Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.15748 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929357623132160 |
|---|---|
| author | Lim, Uzu |
| author_facet | Lim, Uzu |
| contents | In this expository article, we outline a basic theory of group (co)homology and prove a cohomological formulation of the Local Reciprocity Law: $${\rm Gal}(L/K)^{\rm ab} \cong H_T^{-2}({\rm Gal}(L/K),\mathbb{Z}) \cong H_T^{0}({\rm Gal}(L/K),L^\times) \cong \frac{K^\times}{{\rm Nm}_{L/K}(L^\times)}$$ We first recall basic facts about local fields and homological algebra. Then we define group (co)homology, Tate cohomology, and furnish a toolbox. The Local Reciprocity Law is proven in an abstract cohomological setting, then applied to the case of local fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomology of p-adic fields and Local class field theory Lim, Uzu Number Theory 11S31, 11S25 In this expository article, we outline a basic theory of group (co)homology and prove a cohomological formulation of the Local Reciprocity Law: $${\rm Gal}(L/K)^{\rm ab} \cong H_T^{-2}({\rm Gal}(L/K),\mathbb{Z}) \cong H_T^{0}({\rm Gal}(L/K),L^\times) \cong \frac{K^\times}{{\rm Nm}_{L/K}(L^\times)}$$ We first recall basic facts about local fields and homological algebra. Then we define group (co)homology, Tate cohomology, and furnish a toolbox. The Local Reciprocity Law is proven in an abstract cohomological setting, then applied to the case of local fields. |
| title | Cohomology of p-adic fields and Local class field theory |
| topic | Number Theory 11S31, 11S25 |
| url | https://arxiv.org/abs/2405.15748 |