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[Merged by Bors] - feat(RingTheory): let B be a faithfully flat A-algebra, then A is a local ring if B is #39611

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[Merged by Bors] - feat(RingTheory): let B be a faithfully flat A-algebra, then A is a local ring if B is #39611

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13 changes: 13 additions & 0 deletions Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean
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Expand Up @@ -142,3 +142,16 @@ lemma PrimeSpectrum.comap_surjective_of_faithfullyFlat :
@[deprecated (since := "2025-12-10")]
alias PrimeSpectrum.specComap_surjective_of_faithfullyFlat :=
PrimeSpectrum.comap_surjective_of_faithfullyFlat

section IsLocalRing

variable (A B)

instance Module.FaithfullyFlat.isLocalHom : IsLocalHom (algebraMap A B) :=
IsLocalHom.of_comap_surjective (algebraMap A B) PrimeSpectrum.comap_surjective_of_faithfullyFlat

/-- Let `B` be a faithfully flat `A`-algebra, then `A` is a local ring if `B` is. -/
theorem Module.FaithfullyFlat.isLocalRing [IsLocalRing B] : IsLocalRing A :=
(algebraMap A B).domain_isLocalRing

end IsLocalRing
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