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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.KrullDimension.Regular
import Mathlib.RingTheory.Localization.AtPrime.Basic
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Regular.RegularSequence
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Ring
namespace OAI
noncomputable section
namespace PiExponentJets.PolynomialLocalUnmixedness
section RegularPair
variable {S : Type*} [CommRing S]
def pairRelation (f g : S) : (S × S) →ₗ[S] S where
toFun z := f * z.1 + g * z.2
map_add' x y := by dsimp; ring
map_smul' r x := by simp only [Prod.smul_fst, Prod.smul_snd, RingHom.id_apply, smul_eq_mul]; ring
def pairBoundary (f g : S) : S →ₗ[S] S × S where
toFun t := (-g * t, f * t)
map_add' x y := by ext <;> dsimp <;> ring
map_smul' r x := by ext <;> simp only [Prod.smul_fst, Prod.smul_snd, RingHom.id_apply, smul_eq_mul] <;> ring
theorem regular_pair_components (f g : S)
(hreg : RingTheory.Sequence.IsRegular S [f, g]) :
IsSMulRegular S f ∧ IsSMulRegular (S ⧸ Ideal.span {f}) g := by
refine ⟨((RingTheory.Sequence.isRegular_cons_iff S f [g]).mp hreg).1, ?_⟩
have hg := hreg.toIsWeaklyRegular.regular_mod_prev 1 (by simp)
change IsSMulRegular (S ⧸ (Ideal.ofList [f] • (⊤ : Submodule S S))) g at hg
have hideal : (Ideal.ofList [f] • (⊤ : Submodule S S)) = Ideal.span {f} := by
change Ideal.ofList [f] * (⊤ : Ideal S) = Ideal.span {f}
rw [Ideal.mul_top, Ideal.ofList_singleton]
exact ((Submodule.quotEquivOfEq _ _ hideal).isSMulRegular_congr g).mp hg
theorem regular_pair_syzygy (f g : S)
(hreg : RingTheory.Sequence.IsRegular S [f, g]) (a b : S)
(hab : f * a + g * b = 0) :
∃ t : S, a = -g * t ∧ b = f * t := by
obtain ⟨hf, hg⟩ := regular_pair_components f g hreg
have hgb : g * b ∈ Ideal.span ({f} : Set S) := by
apply Ideal.mem_span_singleton.mpr
refine ⟨-a, ?_⟩
linear_combination hab
have hb : b ∈ Ideal.span ({f} : Set S) :=
mem_of_isSMulRegular_quotient_of_smul_mem hg hgb
obtain ⟨t, rfl⟩ := Ideal.mem_span_singleton.mp hb
refine ⟨t, ?_, rfl⟩
apply hf
change f * a = f * (-g * t)
linear_combination hab
theorem regular_pair_range_boundary_eq_kernel (f g : S)
(hreg : RingTheory.Sequence.IsRegular S [f, g]) :
LinearMap.range (pairBoundary f g) = LinearMap.ker (pairRelation f g) := by
ext z
rw [LinearMap.mem_range, LinearMap.mem_ker]
constructor
· rintro ⟨t, rfl⟩
change f * (-g * t) + g * (f * t) = 0
ring
· intro hz
obtain ⟨t, ha, hb⟩ := regular_pair_syzygy f g hreg z.1 z.2 hz
exact ⟨t, Prod.ext ha.symm hb.symm⟩
theorem regular_pair_boundary_injective (f g : S)
(hreg : RingTheory.Sequence.IsRegular S [f, g]) :
Function.Injective (pairBoundary f g) := by
intro x y hxy
apply (regular_pair_components f g hreg).1
exact congrArg Prod.snd hxy
theorem pair_relation_range (f g : S) :
LinearMap.range (pairRelation f g) = Ideal.ofList [f, g] := by
apply le_antisymm
· rintro _ ⟨⟨a, b⟩, rfl⟩
have hf : f ∈ Ideal.ofList [f, g] := Ideal.subset_span (by simp)
have hg : g ∈ Ideal.ofList [f, g] := Ideal.subset_span (by simp)
