openAI-math-dataset / lean /OAI /AlgebraicGeometry /StableCoordinate /DeterminantLeadingReduction.lean
| import OAI.AlgebraicGeometry.StableCoordinate.DeterminantNonzero | |
| import OAI.AlgebraicGeometry.StableCoordinate.DeterminantDomain | |
| import OAI.AlgebraicGeometry.StableCoordinate.LeadingLocallyNilpotent | |
| import OAI.AlgebraicGeometry.StableCoordinate.FactorialClosure | |
| namespace OAI | |
| /-! | |
| # Leading locally nilpotent reduction on the actual determinant ring | |
| This is the auxiliary-grading step in the source's rigidity argument. The | |
| input is a nonzero locally nilpotent derivation on the concrete determinant | |
| quotient preserving fiber weights and killing the exact obstruction `v`. | |
| The output is a nonzero locally nilpotent derivation with the same fiber | |
| weight property fixing `a`, `d`, and `u`. All finite generation, grading, | |
| domain, characteristic, and nonvanishing premises are discharged for this | |
| quotient; none is added to the conclusion as an unproved interface. | |
| -/ | |
| noncomputable section | |
| namespace StableCoordinate.Obstruction | |
| variable (k : Type*) [Field k] [CharZero k] | |
| /-- The actual leading-component reduction in the determinant quotient. | |
| Auxiliary degrees `-1` and `2` of the two parts of `v` are kept separate, | |
| and compatibility with the second grading retains the fiber weights. | |
| Factorial closure then splits the nonzero difference-of-squares factors. | |
| -/ | |
| theorem exists_nonzero_lnd_fixing_a_d_u | |
| (D : Derivation k (C k) (C k)) | |
| (hD : D ≠ 0) (hln : IsLocallyNilpotent D) | |
| (hpres : ∀ (n : ℤ) (x : C k), x ∈ fiberPiece k n → D x ∈ fiberPiece k n) | |
| (hv : D (v k) = 0) : | |
| ∃ E : Derivation k (C k) (C k), E ≠ 0 ∧ IsLocallyNilpotent E ∧ | |
| (∀ (n : ℤ) (x : C k), x ∈ fiberPiece k n → E x ∈ fiberPiece k n) ∧ | |
| E (cX k 0) = 0 ∧ E (cX k 1) = 0 ∧ E (cX k 4) = 0 := by | |
| let : IsDomain (C k) := by | |
| change IsDomain (DeterminantDomain.DeterminantRing k) | |
| infer_instance | |
| let : CharZero (C k) := Algebra.charZero_of_charZero k (C k) | |
| have hv' : D (vLow k + vHigh k) = 0 := by | |
| rw [← v_eq_low_add_high] | |
| exact hv | |
| obtain ⟨m, hm, hnil, hcompatible, htop⟩ := | |
| LeadingComponent.exists_nonzero_locallyNilpotent_component_compatible | |
| (auxiliaryPiece k) (fiberPiece k) (auxiliary_fiber_proj_commute k) | |
| (cX k) auxiliaryWeight (adjoin_range_cX k) (cX_mem_auxiliaryPiece k) | |
| D hD hln hpres (vLow_mem_auxiliaryPiece k) (vHigh_mem_auxiliaryPiece k) | |
| (by decide : (-1 : ℤ) < 2) hv' | |
| let E := LeadingComponent.component (auxiliaryPiece k) D m | |
| have hfixed : E (cX k 0) = 0 ∧ E (cX k 1) = 0 ∧ E (cX k 4) = 0 := | |
| lnd_fixed_of_difference_of_squares E hnil | |
| (cX_a_ne_zero k) (cX_u_ne_zero k) | |
| (au_sub_d_ne_zero k) (au_add_d_ne_zero k) htop | |
| exact ⟨E, hm, hnil, hcompatible, hfixed⟩ | |
| end StableCoordinate.Obstruction | |
| end | |
| end OAI | |