File size: 3,087 Bytes
7f5fab8 | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import OAI.AlgebraicGeometry.CommutingDerivations.DenominatorDescent
import OAI.AlgebraicGeometry.CommutingDerivations.DescentFamily
import OAI.AlgebraicGeometry.CommutingDerivations.LocalizedPartials
namespace OAI
/-!
Construction of a commuting, locally nilpotent, linearly independent family
from a relative localization equivalence and a coefficient-intersection identity.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
variable {K B L σ τ : Type*} [CommRing K] [CommRing B] [CommRing L]
[Algebra K B] [Algebra K L] [Algebra (MvPolynomial σ K) L]
[IsScalarTower K (MvPolynomial σ K) L] [Fintype σ] [Fintype τ]
[IsDomain B] [CharZero B]
theorem exists_commuting_family_of_localization
(c : MvPolynomial σ K) [IsLocalization.Away c L]
(hinj : Function.Injective (algebraMap (MvPolynomial σ K) L))
(e : L ≃ₐ[K] MvPolynomial τ B) (b : B)
(hec : e (algebraMap (MvPolynomial σ K) L c) = C b)
(S : Subalgebra K (MvPolynomial σ K))
(hintersection : ∀ p : MvPolynomial σ K,
(∃ b : B, e (algebraMap (MvPolynomial σ K) L p) = C b) ↔ p ∈ S) :
∃ (N : ℕ) (d : τ → Derivation K (MvPolynomial σ K) (MvPolynomial σ K)),
(∀ i p, algebraMap (MvPolynomial σ K) L (d i p) =
(algebraMap (MvPolynomial σ K) L c)^N *
localizedPartial e i (algebraMap (MvPolynomial σ K) L p)) ∧
(∀ i, d i c = 0) ∧
(∀ i j p, d i (d j p) = d j (d i p)) ∧
(∀ i, LocallyNilpotent (d i)) ∧
LinearIndependent (MvPolynomial σ K) d ∧
(∀ p, (∀ i, d i p = 0) ↔ p ∈ S) := by
classical
let ι : MvPolynomial σ K →ₐ[K] L :=
IsScalarTower.toAlgHom K (MvPolynomial σ K) L
let D : τ → Derivation K L L := localizedPartial e
have hfix (i : τ) : D i (ι c) = 0 :=
localizedPartial_fixes_relative e i (ι c) b hec
have hu : IsUnit (ι c) := IsLocalization.Away.algebraMap_isUnit c
obtain ⟨N, d, hd⟩ := exists_common_denominator_descent c hinj D
have hd' (i : τ) (p : MvPolynomial σ K) :
ι (d i p) = (ι c)^N * D i (ι p) := hd i p
refine ⟨N, d, hd, ?_, ?_, ?_, ?_, ?_⟩
· intro i
exact descended_fixes_parameter ι hinj c N (D i) (d i) (hd' i) (hfix i)
· intro i j p
exact descended_commute ι hinj c N (D i) (D j) (d i) (d j)
(hd' i) (hd' j) (hfix i) (hfix j) (localizedPartials_commute e i j) p
· intro i
exact descended_locallyNilpotent ι hinj c N (D i) (d i) (hd' i) (hfix i)
(localizedPartial_locallyNilpotent e i)
· apply descended_linearIndependent ι hinj c hu N D d hd' hfix
(fun j => e.symm (X j)) (localizedPartial_coordinate e)
intro j
exact IsLocalization.Away.surj c (e.symm (X j))
· intro p
calc
(∀ i, d i p = 0) ↔ ∀ i, D i (ι p) = 0 :=
forall_congr' (fun i => descended_kernel ι hinj c hu N (D i) (d i) (hd' i) p)
_ ↔ ∃ b : B, e (ι p) = C b := localizedPartials_common_kernel e (ι p)
_ ↔ p ∈ S := hintersection p
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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