diff --git a/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean b/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean index c1711ef3e96f65..e3c3d944996113 100644 --- a/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean +++ b/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean @@ -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