Non-commutative algebraic geometry of semi-graded rings
Oswaldo Lezama, Edward Latorre
TL;DR
This paper introduces semi-graded rings, extending graded rings and skew PBW extensions, and explores their properties within non-commutative algebraic geometry, including generalized Hilbert series, projective schemes, and key theorems.
Contribution
It develops the theory of semi-graded rings, extending non-commutative geometric concepts and generalizing important theorems like Serre-Artin-Zhang-Verevkin.
Findings
Properties of generalized Hilbert series and polynomials established
Extension of non-commutative projective schemes to semi-graded rings
Generalization of the Serre-Artin-Zhang-Verevkin theorem
Abstract
In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are included at the end of the paper.
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