openAI-math-dataset / lean /OAI /AlgebraicGeometry /CharacterVarieties /Foundation /MatrixPresentation.lean
| import OAI.AlgebraicGeometry.CharacterVarieties.Foundation.Flags | |
| import Mathlib.RingTheory.MvPolynomial.Basic | |
| import Mathlib.RingTheory.Ideal.Quotient.Operations | |
| import Mathlib.RingTheory.Finiteness.Basic | |
| namespace OAI | |
| noncomputable section | |
| open scoped Classical Matrix | |
| namespace IntegralCharacterVarieties.FiniteMatrixPresentation | |
| variable (R : Type*) [CommRing R] (E : Type*) (ι : E → Type*) | |
| variable [∀ e, Fintype (ι e)] [∀ e, DecidableEq (ι e)] | |
| /-- Every transport and every chosen split frame has its own occurrence index. The sign distinguishes its matrix from its inverse, not a second occurrence. -/ | |
| abbrev Entry := (e : E) × Bool × (ι e × ι e) | |
| abbrev Polynomial := MvPolynomial (Entry E ι) R | |
| def variableMatrix (e : E) (b : Bool) : Matrix (ι e) (ι e) (Polynomial R E ι) := | |
| fun i j => MvPolynomial.X ⟨e,b,i,j⟩ | |
| variable {J : Type*} (equation : J → Polynomial R E ι) | |
| /-- The inverse equations are imposed in both directions over the full ring. Every incidence equation is included individually, even for repeated sides. -/ | |
| inductive Relation : Polynomial R E ι → Prop | |
| | inverse (e : E) (b : Bool) (i j : ι e) : Relation | |
| ((variableMatrix R E ι e b * variableMatrix R E ι e (!b)) i j - | |
| (1 : Matrix (ι e) (ι e) (Polynomial R E ι)) i j) | |
| | incidence (j : J) : Relation (equation j) | |
| def relationIdeal : Ideal (Polynomial R E ι) := | |
| Ideal.span {f | Relation R E ι equation f} | |
| abbrev Coordinate := Polynomial R E ι ⧸ relationIdeal R E ι equation | |
| variable {R E ι} | |
| def universalMatrix (e : E) (b : Bool) : Matrix (ι e) (ι e) (Coordinate R E ι equation) := | |
| (variableMatrix R E ι e b).map (Ideal.Quotient.mk (relationIdeal R E ι equation)) | |
| lemma relation_zero (f : Polynomial R E ι) (hf : Relation R E ι equation f) : | |
| Ideal.Quotient.mk (relationIdeal R E ι equation) f = 0 := | |
| Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span hf) | |
| lemma universal_inverse (e : E) (b : Bool) : | |
| universalMatrix equation e b * universalMatrix equation e (!b) = 1 := by | |
| ext i j | |
| have h := relation_zero equation _ (Relation.inverse e b i j) | |
| rw [map_sub,sub_eq_zero] at h | |
| change ((variableMatrix R E ι e b * variableMatrix R E ι e (!b)).map | |
| (Ideal.Quotient.mk (relationIdeal R E ι equation))) i j = | |
| ((1 : Matrix (ι e) (ι e) (Polynomial R E ι)).map | |
| (Ideal.Quotient.mk (relationIdeal R E ι equation))) i j at h | |
| rw [Matrix.map_mul,Matrix.map_one _ (map_zero _) (map_one _)] at h | |
| exact h | |
| /-- These are general linear transports, not only matrices with nonvanishing determinant at complex points. -/ | |
| def universalUnit (e : E) : (Matrix (ι e) (ι e) (Coordinate R E ι equation))ˣ where | |
| val := universalMatrix equation e false | |
| inv := universalMatrix equation e true | |
| val_inv := universal_inverse equation e false | |
| inv_val := universal_inverse equation e true | |
| variable {S : Type*} [CommRing S] [Algebra R S] | |
| /-- Evaluation retains every occurrence's forward and backward transport. -/ | |
| def matrixEvaluation (g : (e : E) → (Matrix (ι e) (ι e) S)ˣ) : | |
| Polynomial R E ι →ₐ[R] S := | |
| MvPolynomial.aeval (fun ⟨e,b,i,j⟩ => if b then (↑(g e)⁻¹ : Matrix _ _ S) i j | |
| else (↑(g e) : Matrix _ _ S) i j) | |
| @[simp] lemma matrixEvaluation_variable (g : (e : E) → (Matrix (ι e) (ι e) S)ˣ) | |
| (e : E) (b : Bool) : | |
| (variableMatrix R E ι e b).map (matrixEvaluation g).toRingHom = | |
| if b then (↑(g e)⁻¹ : Matrix _ _ S) else (↑(g e) : Matrix _ _ S) := by | |
| ext i j | |
| cases b <;> simp [variableMatrix,matrixEvaluation] | |
| /-- A split-frame solution of the finite incidence equations over S. -/ | |
| abbrev Solution (S : Type*) [CommRing S] [Algebra R S] := | |
