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12dee50 | 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 | import Mathlib
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.LocalComparison
noncomputable section
variable {K R B : Type*} [Field K] [CommRing R] [CommRing B]
[Algebra K R] [Algebra K B]
def principalIntersectionEquiv (π : R →ₐ[K] B) (hπ : Function.Surjective π)
(g : R) (hg : RingHom.ker π.toRingHom = Ideal.span {g}) (f : R) :
(R ⧸ Ideal.span {f,g}) ≃ₐ[K] (B ⧸ Ideal.span {π f}) := by
let q := (Ideal.Quotient.mkₐ K (Ideal.span {π f})).comp π
have hk : RingHom.ker q.toRingHom = Ideal.span {f,g} := by
have hq : RingHom.ker q.toRingHom = (Ideal.span {π f}).comap π.toRingHom := by
ext x
change Ideal.Quotient.mk (Ideal.span {π f}) (π x) = 0 ↔ _
exact Ideal.Quotient.eq_zero_iff_mem
rw [hq]
have he : (Ideal.span {f}).map π.toRingHom = Ideal.span {π f} := by
rw [Ideal.map_span, Set.image_singleton]
rfl
rw [← he, Ideal.comap_map_of_surjective π.toRingHom hπ]
change Ideal.span {f} ⊔ RingHom.ker π.toRingHom = _
rw [hg, ← Ideal.span_union]
congr 1
exact (Ideal.quotientEquivAlgOfEq K hk.symm).trans
(Ideal.quotientKerAlgEquivOfSurjective
((Ideal.Quotient.mk_surjective).comp hπ))
end
end MaximalSeshadri.LocalComparison
end
end
end
end OAI
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