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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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