| import OAI.AlgebraicGeometry.StableCoordinate.FrameLiftUniqueness |
| import OAI.AlgebraicGeometry.StableCoordinate.ObstructionDefinitions |
| import OAI.AlgebraicGeometry.StableCoordinate.BundleCharts |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| A determinant frame in a commutative `k`-algebra, together with an image of |
| `u`, gives an actual algebra homomorphism from the exact source ring |
| `k[a,d,b,c,u]/(ac-bd-1)`. This is the quotient-map construction needed when a |
| normalized polynomial frame is turned into a lifted additive action. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace StableCoordinate.FrameLift |
|
|
| open MvPolynomial |
|
|
| variable {k S T : Type*} [CommRing k] [CommRing S] [CommRing T] |
| [Algebra k S] [Algebra k T] |
|
|
| /-- Evaluation on the five independent determinant-presentation variables. -/ |
| def Frame.polynomialMap (F : Frame S) (u : S) : Obstruction.P k 5 →ₐ[k] S := |
| MvPolynomial.aeval ![F.a, F.d, F.b, F.c, u] |
|
|
| /-- Every determinant frame satisfies the exact source relation. -/ |
| theorem Frame.polynomialMap_relation (F : Frame S) (u : S) : |
| F.polynomialMap (k := k) u (Obstruction.determinantRelation k) = 0 := by |
| simpa [Frame.polynomialMap, Obstruction.determinantRelation] using |
| sub_eq_zero.mpr F.determinant |
|
|
| /-- The determinant relation generates an ideal in the kernel of evaluation. -/ |
| theorem Frame.ideal_le_ker_polynomialMap (F : Frame S) (u : S) : |
| Obstruction.idealC k ≤ RingHom.ker (F.polynomialMap (k := k) u).toRingHom := by |
| rw [Obstruction.idealC, Ideal.span_singleton_le_iff_mem] |
| exact F.polynomialMap_relation u |
|
|
| /-- The actual map out of the source determinant quotient induced by a frame. |
| The image of the fifth coordinate `u` is arbitrary and independently specified. -/ |
| def Frame.quotientMap (F : Frame S) (u : S) : Obstruction.C k →ₐ[k] S := |
| Ideal.Quotient.liftₐ (Obstruction.idealC k) (F.polynomialMap u) |
| (fun _ hp => F.ideal_le_ker_polynomialMap u hp) |
|
|
| @[simp] theorem Frame.quotientMap_cX (F : Frame S) (u : S) (i : Fin 5) : |
| F.quotientMap (k := k) u (Obstruction.cX k i) = ![F.a, F.d, F.b, F.c, u] i := by |
| change F.polynomialMap u (X i) = _ |
| simp [Frame.polynomialMap] |
|
|
| @[simp] theorem Frame.quotientMap_a (F : Frame S) (u : S) : |
| F.quotientMap (k := k) u (Obstruction.cX k 0) = F.a := by simp |
|
|
| @[simp] theorem Frame.quotientMap_d (F : Frame S) (u : S) : |
| F.quotientMap (k := k) u (Obstruction.cX k 1) = F.d := by simp |
|
|
| @[simp] theorem Frame.quotientMap_b (F : Frame S) (u : S) : |
| F.quotientMap (k := k) u (Obstruction.cX k 2) = F.b := by simp |
|
|
| @[simp] theorem Frame.quotientMap_c (F : Frame S) (u : S) : |
| F.quotientMap (k := k) u (Obstruction.cX k 3) = F.c := by simp |
|
|
| @[simp] theorem Frame.quotientMap_u (F : Frame S) (u : S) : |
| F.quotientMap (k := k) u (Obstruction.cX k 4) = u := by simp |
|
|
| /-- Maps from the concrete determinant quotient are determined by all five |
| coordinate images. -/ |
| theorem determinant_algHom_ext {φ ψ : Obstruction.C k →ₐ[k] S} |
| (h : ∀ i : Fin 5, φ (Obstruction.cX k i) = ψ (Obstruction.cX k i)) : φ = ψ := by |
| apply Ideal.Quotient.algHom_ext |
| apply MvPolynomial.algHom_ext |
| exact h |
|
|
| /-- The canonical determinant frame in the exact quotient. -/ |
| def sourceFrame (k : Type*) [CommRing k] : Frame (Obstruction.C k) where |
| a := Obstruction.cX k 0 |
| d := Obstruction.cX k 1 |
| b := Obstruction.cX k 2 |
| c := Obstruction.cX k 3 |
| determinant := sub_eq_zero.mp (Obstruction.c_relation k) |
|
|
| /-- Each algebra map gives its frame of coordinate images. -/ |
| def frameOfHom (φ : Obstruction.C k →ₐ[k] S) : Frame S := |
| (sourceFrame k).map φ.toRingHom |
|
|
| @[simp] theorem frameOfHom_a (φ : Obstruction.C k →ₐ[k] S) : |
| (frameOfHom φ).a = φ (Obstruction.cX k 0) := rfl |
