Project-declaredLean 4.32.0 · mathlib@249c48c2

Is Closed base Change image closure range algebra Map

IsDedekindDomain.HeightOneSpectrum.isClosed_baseChange_image_closure_range_algebraMap

number theory arithmetic geometry Fermat's Last Theorem base change

Plain-language statement

The image of B ⊗[A] 𝓞_v (the closure of B) in ∏_w L_w is closed.

Exact Lean statement

lemma isClosed_baseChange_image_closure_range_algebraMap [FiniteDimensional K L] :
    IsClosed ((baseChange K L B v) ''
        closure (Set.range (algebraMap B (L ⊗[K] adicCompletion K v))))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma isClosed_baseChange_image_closure_range_algebraMap [FiniteDimensional K L] :    IsClosed ((baseChange K L B v) ''        closure (Set.range (algebraMap B (L ⊗[K] adicCompletion K v)))) := by  let S := AddSubgroup.map      (baseChange K L B v).toAddMonoidHom      (tensorAdicCompletionIntegersTo K L B v).range.toSubring.toAddSubgroup  have hSclosed : IsClosed S.carrier := by    apply AddSubgroup.isClosed_of_isOpen    apply (baseChangeRight_isOpenQuotientMap K L B v).isOpenMap    apply (tensorAdicCompletionIntegersTo_isClopen_range K L B v).isOpen  suffices h : (baseChange K L B v) ''    closure (Set.range (algebraMap B (L ⊗[K] adicCompletion K v))) = S.carrier by    rwa [h]  rw [← range_tensorAdicCompletionIntegersTo_eq_closure_range_algebraMap]  rfl

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/DedekindDomain/Completion/BaseChange.lean:514-528

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