Not Mem range algebra Map of residue not Mem
WeierstrassCurve.notMem_range_algebraMap_of_residue_notMem
Plain-language statement
If the residue of an integral element θ of S does not come from the residue field of R, then θ does not come from K either: an element of K integral over the integrally closed R lies in R, and residues are compatible.
Exact Lean statement
theorem notMem_range_algebraMap_of_residue_notMem {S : Type u} [CommRing S] [IsLocalRing S]
[Algebra R S] [Algebra.IsIntegral R S] [IsLocalHom (algebraMap R S)] {L : Type u} [Field L]
[Algebra K L] [Algebra R L] [Algebra S L] [IsScalarTower R S L] [IsScalarTower R K L]
[IsFractionRing S L] {θ : S}
(hθ : residue S θ ∉ Set.range (algebraMap (ResidueField R) (ResidueField S))) :
algebraMap S L θ ∉ Set.range (algebraMap K L)Formal artifact
Lean source
theorem notMem_range_algebraMap_of_residue_notMem {S : Type u} [CommRing S] [IsLocalRing S] [Algebra R S] [Algebra.IsIntegral R S] [IsLocalHom (algebraMap R S)] {L : Type u} [Field L] [Algebra K L] [Algebra R L] [Algebra S L] [IsScalarTower R S L] [IsScalarTower R K L] [IsFractionRing S L] {θ : S} (hθ : residue S θ ∉ Set.range (algebraMap (ResidueField R) (ResidueField S))) : algebraMap S L θ ∉ Set.range (algebraMap K L) := by rintro ⟨a, ha⟩ have haint : _root_.IsIntegral R a := by have h1 : _root_.IsIntegral R (algebraMap S L θ) := _root_.IsIntegral.map (IsScalarTower.toAlgHom R S L) (Algebra.IsIntegral.isIntegral θ) rw [← ha] at h1 exact (isIntegral_algHom_iff (IsScalarTower.toAlgHom R K L) (FaithfulSMul.algebraMap_injective K L)).mp h1 obtain ⟨r, hr⟩ := IsIntegrallyClosed.isIntegral_iff.mp haint refine hθ ⟨residue R r, ?_⟩ rw [show algebraMap (ResidueField R) (ResidueField S) (residue R r) = residue S (algebraMap R S r) by simp only [← ResidueField.algebraMap_residue]] congr 1 apply IsFractionRing.injective S L rw [← ha, ← hr, ← IsScalarTower.algebraMap_apply R S L, ← IsScalarTower.algebraMap_apply R K L]- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/QuadraticTwists/SplitMultiplicativeReduction.lean:164-183
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