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arXiv · 2405.15748

Cohomology of p-adic fields and Local class field theory

Uzu Lim
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Abstract

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.

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Uzu Lim. 2024-05-24. Cohomology of p-adic fields and Local class field theory. https://arxiv.org/abs/2405.15748

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