Valued adic Completion Semialg Hom
IsDedekindDomain.HeightOneSpectrum.Extension.valued_adicCompletionSemialgHom
Plain-language statement
The local ramification index for the extension L_w/K_v is equal to the global ramification index for the extension w/v. In other words, if x in K_v and i:K_v->L_w then w(i(x))=v(x)^e where e is computed globally.
Exact Lean statement
lemma valued_adicCompletionSemialgHom (x) :
Valued.v (adicCompletionSemialgHom K L w x) = Valued.v x ^
(w.1.under A).asIdeal.ramificationIdx' w.1.asIdealFormal artifact
Lean source
lemma valued_adicCompletionSemialgHom (x) : Valued.v (adicCompletionSemialgHom K L w x) = Valued.v x ^ (w.1.under A).asIdeal.ramificationIdx' w.1.asIdeal := by revert x apply funext_iff.mp symm apply (adicCompletion.ofCompletion_surjective K v).injective_comp_right apply UniformSpace.Completion.ext · exact ((Valued.continuous_valuation_of_surjective (v.valuedAdicCompletion_surjective K)).pow _).comp (adicCompletion.continuous_ofCompletion K v) · exact ((Valued.continuous_valuation_of_surjective (w.1.valuedAdicCompletion_surjective L)).comp (adicCompletionSemialgHom_continuous K L w)).comp (adicCompletion.continuous_ofCompletion K v) intro a simp only [Function.comp_apply, adicCompletion.valued_ofCompletion, Valued.valuedCompletion_apply, w.2, adicCompletionSemialgHom_coe, WithVal.equiv_symm_apply, WithVal.valued_toVal, ← valuation_comap A K L B w.1 _] rw [WithVal.valued_toVal]- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/DedekindDomain/Completion/BaseChange.lean:206-224
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