| {g : (e : E) → (Matrix (ι e) (ι e) S)ˣ // ∀ j, matrixEvaluation g (equation j)=0} | |
| lemma ideal_le_ker (g : Solution equation S) : | |
| relationIdeal R E ι equation ≤ RingHom.ker (matrixEvaluation g.val).toRingHom := by | |
| apply Ideal.span_le.mpr | |
| intro f hf | |
| change matrixEvaluation g.val f=0 | |
| cases hf with | |
| | incidence j => exact g.property j | |
| | inverse e b i j => | |
| rw [map_sub,sub_eq_zero] | |
| have hinv : ((variableMatrix R E ι e b * variableMatrix R E ι e (!b)).map | |
| (matrixEvaluation g.val).toRingHom) = 1 := by | |
| rw [Matrix.map_mul,matrixEvaluation_variable,matrixEvaluation_variable] | |
| cases b <;> simp | |
| have he := congrFun (congrFun hinv i) j | |
| change matrixEvaluation g.val | |
| ((variableMatrix R E ι e b * variableMatrix R E ι e (!b)) i j) = (1 : Matrix _ _ S) i j at he | |
| rw [he] | |
| by_cases hij : i=j | |
| · subst j | |
| simp only [Matrix.one_apply_eq,map_one] | |
| · simp only [Matrix.one_apply_ne hij,map_zero] | |
| def solutionToPoint (g : Solution equation S) : Coordinate R E ι equation →ₐ[R] S := | |
| Ideal.Quotient.liftₐ (relationIdeal R E ι equation) (matrixEvaluation g.val) (ideal_le_ker equation g) | |
| @[simp] lemma solutionToPoint_mk (g : Solution equation S) (f : Polynomial R E ι) : | |
| solutionToPoint equation g (Ideal.Quotient.mk (relationIdeal R E ι equation) f) = | |
| matrixEvaluation g.val f := rfl | |
| /-- Reading the universal transports along an ring-valued point. -/ | |
| def pointMatrices (φ : Coordinate R E ι equation →ₐ[R] S) (e : E) : | |
| (Matrix (ι e) (ι e) S)ˣ := | |
| Units.map (RingHom.mapMatrix φ.toRingHom).toMonoidHom (universalUnit equation e) | |
| @[simp] lemma pointMatrices_val (φ : Coordinate R E ι equation →ₐ[R] S) (e : E) : | |
| (↑(pointMatrices equation φ e) : Matrix _ _ S) = | |
| (universalMatrix equation e false).map φ.toRingHom := rfl | |
| @[simp] lemma pointMatrices_inv (φ : Coordinate R E ι equation →ₐ[R] S) (e : E) : | |
| (↑(pointMatrices equation φ e)⁻¹ : Matrix _ _ S) = | |
| (universalMatrix equation e true).map φ.toRingHom := rfl | |
| lemma point_evaluation (φ : Coordinate R E ι equation →ₐ[R] S) : | |
| matrixEvaluation (pointMatrices equation φ) = | |
| φ.comp (Ideal.Quotient.mkₐ R (relationIdeal R E ι equation)) := by | |
| apply MvPolynomial.algHom_ext | |
| rintro ⟨e,b,i,j⟩ | |
| simp only [matrixEvaluation,MvPolynomial.aeval_X] | |
| cases b <;> rfl | |
| def pointToSolution (φ : Coordinate R E ι equation →ₐ[R] S) : Solution equation S := | |
| ⟨pointMatrices equation φ,by | |
| intro j | |
| rw [point_evaluation] | |
| change φ (Ideal.Quotient.mk (relationIdeal R E ι equation) (equation j))=0 | |
| rw [relation_zero equation _ (Relation.incidence j),map_zero]⟩ | |
| /-- Exact representability over arbitrary coefficient rings, including the full ring of integers. No reduction, localization or rank condition is dropped. -/ | |
| def pointsEquiv : Solution equation S ≃ (Coordinate R E ι equation →ₐ[R] S) where | |
| toFun := solutionToPoint equation | |
| invFun := pointToSolution equation | |
| left_inv g := by | |
| apply Subtype.ext | |
| funext e | |
| apply Units.ext | |
| ext i j | |
| change matrixEvaluation g.val (MvPolynomial.X ⟨e,false,i,j⟩) = _ | |
| simp [matrixEvaluation] | |
| right_inv φ := by | |
| apply Ideal.Quotient.algHom_ext | |
| rw [show (solutionToPoint equation (pointToSolution equation φ)).comp | |
| (Ideal.Quotient.mkₐ R (relationIdeal R E ι equation)) = | |
| matrixEvaluation (pointMatrices equation φ) from rfl] | |
| exact point_evaluation equation φ | |
| /-- There are finitely many matrix entries when the diagram is finite. -/ | |
| instance finiteEntry [Finite E] [∀ e, Finite (ι e)] : Finite (Entry E ι) := inferInstance | |
| instance finiteType [Finite E] [∀ e, Finite (ι e)] : | |
| Algebra.FiniteType R (Coordinate R E ι equation) := by | |
| infer_instance | |
| end IntegralCharacterVarieties.FiniteMatrixPresentation | |
| end | |
| end OAI | |