change f * a + g * b ∈ Ideal.ofList [f, g]
exact Ideal.add_mem _ (by simpa [mul_comm] using (Ideal.ofList [f, g]).smul_mem a hf)
(by simpa [mul_comm] using (Ideal.ofList [f, g]).smul_mem b hg)
· apply Ideal.span_le.mpr
intro r hr
have hr : r = f ∨ r = g := by simpa using hr
rcases hr with rfl | rfl
· exact ⟨(1, 0), by simp [pairRelation]⟩
· exact ⟨(0, 1), by simp [pairRelation]⟩
theorem pair_relation_range_eq_quotient_kernel (f g : S) :
LinearMap.range (pairRelation f g) = LinearMap.ker (Ideal.ofList [f, g]).mkQ := by
rw [Submodule.ker_mkQ, pair_relation_range]
def pairToIdeal (f g : S) : (S × S) →ₗ[S] Ideal.ofList [f, g] :=
(pairRelation f g).codRestrict (Ideal.ofList [f, g]) (fun z => by
rw [← pair_relation_range]
exact LinearMap.mem_range_self _ _)
theorem pairToIdeal_surjective (f g : S) : Function.Surjective (pairToIdeal f g) := by
intro x
obtain ⟨z, hz⟩ : ∃ z, pairRelation f g z = (x : S) := by
apply LinearMap.mem_range.mp
rw [pair_relation_range]
exact x.property
exact ⟨z, Subtype.ext hz⟩
theorem regular_pair_range_boundary_eq_kernel_toIdeal (f g : S)
(hreg : RingTheory.Sequence.IsRegular S [f, g]) :
LinearMap.range (pairBoundary f g) = LinearMap.ker (pairToIdeal f g) := by
simpa only [pairToIdeal, LinearMap.ker_codRestrict] using
regular_pair_range_boundary_eq_kernel f g hreg
theorem ideal_subtype_range_eq_quotient_kernel (f g : S) :
LinearMap.range (Ideal.ofList [f, g]).subtype =
LinearMap.ker (Ideal.ofList [f, g]).mkQ := by
rw [Submodule.range_subtype, Submodule.ker_mkQ]
end RegularPair
variable (k : Type*) [Field k] (n : ℕ)
variable (Q : Ideal (MvPolynomial (Fin n) k)) [Q.IsPrime]
abbrev PolynomialLocal := Localization.AtPrime Q
theorem polynomial_local_regular_pair_resolution
(f g : PolynomialLocal k n Q)
(hreg : RingTheory.Sequence.IsRegular (PolynomialLocal k n Q) [f, g]) :
Function.Injective (pairBoundary f g) ∧
LinearMap.range (pairBoundary f g) = LinearMap.ker (pairRelation f g) ∧
LinearMap.range (pairRelation f g) = LinearMap.ker (Ideal.ofList [f, g]).mkQ ∧
Function.Surjective (Ideal.ofList [f, g]).mkQ :=
⟨regular_pair_boundary_injective f g hreg,
regular_pair_range_boundary_eq_kernel f g hreg,
pair_relation_range_eq_quotient_kernel f g, Submodule.mkQ_surjective _⟩
theorem polynomial_local_regular_pair_dimension_drop
(f g : PolynomialLocal k n Q)
(hreg : RingTheory.Sequence.IsRegular (PolynomialLocal k n Q) [f, g]) :
ringKrullDim (PolynomialLocal k n Q ⧸ Ideal.ofList [f, g]) + 2 =
ringKrullDim (PolynomialLocal k n Q) := by
simpa using ringKrullDim_add_length_eq_ringKrullDim_of_isRegular [f, g] hreg
def PolynomialLocalPairUnmixednessTarget : Prop :=
∀ f g : PolynomialLocal k n Q,
RingTheory.Sequence.IsRegular (PolynomialLocal k n Q) [f, g] →
∀ P : Ideal (PolynomialLocal k n Q),
IsAssociatedPrime P (PolynomialLocal k n Q ⧸ Ideal.ofList [f, g]) →
P.height = 2
end PiExponentJets.PolynomialLocalUnmixedness
end
end OAI
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