openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /CoefficientLocalizationMaximal.lean
| import Mathlib.Algebra.Field.Equiv | |
| import Mathlib.RingTheory.Ideal.Quotient.Operations | |
| import Mathlib.RingTheory.Localization.Ideal | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.CoordinateTranscendenceLocalization | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentJets.PolynomialLocalResidueResolution | |
| section MaximalKernel | |
| variable {F A L : Type*} [Field F] [CommRing A] [Field L] | |
| variable [Algebra F A] [Algebra F L] [Algebra.IsAlgebraic F L] | |
| theorem algebraic_field_kernel_isMaximal (f : A →ₐ[F] L) : | |
| (RingHom.ker f.toRingHom).IsMaximal := by | |
| apply Ideal.Quotient.maximal_of_isField | |
| exact MulEquiv.isField (Subalgebra.isField_of_algebraic f.range) | |
| (Ideal.quotientKerEquivRange f).toRingEquiv.toMulEquiv | |
| end MaximalKernel | |
| variable {R A L : Type*} [CommRing R] [CommRing A] [CommRing L] | |
| variable (M : Submonoid R) [Algebra R A] [IsLocalization M A] | |
| include M in | |
| theorem extended_kernel_eq (f : R →+* L) (g : A →+* L) | |
| (h : g.comp (algebraMap R A) = f) : | |
| (RingHom.ker f).map (algebraMap R A) = RingHom.ker g := by | |
| have hk : (RingHom.ker g).comap (algebraMap R A) = RingHom.ker f := by | |
| ext x | |
| change g (algebraMap R A x) = 0 ↔ f x = 0 | |
| have he := RingHom.congr_fun h x | |
| change g (algebraMap R A x) = f x at he | |
| rw [he] | |
| rw [← hk] | |
| exact IsLocalization.map_under M A (RingHom.ker g) | |
| end PiExponentJets.PolynomialLocalResidueResolution | |
| end | |
| end OAI | |