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f205175 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.EquivariantCoordinates
variable {Λ N T V : Type*} [Group Λ] [Group N] [Group T] [MulAction Λ V]
lemma hom_eq_of_smul_eq (a : Λ →* T)
(hstab : ∀ (g : Λ) (v : V), g • v=v → a g=1) (g h : Λ) (v : V)
(heq : g • v=h • v) : a g=a h := by
have hs : (g⁻¹*h) • v=v := by rw [mul_smul,← heq,inv_smul_smul]
have hh := hstab (g⁻¹*h) v hs
rw [map_mul,map_inv,inv_mul_eq_one] at hh
exact hh
lemma stabilizer_killed [IsMulTorsionFree T] (ρ : Λ →* N) (φ : N →* T)
(hfinite : ∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set N).Finite) :
∀ (g : Λ) (v : V), g • v=v → (φ.comp ρ) g=1 := by
intro g v hg
let H := (MulAction.stabilizer Λ v).map ρ
have hmem : ρ g ∈ H := ⟨g,hg,rfl⟩
let : Finite H := (hfinite v).to_subtype
have hh := isOfFinOrder_of_finite (⟨ρ g,hmem⟩ : H)
have hm := ((φ.comp H.subtype).isOfFinOrder hh).eq_one'
exact hm
theorem exists_equivariant (a : Λ →* T)
(hstab : ∀ (g : Λ) (v : V), g • v=v → a g=1) :
∃ π : V → T, ∀ g v, π (g • v)=a g*π v := by
let q : V → MulAction.orbitRel.Quotient Λ V := Quotient.mk''
let r : V → V := fun v => (q v).out
have hex : ∀ v, ∃ g : Λ, g • r v=v := by
intro v
exact Quotient.exact (Quotient.out_eq' (q v)).symm
choose b hb using hex
let π : V → T := fun v => a (b v)
refine ⟨π,?_⟩
intro g v
have hr : r (g • v)=r v := by
apply congrArg Quotient.out
exact MulAction.orbitRel.Quotient.quotient_smul_eq
have hh : b (g • v) • r v=(g*b v) • r v := by
rw [mul_smul,hb,← hr,hb]
exact (hom_eq_of_smul_eq a hstab _ _ _ hh).trans (map_mul a g (b v))
lemma surjective_of_equivariant [Nonempty V] (a : Λ →* T)
(ha : Function.Surjective a) (π : V → T)
(hπ : ∀ g v, π (g • v)=a g*π v) : Function.Surjective π := by
intro t
obtain ⟨v⟩ := ‹Nonempty V›
obtain ⟨g,hg⟩ := ha (t*(π v)⁻¹)
exact ⟨g • v,by rw [hπ,hg,inv_mul_cancel_right]⟩
lemma same_level_kernel_orbit (a : Λ →* T) (π : V → T)
(hπ : ∀ g v, π (g • v)=a g*π v) (x y : V)
(hxy : y ∈ MulAction.orbit Λ x) (heq : π y=π x) :
y ∈ MulAction.orbit a.ker x := by
obtain ⟨g,hg⟩ := hxy
change g • x=y at hg
have hga : a g=1 := by
have hh := hπ g x
rw [hg,heq] at hh
have hmul : a g*π x=1*π x := by simpa only [one_mul] using hh.symm
exact mul_right_cancel hmul
exact ⟨⟨g,hga⟩,hg⟩
theorem finite_fibers (a : Λ →* T) (π : V → T)
(hπ : ∀ g v, π (g • v)=a g*π v)
(hker : ∀ v : V, (MulAction.orbit a.ker v).Finite)
(S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s)
(t : T) : (π ⁻¹' {t}).Finite := by
have hf : ∀ s : V, {v | π v=t ∧ v∈MulAction.orbit Λ s}.Finite := by
intro s
by_cases he : ∃ y, π y=t ∧ y∈MulAction.orbit Λ s
· obtain ⟨y,hyt,hys⟩ := he
apply (hker y).subset
rintro v ⟨hvt,hvs⟩
apply same_level_kernel_orbit a π hπ y v _ (hvt.trans hyt.symm)
rw [MulAction.orbit_eq_iff.mpr hys]
exact hvs
· convert Set.finite_empty using 1
ext v
simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false]
exact fun h => he ⟨v,h⟩
apply (hS.biUnion fun s _ => hf s).subset
intro v hv
obtain ⟨s,hs,hvs⟩ := hcover v
exact Set.mem_iUnion₂.mpr ⟨s,hs,hv,hvs⟩
end CriticalPercolation.EquivariantCoordinates
end
end
end
end OAI
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