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Maps between schematic semi-graded rings

Andrés Chacón, María Camila Ramírez, Armando Reyes

Abstract

Motivated by Smith's work \cite{Smith2003, Smith2016} on maps between non-commu\-tative projective spaces of the form ${\rm Proj}_{nc} A$ in the setting of non-commutative projective geometry developed by Rosenberg and Van den Bergh, and the notion of schematicness introduced by Van Oystaeyen and Willaert \cite{VanOystaeyenWillaert1995} to $\mathbb{N}$-graded rings with the aim of formulating a non-commutative scheme theory à la Grothendieck \cite{EGAII1961}, in this paper we consider a first approach to maps in the Smith's sense in the more general setting of non-commutative projective spaces over semi-graded rings defined by Lezama and Latorre \cite{LezamaLatorre2017}. We extend Smith's key result \cite[Theorem 3.2]{Smith2003}, \cite[Theorem 1.2]{Smith2016} from the category of schematic $\mathbb{N}$-graded rings to the category of schematic semi-graded rings.

Maps between schematic semi-graded rings

Abstract

Motivated by Smith's work \cite{Smith2003, Smith2016} on maps between non-commu\-tative projective spaces of the form in the setting of non-commutative projective geometry developed by Rosenberg and Van den Bergh, and the notion of schematicness introduced by Van Oystaeyen and Willaert \cite{VanOystaeyenWillaert1995} to -graded rings with the aim of formulating a non-commutative scheme theory à la Grothendieck \cite{EGAII1961}, in this paper we consider a first approach to maps in the Smith's sense in the more general setting of non-commutative projective spaces over semi-graded rings defined by Lezama and Latorre \cite{LezamaLatorre2017}. We extend Smith's key result \cite[Theorem 3.2]{Smith2003}, \cite[Theorem 1.2]{Smith2016} from the category of schematic -graded rings to the category of schematic semi-graded rings.
Paper Structure (5 sections, 13 theorems, 13 equations)

This paper contains 5 sections, 13 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Maps between schematic semi-graded rings
  4. Acknowledgments
  5. Conclusions and future work

Key Result

Proposition 3.1

Let $f:N\rightarrow M$ be a morphism of $\mathsf{SGR}-R$ and $X\subseteq N$. Then $f(\langle X\rangle^{\mathsf{SG}})=\langle f(X)\rangle^{\mathsf{SG}}$.

Theorems & Definitions (32)

  • Definition 2.1: LezamaLatorre2017
  • Definition 2.2: LezamaLatorre2017
  • Remark 2.3: ChaconReyes2022
  • Definition 2.4: ChaconReyes2022
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 22 more