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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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHypersurfaceEquidimension
import OAI.NumberTheory.PiExponent.LocalAlgebra.RetainedMinimalPrimeFamily
namespace OAI
namespace PiExponentJets.W22.RetainedPrimeCycle
attribute [local instance] MvPolynomial.gradedAlgebra
universe u
variable {k σ : Type u} [Field k] [Fintype σ]
theorem retainedChild_dimension
(C : RetainedPrimeCycle.{u,u,u} k σ) (f : MvPolynomial σ k)
(havoid : ∀ a, f ∉ (C.component a).val) (s : ℕ)
(hparent : ∀ a,
(actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1)
(T : Ideal (MvPolynomial σ k)) (a : C.Index)
(Q : RetainedMinimalPrime (Ideal.span {f} ⊔ (C.component a).val) T) :
ringKrullDim (MvPolynomial σ k ⧸ Q.val) = (s+1 : ℕ) := by
obtain ⟨N, hN⟩ := actualHP_eventually (C.component a).val (C.component a).homogeneous
have hPdim := W24.prime_hilbert_natDegree_add_one_eq_krullDim (C.component a).val
(C.component a).homogeneous (C.component a).coordinate (C.component a).coordinate_not_mem
(actualHP (C.component a).val (C.component a).homogeneous) (N+1)
(fun n hn => (hN n (by omega)).symm)
have hdim : ringKrullDim (MvPolynomial σ k ⧸ (C.component a).val) = (s+2 : ℕ) := by
calc
_ = ((actualHP (C.component a).val (C.component a).homogeneous).natDegree + 1 :
WithBot ℕ∞) := hPdim.symm
_ = ((s+2 : ℕ) : WithBot ℕ∞) := by
rw [← Nat.cast_one, ← Nat.cast_add]
exact congrArg (fun n : ℕ => (n : WithBot ℕ∞)) (by rw [hparent a])
exact W24.minimal_prime_hypersurface_dimension k σ (C.component a).val Q.val f
(havoid a) (by simpa only [sup_comm] using Q.property.1) s hdim
noncomputable def equidimensionalRetainedCut
(C : RetainedPrimeCycle.{u,u,u} k σ)
(f : MvPolynomial σ k) (d : ℕ) (hd : 0 < d) (hf : f.IsHomogeneous d)
(havoid : ∀ a, f ∉ (C.component a).val)
(s : ℕ) (hparent : ∀ a,
(actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1)
(T : Ideal (MvPolynomial σ k)) (v : σ) (hv : MvPolynomial.X v ∉ T) : ProperCut C :=
canonicalRetainedCut C f d hd hf havoid s hparent T v hv
(retainedChild_dimension C f havoid s hparent T)
theorem equidimensionalRetainedCut_covers
(C : RetainedPrimeCycle.{u,u,u} k σ)
(f : MvPolynomial σ k) (d : ℕ) (hd : 0 < d) (hf : f.IsHomogeneous d)
(havoid : ∀ a, f ∉ (C.component a).val)
(s : ℕ) (hparent : ∀ a,
(actualHP (C.component a).val (C.component a).homogeneous).natDegree = s+1)
(T : Ideal (MvPolynomial σ k)) (v : σ) (hv : MvPolynomial.X v ∉ T) :
(equidimensionalRetainedCut C f d hd hf havoid s hparent T v hv).CoversBelow T :=
canonicalRetainedCut_covers C f d hd hf havoid s hparent T v hv
(retainedChild_dimension C f havoid s hparent T)
end PiExponentJets.W22.RetainedPrimeCycle
end OAI
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