On Congruence Theorem for valued division algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915345308057600 |
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| author | Khanh, Huynh Viet Khoa, Nguyen Duc Anh |
| author_facet | Khanh, Huynh Viet Khoa, Nguyen Duc Anh |
| contents | Let $K$ be a field equipped with a Henselian valuation, and let $D$ be a tame central division algebra over the field $K$. Denote by $\mathrm{TK}_1(D)$ the torsion subgroup of the Whitehead group ${\rm K}_1(D) = D^*/D'$, where $D^*$ is the multiplicative group of $D$ and $D'$ is its derived subgroup. Let ${\bf G}$ be the subgroup of $D^*$ such that $\mathrm{TK}_1(D) = {\bf G}/D'$. In this note, we prove that either $(1 + M_D) \cap {\bf G} \subseteq D'$, or the residue field $\overline{K}$ has characteristic $p > 0$ and the group ${\bf H} := ((1 + M_D) \cap {\bf G})D'/D'$ is a $p$-group. Additionally, we provide examples of valued division algebras with non-trivial ${\bf H}$. This illustrates that, in contrast to the reduced Whitehead group \({\rm SK}_1(D)\), a complete analogue of the Congruence Theorem does not hold for \({\rm TK}_1(D)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Congruence Theorem for valued division algebras Khanh, Huynh Viet Khoa, Nguyen Duc Anh Rings and Algebras Group Theory 16W60, 19B99, 16K20 Let $K$ be a field equipped with a Henselian valuation, and let $D$ be a tame central division algebra over the field $K$. Denote by $\mathrm{TK}_1(D)$ the torsion subgroup of the Whitehead group ${\rm K}_1(D) = D^*/D'$, where $D^*$ is the multiplicative group of $D$ and $D'$ is its derived subgroup. Let ${\bf G}$ be the subgroup of $D^*$ such that $\mathrm{TK}_1(D) = {\bf G}/D'$. In this note, we prove that either $(1 + M_D) \cap {\bf G} \subseteq D'$, or the residue field $\overline{K}$ has characteristic $p > 0$ and the group ${\bf H} := ((1 + M_D) \cap {\bf G})D'/D'$ is a $p$-group. Additionally, we provide examples of valued division algebras with non-trivial ${\bf H}$. This illustrates that, in contrast to the reduced Whitehead group \({\rm SK}_1(D)\), a complete analogue of the Congruence Theorem does not hold for \({\rm TK}_1(D)\). |
| title | On Congruence Theorem for valued division algebras |
| topic | Rings and Algebras Group Theory 16W60, 19B99, 16K20 |
| url | https://arxiv.org/abs/2503.17714 |