Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916684005113856 |
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| author | Wang, Yidi |
| author_facet | Wang, Yidi |
| contents | In this manuscript, we present a partial generalization of the field patching technique initially proposed by Harbater-Hartmann to Hensel semi-global fields, i.e., function fields of curves over excellent henselian discretely valued fields. More specifically, we show that patching holds for étale algebras over such fields and a suitable set of overfields. Within this new framework, we further establish a local-global principle for higher degree Galois cohomology groups over Hensel semi-global fields. As an application, we extend a recent result regarding a uniform period-index bound for higher degree Galois cohomology classes by Harbater-Hartmann-Krashen to Hensel semi-global fields. Additionally, we prove such a bound for coefficient groups of non-prime orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20119 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields Wang, Yidi Number Theory 11E72, 12G05, 12E30 (primary), 14B12, 14H25 (secondary) In this manuscript, we present a partial generalization of the field patching technique initially proposed by Harbater-Hartmann to Hensel semi-global fields, i.e., function fields of curves over excellent henselian discretely valued fields. More specifically, we show that patching holds for étale algebras over such fields and a suitable set of overfields. Within this new framework, we further establish a local-global principle for higher degree Galois cohomology groups over Hensel semi-global fields. As an application, we extend a recent result regarding a uniform period-index bound for higher degree Galois cohomology classes by Harbater-Hartmann-Krashen to Hensel semi-global fields. Additionally, we prove such a bound for coefficient groups of non-prime orders. |
| title | Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields |
| topic | Number Theory 11E72, 12G05, 12E30 (primary), 14B12, 14H25 (secondary) |
| url | https://arxiv.org/abs/2310.20119 |