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ae00908 | 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 | import Mathlib
import OAI.Algebra.MarkedTensor.ReferenceWords
import OAI.Algebra.MarkedTensor.TripleProducts
namespace OAI
/-! Cubic nilpotence for reference holonomies and quotient ideals. -/
noncomputable section
open scoped BigOperators commutatorElement
open Matrix
namespace BoundaryOnly.Holonomy
open Matrix
open scoped BigOperators commutatorElement
variable {R E : Type*} [CommRing R] [Algebra ℚ R]
noncomputable def referenceOver (R : Type*) [CommRing R] [Algebra ℚ R]
(c : shortWords) : SpecialLinearGroup (Fin 2) R :=
SpecialLinearGroup.map (algebraMap ℚ R) (referenceRho c)
theorem cubic_nilpotence_reference [IsNoetherianRing R] [IsLocalRing R]
(z : E → SpecialLinearGroup (Fin 2) R)
(u : shortWords → SpecialLinearGroup (Fin 2) R)
(hz : ∀ e, mat (z e)-1 ∈ (IsLocalRing.maximalIdeal R).matrix (Fin 2))
(hu : ∀ c, mat (u c)-mat (referenceOver R c) ∈
(IsLocalRing.maximalIdeal R).matrix (Fin 2))
(hrel : ∀ e f g a b c,
⁅u a*z e*(u a)⁻¹,⁅u b*z f*(u b)⁻¹,u c*z g*(u c)⁻¹⁆⁆ = 1) :
deviationIdeal z^3 = ⊥ := by
apply cubic_nilpotence_local z u (referenceOver R) hz hu _ hrel
exact adjoint_span_base_change (fun c : shortWords ↦ referenceRho c)
finite_reference_adjoint_span
omit [Algebra ℚ R] in
lemma cube_sum_pullback {S T : Type*} [CommRing S] (r : Ideal S)
(q : S →+* R) (hq : RingHom.ker q = r)
(N : Ideal R) (hcube : N^3 = ⊥) (W : T → Ideal S)
(hW : ∀ i, (W i).map q ≤ N) : (⨆ i, W i)^3 ≤ r := by
have hm : (⨆ i, W i).map q ≤ N := by
rw [Ideal.map_iSup]
exact iSup_le hW
have hp : ((⨆ i, W i)^3).map q ≤ ⊥ := by
rw [Ideal.map_pow, ← hcube]
exact pow_le_pow_left' hm 3
have h := (Ideal.map_le_iff_le_comap).mp hp
simpa only [← RingHom.ker_eq_comap_bot, hq] using h
omit [Algebra ℚ R] in
lemma slope_deviation_mem (z : E → SpecialLinearGroup (Fin 2) R)
(e : E) (c : SpecialLinearGroup (Fin 2) R) (invert : Bool) :
mat (c*(if invert then (z e)⁻¹ else z e)*c⁻¹)-1 ∈
(deviationIdeal z).matrix (Fin 2) := by
have h : mat (if invert then (z e)⁻¹ else z e)-1 ∈
(deviationIdeal z).matrix (Fin 2) := by
cases invert
· exact deviation_mem z e
· exact inv_deviation (deviation_mem z e)
rw [conjugate_deviation]
exact matrix_mul_right ((deviationIdeal z).matrix (Fin 2) |>.mul_mem_left _ h) _
end BoundaryOnly.Holonomy
namespace BoundaryOnly.Holonomy
open Matrix
open scoped commutatorElement
variable {R S E : Type*} [CommRing R] [CommRing S]
lemma map_deviation_entry (f : R →+* S) (v : SpecialLinearGroup (Fin 2) R)
(i j : Fin 2) :
(mat (SpecialLinearGroup.map f v)-1) i j = f ((mat v-1) i j) := by
simp only [Matrix.sub_apply, map_sub]
change f (mat v i j) - (1 : Matrix (Fin 2) (Fin 2) S) i j =
f (mat v i j) - f ((1 : Matrix (Fin 2) (Fin 2) R) i j)
simp [Matrix.one_apply]
