Lezama and Latorre \cite{LezamaLatorre2017} following the ideas about schematicness introduced by Van Oystaeyen and Willaert \cite{VanOystaeyenWillaert1995} for $\mathbb{N}$-graded algebras. With this theory, we prove the Serre-Artin-Zhang-Verevkin theorem \cite{ArtinZhang1994, EGAII1961, Hartshorne1977, Serre1955, Verevkin1992a, Verevkin1992} 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 \cite{Lezama2021, LezamaLatorre2017} from a different point of view.