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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 | import Mathlib
import OAI.Algebra.FinitisticAsymmetry.ZeroDimensions
import OAI.Algebra.FinitisticAsymmetry.Triangular.Socle
namespace OAI
section
section
noncomputable section
open CategoryTheory
namespace LittleFinitistic
universe u
variable (A : Type u) [Ring A]
namespace CummingsConstruction
variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C]
variable (χ : C →ₐ[k] (ι → k))
lemma mem_ker_augmentation (x : T χ) :
x ∈ RingHom.ker (augmentation χ).toRingHom ↔ χ x.fst.1 = 0 ∧ newAugmentation x.fst.2 = 0 := by
change (Sum.elim (χ x.fst.1) (newAugmentation x.fst.2) : Sum ι ι → k) = 0 ↔ _
constructor
· intro h
exact ⟨funext fun i => congrFun h (.inl i), funext fun i => congrFun h (.inr i)⟩
· rintro ⟨h₁, h₂⟩
ext i
cases i <;> simp [h₁, h₂]
def pureOldRadical : Ideal (T χ) where
carrier := {x | x.fst.2 = 0 ∧ (SimpleSum.value (TriangularConstruction.offValue x.snd)) = 0 ∧ χ x.fst.1 = 0}
zero_mem' := ⟨rfl, rfl, map_zero χ⟩
add_mem' := by
rintro x y ⟨hxe, hxu, hxc⟩ ⟨hye, hyu, hyc⟩
change x.fst.2 + y.fst.2 = 0 ∧
SimpleSum.value (TriangularConstruction.offValue x.snd) + SimpleSum.value (TriangularConstruction.offValue y.snd) = 0 ∧ χ (x.fst.1 + y.fst.1) = 0
simp [hxe, hye, hxu, hyu, hxc, hyc, map_add]
smul_mem' := by
rintro r x ⟨hxe, hxu, hxc⟩
change r.fst.2 * x.fst.2 = 0 ∧
newAugmentation r.fst.2 * SimpleSum.value (TriangularConstruction.offValue x.snd) +
χ x.fst.1 * SimpleSum.value (TriangularConstruction.offValue r.snd) = 0 ∧ χ (r.fst.1 * x.fst.1) = 0
simp [hxe, hxu, hxc, map_mul]
lemma ker_square_pureOldRadical :
(RingHom.ker (augmentation χ).toRingHom) ^ 2 ≤ pureOldRadical χ := by
rw [Submodule.pow_succ, Submodule.pow_one]
apply Ideal.mul_le.mpr
intro x hx y hy
obtain ⟨hxc, hxe⟩ := (mem_ker_augmentation χ x).mp hx
obtain ⟨hyc, hye⟩ := (mem_ker_augmentation χ y).mp hy
change x.fst.2 * y.fst.2 = 0 ∧
newAugmentation x.fst.2 * SimpleSum.value (TriangularConstruction.offValue y.snd) +
χ y.fst.1 * SimpleSum.value (TriangularConstruction.offValue x.snd) = 0 ∧ χ (x.fst.1 * y.fst.1) = 0
refine ⟨?_, by simp [hxe, hyc], by simp [map_mul, hxc]⟩
ext i : 1
have hx0 : (x.fst.2 i).fst = 0 := congrFun hxe i
have hy0 : (y.fst.2 i).fst = 0 := congrFun hye i
apply TrivSqZeroExt.ext <;> simp [hx0, hy0]
lemma pow_le_comap_pow_noncomm {R S : Type*} [Ring R] [Ring S] (f : R →+* S)
(I : Ideal R) (J : Ideal S) (h : I ≤ J.comap f) (n : ℕ) : I ^ n ≤ (J ^ n).comap f := by
induction n with
| zero => simp [Submodule.pow_zero, Ideal.one_eq_top]
| succ n ih =>
rw [Submodule.pow_succ, Submodule.pow_succ]
apply Ideal.mul_le.mpr
intro x hx y hy
change f (x * y) ∈ J ^ n * J
rw [map_mul]
exact Ideal.mul_mem_mul (ih hx) (h hy)
lemma augmentation_ker_nilpotent (hJ : ∃ n : ℕ, (RingHom.ker χ.toRingHom) ^ n = ⊥) :
∃ n : ℕ, (RingHom.ker (augmentation χ).toRingHom) ^ n = ⊥ := by
obtain ⟨n, hn⟩ := hJ
let K := RingHom.ker (augmentation χ).toRingHom
let J := RingHom.ker χ.toRingHom
have hcomap : K ≤ J.comap (TriangularConstruction.toC C (NewCorner k ι) (SimpleSum χ)) := by
intro x hx
exact ((mem_ker_augmentation χ x).mp hx).1
have hpow := pow_le_comap_pow_noncomm _ K J hcomap (n + 2)
have hjn : J ^ (n + 2) = ⊥ := by
apply le_bot_iff.mp
exact (Ideal.pow_le_pow_right (I := J) (Nat.le_add_right n 2)).trans hn.le
rw [hjn] at hpow
refine ⟨n + 2, le_bot_iff.mp ?_⟩
intro x hx
have hc : x.fst.1 = 0 := hpow hx
have hp := ker_square_pureOldRadical χ ((Ideal.pow_le_pow_right (I := K) (by omega : 2 ≤ n + 2)) hx)
change x = 0
apply TrivSqZeroExt.ext
· exact Prod.ext hc hp.1
· exact hp.2.1
variable [DecidableEq ι]
lemma left_dimensions_zero [Finite ι] [Nontrivial C]
(hχ : Function.Surjective χ) (hJ : ∃ n : ℕ, (RingHom.ker χ.toRingHom) ^ n = ⊥) :
littleFinitisticDimension (T χ) = 0 ∧ bigFinitisticDimension (T χ) = 0 := by
let : Nontrivial (T χ) := inferInstanceAs (Nontrivial ((C × NewCorner k ι) ×
TriangularConstruction.OffDiagonal C (NewCorner k ι) (SimpleSum χ)))
exact dimensions_zero_of_splitBasic_rightSocle (augmentation χ)
(augmentation_surjective χ hχ) (augmentation_ker_nilpotent χ hJ)
(socle χ) (socle_ne_zero χ) (socle_mul χ)
end CummingsConstruction
end LittleFinitistic
end
end
end
end OAI
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