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Complements of the point schemes of noncommutative projective lines

Jackson Ryder

TL;DR

This work analyzes the affine open complement of the point scheme in Chan–Nyman's noncommutative P^1 for simple rank-2 bimodules. By inverting a normal degree-2 family $g$, they form the $\ ext{Z}$-indexed localization $\Lambda = \text{S}^{nc}(V)[g^{-1}]$ and study $\Lambda_{00}$, proving it is a noncommutative Dedekind domain of GK-dimension $1$. A detailed ideal-theoretic analysis shows a dichotomy: when $V$ is algebraic (finite order of $\sigma$) the ring is finite over its center, while if $V$ is non-algebraic the ring is simple; this mirrors analogous open subsets of noncommutative quadrics. The authors combine explicit $\mathbb{Z}$-indexed algebra decompositions with Ore localization and Dedekind-domain theory to identify $\Lambda_{00}$ as a key noncommutative affine curve and to illustrate a new simple Dedekind-type example beyond skew polynomial or Laurent rings.

Abstract

Recently, Chan and Nyman constructed noncommutative projective lines via a noncommutative symmetric algebra for a bimodule $V$ over a pair of fields. These noncommutative projective lines of contain a canonical closed subscheme (the point scheme) determined by a normal family of elements in the noncommutative symmetric algebra. We study the complement of this subscheme when $V$ is simple, the coordinate ring of which is obtained by inverting said normal family. We show that this localised ring is a noncommutative Dedekind domain of Gelfand-Kirillov dimension 1. Furthermore, the question of simplicity of these Dedekind domains is answered by a similar dichotomy to an analogous open subscheme of the noncommutative quadrics of Artin, Tate and Van den Bergh.

Complements of the point schemes of noncommutative projective lines

TL;DR

This work analyzes the affine open complement of the point scheme in Chan–Nyman's noncommutative P^1 for simple rank-2 bimodules. By inverting a normal degree-2 family , they form the -indexed localization and study , proving it is a noncommutative Dedekind domain of GK-dimension . A detailed ideal-theoretic analysis shows a dichotomy: when is algebraic (finite order of ) the ring is finite over its center, while if is non-algebraic the ring is simple; this mirrors analogous open subsets of noncommutative quadrics. The authors combine explicit -indexed algebra decompositions with Ore localization and Dedekind-domain theory to identify as a key noncommutative affine curve and to illustrate a new simple Dedekind-type example beyond skew polynomial or Laurent rings.

Abstract

Recently, Chan and Nyman constructed noncommutative projective lines via a noncommutative symmetric algebra for a bimodule over a pair of fields. These noncommutative projective lines of contain a canonical closed subscheme (the point scheme) determined by a normal family of elements in the noncommutative symmetric algebra. We study the complement of this subscheme when is simple, the coordinate ring of which is obtained by inverting said normal family. We show that this localised ring is a noncommutative Dedekind domain of Gelfand-Kirillov dimension 1. Furthermore, the question of simplicity of these Dedekind domains is answered by a similar dichotomy to an analogous open subscheme of the noncommutative quadrics of Artin, Tate and Van den Bergh.

Paper Structure

This paper contains 16 sections, 35 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. $\mathbb{Z}$-indexed algebras and noncommutative $\mathbb{P}^1$'s
  3. Indexed algebras
  4. Noncommutative projective schemes
  5. Noncommutative symmetric algebras and $\mathbb{P}^1$s
  6. Bimodule decompositions of tensor and noncommutative symmetric algebras
  7. A $\mathbb{Z}$-indexed algebra isomorphic to $T$
  8. A bimodule decomposition of $\mathbb S^{nc}(F)$
  9. $\Lambda_{00}$ is a noncommutative affine curve
  10. Two-sided ideals in $\mathbb{Z}$-indexed algebras
  11. $\Lambda_{00}$ is a Dedekind domain
  12. $\Lambda_{00}$ has dimension 1
  13. Algebraicity
  14. Algebraicity and the group $H$
  15. $\Lambda_{00}$ for algebraic two-sided vector spaces
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is an isomorphism of $\mathbb{Z}$-indexed algebras $T(F) \xrightarrow{\sim}\mathcal{T}$ between the tensor algebra $T(F)$ and $\mathcal{T}$. Moreover, for all $i,j \in \mathbb{Z}$ there are $K_i-K_j$-bilinear isomorphisms

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 3.1
  • ...and 64 more