Arithmetical invariants of local quaternion orders
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866917195098882048 |
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| author | Baeth, Nicholas R. Smertnig, Daniel |
| author_facet | Baeth, Nicholas R. Smertnig, Daniel |
| contents | Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $ρ(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$, $v_j$ atoms of $R^\bullet$ and $k$, $l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $ρ(R^\bullet)$ and show that the set of distances $Δ(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1706_00572 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Arithmetical invariants of local quaternion orders Baeth, Nicholas R. Smertnig, Daniel Rings and Algebras 16H10, 11R27, 11S45 Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $ρ(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$, $v_j$ atoms of $R^\bullet$ and $k$, $l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $ρ(R^\bullet)$ and show that the set of distances $Δ(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples. |
| title | Arithmetical invariants of local quaternion orders |
| topic | Rings and Algebras 16H10, 11R27, 11S45 |
| url | https://arxiv.org/abs/1706.00572 |