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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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib.Data.Fintype.Option
import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertDenominatorReduction
import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertZeroVariables
import OAI.NumberTheory.PiExponent.LocalAlgebra.StabilizedColonPowers
import OAI.NumberTheory.PiExponent.Polynomials.CoordinateQuotientGrading
namespace OAI
namespace PiExponentJets.W64
universe u v
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k : Type v} [Field k]
theorem iteratedColon_eq_colon_power {σ : Type u} [Finite σ]
(I : Ideal (MvPolynomial σ k)) (f : MvPolynomial σ k) (n : ℕ) :
iteratedColon I f n = I.colon {f^n} := by
induction n with
| zero => simp only [iteratedColon_zero, PiExponentJets.W06.colon_pow_zero]
| succ n ih => rw [iteratedColon_succ, ih, PiExponentJets.W06.colon_pow_succ]
theorem sectionHilbertSeries_rename {σ τ : Type u} [Finite σ] [Finite τ]
(e : σ ≃ τ) (I : Ideal (MvPolynomial σ k)) :
sectionHilbertSeries I =
sectionHilbertSeries (I.map (MvPolynomial.rename e).toRingHom) := by
ext n
simp only [coeff_sectionHilbertSeries,
PiExponentSiegel.W08.renameQuotientSection_finrank e I n]
theorem sectionHilbertSeries_coordinate {σ : Type u} [Finite σ]
(I : Ideal (MvPolynomial (Option σ) k)) (hX : MvPolynomial.X none ∈ I) :
sectionHilbertSeries I =
sectionHilbertSeries (PiExponentSiegel.W08.coordinateImageIdeal I) := by
ext n
simp only [coeff_sectionHilbertSeries,
PiExponentSiegel.W08.coordinateQuotientSection_finrank I hX n]
theorem exists_homogeneous_hilbert_numerator
(σ : Type u) [Fintype σ] (I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
∃ P : Polynomial ℤ, sectionHilbertSeries I =
(P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card σ)).val := by
classical
refine Fintype.induction_empty_option
(P := fun τ _ => ∀ J : Ideal (MvPolynomial τ k),
J.IsHomogeneous (MvPolynomial.homogeneousSubmodule τ k) →
∃ P : Polynomial ℤ, sectionHilbertSeries J =
(P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card τ)).val)
?_ ?_ ?_ σ I hI
· intro α β _ e ih J hJ
let : Fintype α := Fintype.ofEquiv β e.symm
obtain ⟨P, hP⟩ := ih (J.map (MvPolynomial.rename e.symm).toRingHom)
(PiExponentSiegel.W08.renameImageIdeal_homogeneous e.symm J hJ)
refine ⟨P, ?_⟩
rw [sectionHilbertSeries_rename e.symm J]
simpa only [Fintype.card_congr e] using hP
· intro J _
exact exists_hilbert_numerator_no_variables (by simp) J
· intro α _ ih J hJ
let f : MvPolynomial (Option α) k := MvPolynomial.X none
have hf : f.IsHomogeneous 1 := MvPolynomial.isHomogeneous_X k none
obtain ⟨m, hm⟩ := PiExponentJets.W06.exists_colon_pow_stable J f
have hstable : iteratedColon J f (m+1) = iteratedColon J f m := by
simpa only [iteratedColon_eq_colon_power] using hm
let Jlower : ℕ → Ideal (MvPolynomial (Option α) k) :=
fun j => Ideal.span {f} ⊔ iteratedColon J f j
have hlower : ∀ j, (Jlower j).IsHomogeneous
(MvPolynomial.homogeneousSubmodule (Option α) k) := by
intro j
apply Ideal.IsHomogeneous.sup
· apply Ideal.homogeneous_span (MvPolynomial.homogeneousSubmodule (Option α) k)
intro g hg
obtain rfl := Set.mem_singleton_iff.mp hg
exact ⟨1, hf⟩
· exact iteratedColon_homogeneous J hJ f hf j
have hX : ∀ j, MvPolynomial.X none ∈ Jlower j := by
intro j
have hinc : Ideal.span {f} ≤ Jlower j := le_sup_left
exact hinc (Ideal.subset_span (Set.mem_singleton f))
choose P hP using fun j => ih (PiExponentSiegel.W08.coordinateImageIdeal (Jlower j))
(PiExponentSiegel.W08.coordinateImageIdeal_homogeneous (Jlower j) (hlower j))
have hseries : ∀ j, sectionHilbertSeries (Ideal.span {f} ⊔ iteratedColon J f j) =
(P j : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card α)).val := by
intro j
rw [show Ideal.span {f} ⊔ iteratedColon J f j = Jlower j from rfl,
sectionHilbertSeries_coordinate (Jlower j) (hX j)]
exact hP j
refine ⟨colonHilbertNumerator P m, ?_⟩
simpa only [Fintype.card_option] using
stabilized_colon_polynomial_numerator J hJ f hf m (Fintype.card α) hstable P hseries
end PiExponentJets.W64
end OAI
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