On function fields of curves over higher local fields and their division LFD-algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915373759070208 |
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| author | Chipchakov, Ivan D. |
| author_facet | Chipchakov, Ivan D. |
| contents | Let $K _{m}$ be an $m$-local field with an $m$-th residue field $K _{0}$, for some integer $m > 0$, and let $K/K _{m}$ be a field extension of transcendence degree trd$(K/K _{m}) \le 1$. This paper shows that if $K _{0}$ is a field of finite Diophantine dimension (for example, a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $E$), then the absolute Brauer $p$-dimension abrd$_{p}(K)$ of $K$ is finite, for every prime number $p$. Thus it turns out that if $R$ is an associative locally finite-dimensional (abbr., LFD) central division $K$-algebra, then it is a normally locally finite algebra over $K$, that is, every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R}_{Y}$ of $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On function fields of curves over higher local fields and their division LFD-algebras Chipchakov, Ivan D. Number Theory Rings and Algebras 12J10, 12F20, 16K40 (primary), 11S15, 16K20 (secondary) Let $K _{m}$ be an $m$-local field with an $m$-th residue field $K _{0}$, for some integer $m > 0$, and let $K/K _{m}$ be a field extension of transcendence degree trd$(K/K _{m}) \le 1$. This paper shows that if $K _{0}$ is a field of finite Diophantine dimension (for example, a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $E$), then the absolute Brauer $p$-dimension abrd$_{p}(K)$ of $K$ is finite, for every prime number $p$. Thus it turns out that if $R$ is an associative locally finite-dimensional (abbr., LFD) central division $K$-algebra, then it is a normally locally finite algebra over $K$, that is, every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R}_{Y}$ of $R$. |
| title | On function fields of curves over higher local fields and their division LFD-algebras |
| topic | Number Theory Rings and Algebras 12J10, 12F20, 16K40 (primary), 11S15, 16K20 (secondary) |
| url | https://arxiv.org/abs/2507.04863 |