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Non-commutative Grassmann variety as a moduli space

Yujiro Kawamata

TL;DR

This work introduces a non-commutative analogue of the Grassmannian $G(2,4)$ by constructing an NC moduli space $NCG(2,4)$ as an NC scheme built from gluing local NC charts. The core idea is to arrange local models $R_ u$ whose abelianizations recover the classical open sets and whose completions at closed points realize semi-universal NC deformations of the corresponding subspaces, while a universal NC deformation family is globally parameterized. The paper provides explicit gluing data for the case $(m,n)=(2,4)$, showing how overlaps and localization behave to ensure a coherent global moduli space. This construction highlights how NC deformations can form a global moduli-theoretic object that extends classical Grassmann geometry, albeit with potential non-Noetherian behavior and mixed commutativity across charts.

Abstract

We construct a non-commutative version of the Grassmann variety $G(2,4)$ as a non-commutative moduli space of linear subspaces in a projective space.

Non-commutative Grassmann variety as a moduli space

TL;DR

This work introduces a non-commutative analogue of the Grassmannian by constructing an NC moduli space as an NC scheme built from gluing local NC charts. The core idea is to arrange local models whose abelianizations recover the classical open sets and whose completions at closed points realize semi-universal NC deformations of the corresponding subspaces, while a universal NC deformation family is globally parameterized. The paper provides explicit gluing data for the case , showing how overlaps and localization behave to ensure a coherent global moduli space. This construction highlights how NC deformations can form a global moduli-theoretic object that extends classical Grassmann geometry, albeit with potential non-Noetherian behavior and mixed commutativity across charts.

Abstract

We construct a non-commutative version of the Grassmann variety as a non-commutative moduli space of linear subspaces in a projective space.

Paper Structure

This paper contains 4 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. NC scheme
  3. local model of NC Grassmann
  4. $G(2,4)$

Key Result

Theorem 1.1

Assume that $m = 2$ and $n = 4$. Then there exists an NC scheme $NCG(2,4)$ which has the above properties (1), (2) and (3).

Theorems & Definitions (12)

  • Theorem 1.1
  • Definition 3.1
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • proof
  • Lemma 4.4
  • proof
  • Definition 4.5
  • ...and 2 more