On the computation of Kähler differentials and characterizations of Galois extensions with independent defect
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916652894912512 |
|---|---|
| author | Cutkosky, Steven Dale Kuhlmann, Franz-Viktor Rzepka, Anna |
| author_facet | Cutkosky, Steven Dale Kuhlmann, Franz-Viktor Rzepka, Anna |
| contents | For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind differentials. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs.\ independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their rank or value groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the computation of Kähler differentials and characterizations of Galois extensions with independent defect Cutkosky, Steven Dale Kuhlmann, Franz-Viktor Rzepka, Anna Commutative Algebra 12J10, 12J25 For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind differentials. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs.\ independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their rank or value groups. |
| title | On the computation of Kähler differentials and characterizations of Galois extensions with independent defect |
| topic | Commutative Algebra 12J10, 12J25 |
| url | https://arxiv.org/abs/2305.10022 |