Cohomology of p-adic fields and Local class field theory

2024/05/24 by Uzu Lim, Lim, Uzu
Mathematics · #Algebraic Geometry and Number Theory #advanced mathematical theories #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2405.15748

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} ≅ H_T^-2(\rm Gal(L/K),ℤ) ≅ H_T⁰(\rm Gal(L/K),L^×) ≅ \fracK^×\rm Nm_L/K(L^×) 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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