On certain definable coarsenings of valuation rings and their applications
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915659084988416 |
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| author | Kuhlmann, Franz-Viktor |
| author_facet | Kuhlmann, Franz-Viktor |
| contents | We show how suitable extensions $(L|K,v)$ of prime degree of valued fields give rise to definable coarsenings of the valuation rings of $L$ and $K$. In the case of Artin-Schreier and Kummer extensions with wild ramification, we can also define the ramification ideal. We demonstrate the use of the coarsenings on $L$, their maximal ideals, and the ramification ideals for the classification of defects and for the presentation of the Kähler differentials of the extension of the valuation rings of $(L|K,v)$, and their annihilators. Finally, we give a construction that realizes predescribed convex subgroups of suitable value groups as those that are associated with Galois extensions of degree $p$ with independent defect, which in turn give rise to definable coarsenings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06391 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain definable coarsenings of valuation rings and their applications Kuhlmann, Franz-Viktor Logic Commutative Algebra 12J10, 12L12 We show how suitable extensions $(L|K,v)$ of prime degree of valued fields give rise to definable coarsenings of the valuation rings of $L$ and $K$. In the case of Artin-Schreier and Kummer extensions with wild ramification, we can also define the ramification ideal. We demonstrate the use of the coarsenings on $L$, their maximal ideals, and the ramification ideals for the classification of defects and for the presentation of the Kähler differentials of the extension of the valuation rings of $(L|K,v)$, and their annihilators. Finally, we give a construction that realizes predescribed convex subgroups of suitable value groups as those that are associated with Galois extensions of degree $p$ with independent defect, which in turn give rise to definable coarsenings. |
| title | On certain definable coarsenings of valuation rings and their applications |
| topic | Logic Commutative Algebra 12J10, 12L12 |
| url | https://arxiv.org/abs/2512.06391 |