feat(RingTheory/DedekindDomain/Ideal/Lemmas): add nontrivial_heightOneSpectrum#35533
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mariainesdff wants to merge 4 commits intomasterfrom
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feat(RingTheory/DedekindDomain/Ideal/Lemmas): add nontrivial_heightOneSpectrum#35533mariainesdff wants to merge 4 commits intomasterfrom
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PR summary 4e2c52aae1Import changes for modified filesNo significant changes to the import graph Import changes for all files
Declarations diff
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./scripts/pr_summary/declarations_diff.sh long <optional_commit>The doc-module for No changes to technical debt.You can run this locally as
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tb65536
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I think
open IsDedekindDomain in
lemma Algebra.IsIntegral.nontrivial_heightOneSpectrum [IsDomain A] [Algebra R A]
[FaithfulSMul R A] [Algebra.IsIntegral R A] [Nontrivial (HeightOneSpectrum R)] :
Nontrivial (HeightOneSpectrum A) := by
have := (FaithfulSMul.algebraMap_injective R A).isDomain
let f (p : HeightOneSpectrum A) : HeightOneSpectrum R :=
⟨p.asIdeal.under R, inferInstance, mt Ideal.eq_bot_of_comap_eq_bot p.ne_bot⟩
have : Function.Surjective f := fun ⟨p, _, hp⟩ ↦ by
obtain ⟨P, hP, rfl⟩ := p.exists_ideal_over_prime_of_isIntegral_of_isDomain (S := A) (by simp)
exact ⟨⟨P, hP, by aesop⟩, rfl⟩
exact this.nontrivialis a better proof and I am on the fence about having the subjectivity as its own separate lemma or not.
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I have switched to this proof. |
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| let f (p : HeightOneSpectrum A) : HeightOneSpectrum R := | ||
| ⟨p.asIdeal.under R, inferInstance, mt Ideal.eq_bot_of_comap_eq_bot p.ne_bot⟩ |
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Should this be a separate definition? It's strange we can't express primes below in the height one spectrum language.
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Co-authored by: @xgenereux.