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

Abstract

This paper studies $B(r)$, the space of $r$-tuples of $2\times 2$ complex matrices that generate ${\mathrm{Mat}}_{2\times 2}(\mathbf{C})$ 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.

arXiv comment13 pages, 1 figure