| 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 |
|
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| variable [DecidableEq ι] |
|
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| 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 |
|
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| end OAI |
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