Abelian extensions of finite degree‌ Sample Clauses

Abelian extensions of finite degree‌. In this subsection we fix a finite abelian extension 𝐸/𝐾 , a finite set of places 𝑈 of 𝐾 containing 𝑆∞ (𝐾 ) ∪ 𝑆ram(𝐸/𝐾 ) and a finite set of places Σ that is disjoint from 𝑈 . In the sequel we will make particular use of the Dirichlet regulator isomorphism of k[𝐸 ]-modules 𝐸,𝑈 ,Σ 𝜆𝐸,𝑈 : R ⊗Z U× –→~ R ⊗Z 𝑋𝐸,𝑈 ≥ ) –→ We remark that for any integer 𝑟 0 this isomorphism gives rise to an isomorphism of exterior powers 𝜆 ( ) 𝑟 : 𝑟 R[𝐸 ] (R ⊗Z × U 𝐸,𝑈 ,Σ ~ 𝑟 R[𝐸 ] (R ⊗Z 𝑋𝐸,𝑈 ) (2.40) ˆ
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