论文标题
$ 2 \ times 2 $矩阵代数的发电机空间
Spaces of Generators for the $2 \times 2$ Matrix Algebra
论文作者
论文摘要
本文研究$ b（r）$，$ r $ $ 2 \ times 2 $复杂矩阵的空间，生成$ \ permatatorname {mat} _ {2 \ times 2}（\ mathbf c）$作为代数，被认为是basisis的代数。我们表明，$ b（2）$是同质的，相当于$ s^1 \ times^{\ mathbf z/2 \ mathbf z} s^2 $。对于$ r> 2 $，我们确定$ b（r）$的合理共同体小于$ 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.
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.
