Continuity of relative adele norms

The placewise relative norms define continuous maps of finite, infinite, and full adele rings. For finite adeles, coordinatewise continuity alone is insufficient: the restricted-product topology is finer than the product topology.

These continuity results induce continuous norm maps on ideles with their units topology, and hence on idele classes.

References

• J. Neukirch, Algebraic Number Theory, Chapter VI, §2.

theorem TauCeti.GlobalNumberFields.continuous_finiteAdeleNorm (K : Type u_1) (L : Type u_2) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] : Continuous ⇑(finiteAdeleNorm K L)

The relative norm of finite adeles is continuous for the restricted-product topology.

theorem TauCeti.GlobalNumberFields.continuous_infiniteAdeleNorm (K : Type u_1) (L : Type u_2) [Field K] [Field L] [NumberField L] [Algebra K L] : Continuous ⇑(infiniteAdeleNorm K L)

The relative norm of infinite adeles is continuous.

theorem TauCeti.GlobalNumberFields.continuous_adeleNorm (K : Type u_1) (L : Type u_2) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] : Continuous ⇑(adeleNorm K L)

The relative norm of full adeles is continuous.