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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2201.00700 |
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| _version_ | 1866913183367692288 |
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| author | Gant, W. S. Williams, Ben |
| author_facet | Gant, W. S. Williams, Ben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00700 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spaces of Generators for the $2 \times 2$ Matrix Algebra Gant, W. S. Williams, Ben Rings and Algebras 14F25 (Primary) 15A99, 16S15, 55R40 (Secondary) 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. |
| title | Spaces of Generators for the $2 \times 2$ Matrix Algebra |
| topic | Rings and Algebras 14F25 (Primary) 15A99, 16S15, 55R40 (Secondary) |
| url | https://arxiv.org/abs/2201.00700 |