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

TL;DR

This paper investigates the topological structure of spaces of generators for 2x2 matrix algebras, computes their rational cohomology, and applies these results to construct Azumaya algebras over rings of various dimensions.

Contribution

It characterizes the homotopy type of generator spaces for 2x2 matrices and determines their rational cohomology, extending understanding of algebraic generators and their topological properties.

Findings

B(2) is homotopy equivalent to a twisted product of spheres.

Rational cohomology of B(r) is computed for degrees less than 4r-6 when r>2.

Existence of rings with Azumaya algebras requiring a minimum number of generators.

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.