On Köthe's normality question for locally finite-dimensional central division algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909967897853952 |
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| author | Chipchakov, Ivan D. |
| author_facet | Chipchakov, Ivan D. |
| contents | This paper considers Köthe's question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R} _{Y}$ of $R$. It shows that the answer to the posed question is negative if $K$ is a purely transcendental extension of infinite transcendence degree over an algebraically closed field $k$. On the other hand, central division LFD-algebras over $K$ turn out to be NLF in the following special cases: (i) $K$ is a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $K _{0}$; (ii) $K$ is a higher-dimensional local field with last residue field equal to $K _{0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Köthe's normality question for locally finite-dimensional central division algebras Chipchakov, Ivan D. Rings and Algebras 2020: 16K40, 12J10, 12F20 (primary), 11S15, 16K20 (secondary) This paper considers Köthe's question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R} _{Y}$ of $R$. It shows that the answer to the posed question is negative if $K$ is a purely transcendental extension of infinite transcendence degree over an algebraically closed field $k$. On the other hand, central division LFD-algebras over $K$ turn out to be NLF in the following special cases: (i) $K$ is a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $K _{0}$; (ii) $K$ is a higher-dimensional local field with last residue field equal to $K _{0}$. |
| title | On Köthe's normality question for locally finite-dimensional central division algebras |
| topic | Rings and Algebras 2020: 16K40, 12J10, 12F20 (primary), 11S15, 16K20 (secondary) |
| url | https://arxiv.org/abs/2403.14564 |