| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.Differentiation |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| |
| namespace BoundaryOnly.FormalObstruction.AlgebraicReplacement |
| |
| theorem sup_pow_pred_le {R : Type*} [CommRing R] (I J : Ideal R) |
| (n q : ℕ) (hq : 1 ≤ q) : (I ⊔ J) ^ (n + q - 1) ≤ I ^ n ⊔ J ^ q := by |
| rw [← Ideal.add_eq_sup, ← Ideal.add_eq_sup, add_pow, Ideal.sum_eq_sup] |
| apply Finset.sup_le |
| intro i hi |
| by_cases hn : n ≤ i |
| · exact (Ideal.mul_le_left.trans (Ideal.mul_le_left.trans |
| ((Ideal.pow_le_pow_right hn).trans le_sup_left))) |
| · refine (Ideal.mul_le_left.trans (Ideal.mul_le_right.trans |
| ((Ideal.pow_le_pow_right ?_).trans le_sup_right))) |
| omega |
|
|
| theorem shifted_ideal_cofinal {R : Type*} [CommRing R] |
| (J H S : Ideal R) (q : ℕ) (hq : 1 ≤ q) (hS : S ^ q = ⊥) |
| (hs : J ⊔ S = H ⊔ S) (N : ℕ) : |
| J ^ (N + q - 1) ≤ H ^ N ∧ H ^ (N + q - 1) ≤ J ^ N := by |
| constructor |
| · calc |
| J ^ (N + q - 1) ≤ (J ⊔ S) ^ (N + q - 1) := pow_le_pow_left' le_sup_left _ |
| _ = (H ⊔ S) ^ (N + q - 1) := by rw [hs] |
| _ ≤ H ^ N ⊔ S ^ q := sup_pow_pred_le H S N q hq |
| _ = H ^ N := by rw [hS,sup_bot_eq] |
| · calc |
| H ^ (N + q - 1) ≤ (H ⊔ S) ^ (N + q - 1) := pow_le_pow_left' le_sup_left _ |
| _ = (J ⊔ S) ^ (N + q - 1) := by rw [hs] |
| _ ≤ J ^ N ⊔ S ^ q := sup_pow_pred_le J S N q hq |
| _ = J ^ N := by rw [hS,sup_bot_eq] |
|
|
| namespace TaylorTarget |
| variable (A α : Type*) [CommRing A] |
|
|
| abbrev variablesIdeal : Ideal (MvPolynomial α A) := MvPolynomial.idealOfVars α A |
| abbrev Ring (q : ℕ) := MvPolynomial α A ⧸ variablesIdeal A α ^ q |
|
|
| def mk (q : ℕ) : MvPolynomial α A →+* Ring A α q := Ideal.Quotient.mk _ |
|
|
| def nilIdeal (q : ℕ) : Ideal (Ring A α q) := (variablesIdeal A α).map (mk A α q) |
|
|
| lemma variables_pow (q : ℕ) : nilIdeal A α q ^ q = ⊥ := by |
| rw [nilIdeal,← Ideal.map_pow] |
| exact Ideal.map_quotient_self _ |
|
|
| lemma variables_fg [Finite α] (q : ℕ) : (nilIdeal A α q).FG := |
| (MvPolynomial.idealOfVars_fg α A).map _ |
|
|
| lemma variables_ker : variablesIdeal A α = RingHom.ker MvPolynomial.constantCoeff := by |
| apply le_antisymm |
| · rw [Ideal.span_le] |
| rintro _ ⟨i,rfl⟩ |
| simp |
| · intro p hp |
| rw [← pow_one (variablesIdeal A α),MvPolynomial.mem_pow_idealOfVars_iff] |
| intro e he |
| have he0 : e ≠ 0 := by |
| intro h |
| subst e |
| exact (MvPolynomial.mem_support_iff.mp he) hp |
| exact Nat.one_le_iff_ne_zero.mpr (by simpa [Finsupp.degree_eq_zero_iff] using he0) |
|
|
| def reduction (q : ℕ) (hq : 1 ≤ q) : Ring A α q →+* A := |
| Ideal.Quotient.lift _ MvPolynomial.constantCoeff (by |
| change variablesIdeal A α ^ q ≤ RingHom.ker MvPolynomial.constantCoeff |
| rw [← variables_ker] |
| exact Ideal.pow_le_self (by omega)) |
|
|
| @[simp] lemma reduction_mk (q : ℕ) (hq : 1 ≤ q) (p : MvPolynomial α A) : |
| reduction A α q hq (mk A α q p) = MvPolynomial.constantCoeff p := rfl |
|
|
| lemma reduction_surjective (q : ℕ) (hq : 1 ≤ q) : |
| Function.Surjective (reduction A α q hq) := fun a ↦ |
| ⟨mk A α q (MvPolynomial.C a),by simp⟩ |
|
|
| lemma reduction_ker (q : ℕ) (hq : 1 ≤ q) : |
| RingHom.ker (reduction A α q hq) = nilIdeal A α q := by |
| apply Ideal.comap_injective_of_surjective (mk A α q) (Ideal.Quotient.mk_surjective) |
| rw [RingHom.comap_ker] |
| change RingHom.ker MvPolynomial.constantCoeff = _ |
| rw [← variables_ker,nilIdeal] |
| unfold mk |
| rw [Ideal.comap_map_of_surjective _ (Ideal.Quotient.mk_surjective)] |
| change _ = variablesIdeal A α ⊔ _ |
| rw [← RingHom.ker_eq_comap_bot,Ideal.mk_ker] |
| exact (sup_eq_left.mpr (Ideal.pow_le_self (by omega))).symm |
|
|
| theorem finite_shifted (R : Type*) [CommRing R] [Algebra R A] |
| [Module.Finite R A] [Finite α] (q : ℕ) (hq : 1 ≤ q) |
| [halg : Algebra R (Ring A α q)] |
| (hred : ∀ r : R, reduction A α q hq (algebraMap R (Ring A α q) r) = |
| algebraMap R A r) : |
| @Module.Finite R (Ring A α q) _ _ halg.toModule := by |
| let f : Ring A α q →ₐ[R] A := |
| { reduction A α q hq with commutes' := hred } |
| apply Module.finite_of_surjective_of_ker_le_nilradical f |
| (reduction_surjective A α q hq) |
| · change RingHom.ker (reduction A α q hq) ≤ _ |
| rw [reduction_ker] |
| exact (variables_fg A α q).isNilpotent_iff_le_nilradical.mp ⟨q,variables_pow A α q⟩ |
| · change (RingHom.ker (reduction A α q hq)).FG |
| rw [reduction_ker] |
| exact variables_fg A α q |
| |
| end TaylorTarget |
| end BoundaryOnly.FormalObstruction.AlgebraicReplacement |
| |
| end |
| |
| end OAI |
| |