theorem cubic_nilpotence_mod_ideal [Algebra ℚ R] [IsNoetherianRing R] [IsLocalRing R]
(r : Ideal R) (hr : r ≤ IsLocalRing.maximalIdeal R)
(z : E → SpecialLinearGroup (Fin 2) R)
(u : shortWords → SpecialLinearGroup (Fin 2) R)
(hz : ∀ e, mat (z e)-1 ∈ (IsLocalRing.maximalIdeal R).matrix (Fin 2))
(hu : ∀ c, mat (u c)-mat (referenceOver R c) ∈
(IsLocalRing.maximalIdeal R).matrix (Fin 2))
(hrel : ∀ e f g a b c,
mat ⁅u a*z e*(u a)⁻¹,⁅u b*z f*(u b)⁻¹,u c*z g*(u c)⁻¹⁆⁆ - 1 ∈
r.matrix (Fin 2)) :
deviationIdeal z^3 ≤ r := by
have hr' : r ≠ ⊤ := ne_top_of_le_ne_top (IsLocalRing.maximalIdeal.isMaximal R).ne_top hr
let : Nontrivial (R ⧸ r) := Ideal.Quotient.nontrivial_iff.mpr hr'
let q : R →+* R ⧸ r := Ideal.Quotient.mk r
let : IsLocalRing (R ⧸ r) := IsLocalRing.of_surjective' q Ideal.Quotient.mk_surjective
let : IsLocalHom q := IsLocalHom.of_surjective q Ideal.Quotient.mk_surjective
let z' : E → SpecialLinearGroup (Fin 2) (R ⧸ r) := fun e ↦ SpecialLinearGroup.map q (z e)
let u' : shortWords → SpecialLinearGroup (Fin 2) (R ⧸ r) := fun c ↦ SpecialLinearGroup.map q (u c)
have hm {x : R} (hx : x ∈ IsLocalRing.maximalIdeal R) :
q x ∈ IsLocalRing.maximalIdeal (R ⧸ r) :=
IsLocalRing.map_maximalIdeal_le q (Ideal.mem_map_of_mem q hx)
have hz' (e) : mat (z' e)-1 ∈ (IsLocalRing.maximalIdeal (R ⧸ r)).matrix (Fin 2) := by
intro i j
rw [show z' e = SpecialLinearGroup.map q (z e) from rfl, map_deviation_entry]
exact hm (hz e i j)
have href (c) : referenceOver (R ⧸ r) c = SpecialLinearGroup.map q (referenceOver R c) := by
apply Matrix.SpecialLinearGroup.ext
intro i j
rfl
have hu' (c) : mat (u' c)-mat (referenceOver (R ⧸ r) c) ∈
(IsLocalRing.maximalIdeal (R ⧸ r)).matrix (Fin 2) := by
intro i j
rw [href]
change q (mat (u c) i j) - q (mat (referenceOver R c) i j) ∈ _
rw [← map_sub]
exact hm (hu c i j)
have hrel' (e f g a b c) :
⁅u' a*z' e*(u' a)⁻¹,⁅u' b*z' f*(u' b)⁻¹,u' c*z' g*(u' c)⁻¹⁆⁆ = 1 := by
have hh : SpecialLinearGroup.map q
⁅u a*z e*(u a)⁻¹,⁅u b*z f*(u b)⁻¹,u c*z g*(u c)⁻¹⁆⁆ = 1 := by
apply Matrix.SpecialLinearGroup.ext
intro i j
apply sub_eq_zero.mp
change (mat (SpecialLinearGroup.map q _)-1) i j = 0
rw [map_deviation_entry]
exact Ideal.Quotient.eq_zero_iff_mem.mpr (hrel e f g a b c i j)
simpa only [map_commutatorElement, map_mul, map_inv] using hh
have hc := cubic_nilpotence_reference z' u' hz' hu' hrel'
have hd : (deviationIdeal z).map q ≤ deviationIdeal z' := by
apply Ideal.map_le_iff_le_comap.mpr
apply Ideal.span_le.mpr
rintro x ⟨⟨e,i,j⟩,rfl⟩
change q ((mat (z e)-1) i j) ∈ deviationIdeal z'
rw [← map_deviation_entry]
exact deviation_mem z' e i j
have hpow : ((deviationIdeal z)^3).map q ≤ ⊥ := by
rw [Ideal.map_pow, ← hc]
exact pow_le_pow_left' hd 3
simpa only [← RingHom.ker_eq_comap_bot, q, Ideal.mk_ker] using
(Ideal.map_le_iff_le_comap.mp hpow)
end BoundaryOnly.Holonomy
end
end OAI
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