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Spaces of Generators for the $2 \times 2$ Matrix Algebra

W. S. Gant, Ben Williams

TL;DR

The paper analyzes spaces $U(r)$ of $r$-tuples of $2\times2$ matrices that generate $\mathrm{Mat}_{2\times2}(\mathbb{C})$ and their quotients $B(r)=U(r)/\operatorname{PGL}(2;\mathbb{C})$, showing $B(r)$ closely approximates $B\operatorname{PGL}(2)$ and establishing connectivity properties that enable low-degree cohomology comparisons. For $r=2$ it gives a complete homotopy type $B(2)\simeq S^1\times^{\mathbb{Z}/2\mathbb{Z}}S^2$, while for $r>2$ it determines the rational cohomology of $B(r)$ up to degree $4r-6$ via a Serre spectral sequence analysis and a careful study of the invariant theory of $\operatorname{PGL}(2;\mathbb{C})$. The results yield obstructions to generating degree-2 Azumaya algebras by a small number of elements, leading to an explicit bound: for a ring $R$ of Krull dimension $d$, there exists a degree-2 Azumaya algebra over $R$ that cannot be generated by fewer than $2\left\lfloor\tfrac{d}{4}\right\rfloor+2$ elements. This improves prior bounds and holds over any characteristic-zero base via Lefschetz-type arguments.

Abstract

This paper studies $B(r)$, the space of $r$-tuples of $2 \times 2$ complex matrices that generate $\operatorname{Mat}_{2 \times 2}(\mathbf C)$ as an algebra, considered up to change-of-basis. We show that $B(2)$ is homotopy equivalent to $S^1 \times^{\mathbf Z/2\mathbf Z} S^2$. For $r>2$, we determine the rational cohomology of $B(r)$ for degrees less than $4r-6$. As an application, we use the machinery of arXiv:2012.07900 to prove that for all natural numbers $d$, there exists a ring $R$ of Krull dimension $d$ and a degree-$2$ Azumaya algebra $A$ over $R$ that cannot be generated by fewer than $2\lfloor d/4 \rfloor + 2$ elements.

Spaces of Generators for the $2 \times 2$ Matrix Algebra

TL;DR

The paper analyzes spaces of -tuples of matrices that generate and their quotients , showing closely approximates and establishing connectivity properties that enable low-degree cohomology comparisons. For it gives a complete homotopy type , while for it determines the rational cohomology of up to degree via a Serre spectral sequence analysis and a careful study of the invariant theory of . The results yield obstructions to generating degree-2 Azumaya algebras by a small number of elements, leading to an explicit bound: for a ring of Krull dimension , there exists a degree-2 Azumaya algebra over that cannot be generated by fewer than elements. This improves prior bounds and holds over any characteristic-zero base via Lefschetz-type arguments.

Abstract

This paper studies , the space of -tuples of complex matrices that generate as an algebra, considered up to change-of-basis. We show that is homotopy equivalent to . For , we determine the rational cohomology of for degrees less than . As an application, we use the machinery of arXiv:2012.07900 to prove that for all natural numbers , there exists a ring of Krull dimension and a degree- Azumaya algebra over that cannot be generated by fewer than elements.
Paper Structure (8 sections, 24 theorems, 41 equations, 1 figure)

This paper contains 8 sections, 24 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant theory
  4. Connectivity
  5. The case of r=2
  6. Cohomology of U(r)
  7. The case of odd r
  8. The case of even r and the proof of Theorem \ref{['th:fakeMainTheorem']}

Key Result

Proposition 1

With notation as above, there exists a bijective correspondence

Figures (1)

  • Figure 1: The $\mathrm{E}_{2r-2}$-page of the Serre spectral sequence converging to $\mathrm{H}^*(B(r))$ when $r$ is odd. All indicated classes generate a term isomorphic to $\mathbx{Q}$. The empty terms are all $0$.

Theorems & Definitions (42)

  • Proposition 1
  • proof
  • Theorem 2: Theorem 1.5(b) of First2022
  • Theorem 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 32 more