| @[simp] theorem frameOfHom_d (φ : Obstruction.C k →ₐ[k] S) : |
| (frameOfHom φ).d = φ (Obstruction.cX k 1) := rfl |
| @[simp] theorem frameOfHom_b (φ : Obstruction.C k →ₐ[k] S) : |
| (frameOfHom φ).b = φ (Obstruction.cX k 2) := rfl |
| @[simp] theorem frameOfHom_c (φ : Obstruction.C k →ₐ[k] S) : |
| (frameOfHom φ).c = φ (Obstruction.cX k 3) := rfl |
|
|
| @[simp] theorem quotientMap_sourceFrame : |
| (sourceFrame k).quotientMap (Obstruction.cX k 4) = AlgHom.id k (Obstruction.C k) := by |
| apply determinant_algHom_ext |
| intro i |
| fin_cases i <;> simp [sourceFrame] |
| |
| /-- Naturality of the quotient-map construction under a target algebra map. -/ |
| theorem Frame.quotientMap_natural (F : Frame S) (u : S) (φ : S →ₐ[k] T) : |
| φ.comp (F.quotientMap u) = (F.map φ.toRingHom).quotientMap (φ u) := by |
| apply determinant_algHom_ext |
| intro i |
| fin_cases i <;> simp [Frame.map] |
| |
| @[simp] theorem quotientMap_frameOfHom (φ : Obstruction.C k →ₐ[k] S) : |
| (frameOfHom φ).quotientMap (φ (Obstruction.cX k 4)) = φ := by |
| apply determinant_algHom_ext |
| intro i |
| fin_cases i <;> simp |
| |
| /-- Equality of the determinant frame and the fifth-coordinate image gives |
| actual equality of quotient algebra homomorphisms. -/ |
| theorem determinant_algHom_ext_frame {φ ψ : Obstruction.C k →ₐ[k] S} |
| (hF : frameOfHom φ = frameOfHom ψ) |
| (hu : φ (Obstruction.cX k 4) = ψ (Obstruction.cX k 4)) : φ = ψ := by |
| rw [← quotientMap_frameOfHom φ, ← quotientMap_frameOfHom ψ, hF, hu] |
|
|
| /-- Algebra maps from the quadric quotient are determined by its four generators. -/ |
| theorem quadric_algHom_ext {φ ψ : Obstruction.B k →ₐ[k] S} |
| (h : ∀ i : Fin 4, φ (Obstruction.bX k i) = ψ (Obstruction.bX k i)) : φ = ψ := by |
| apply Ideal.Quotient.algHom_ext |
| apply MvPolynomial.algHom_ext |
| exact h |
|
|
| /-- The existing bundle base map with the canonical source quotient aliases. -/ |
| def sourceBaseMap (k : Type*) [CommRing k] : Obstruction.B k →ₐ[k] Obstruction.C k := |
| BundleCharts.baseMap k |
|
|
| /-- The already-constructed base map, expressed in the canonical source names. -/ |
| @[simp] theorem baseMap_bX (i : Fin 4) : |
| sourceBaseMap k (Obstruction.bX k i) = |
| ![Obstruction.cX k 0 * Obstruction.cX k 2, |
| Obstruction.cX k 1 * Obstruction.cX k 3, |
| Obstruction.cX k 1 * Obstruction.cX k 2, Obstruction.cX k 4] i := |
| BundleCharts.baseMap_qgen k i |
|
|
| /-- A frame lying over the four images of a base map gives an actual extension |
| of that base map to the determinant quotient. -/ |
| theorem Frame.quotientMap_comp_baseMap (F : Frame S) (u : S) |
| (φ : Obstruction.B k →ₐ[k] S) |
| (hx : F.a * F.b = φ (Obstruction.bX k 0)) |
| (hy : F.d * F.c = φ (Obstruction.bX k 1)) |
| (hz : F.d * F.b = φ (Obstruction.bX k 2)) |
| (hu : u = φ (Obstruction.bX k 3)) : |
| (F.quotientMap u).comp (sourceBaseMap k) = φ := by |
| apply quadric_algHom_ext |
| intro i |
| fin_cases i |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hx |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hy |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hz |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hu |
|
|
| /-- Restriction to the source base records exactly the three frame products |
| and the independent coordinate `u`. -/ |
| theorem comp_baseMap_eq_iff {φ ψ : Obstruction.C k →ₐ[k] S} : |
| φ.comp (sourceBaseMap k) = ψ.comp (sourceBaseMap k) ↔ |
| SameBase (frameOfHom φ) (frameOfHom ψ) ∧ |
| φ (Obstruction.cX k 4) = ψ (Obstruction.cX k 4) := by |
| constructor |
| · intro h |
| refine ⟨⟨?_, ?_, ?_⟩, ?_⟩ |
| · simpa [AlgHom.comp_apply, baseMap_bX] using AlgHom.congr_fun h (Obstruction.bX k 0) |
| · simpa [AlgHom.comp_apply, baseMap_bX] using AlgHom.congr_fun h (Obstruction.bX k 1) |
| · simpa [AlgHom.comp_apply, baseMap_bX] using AlgHom.congr_fun h (Obstruction.bX k 2) |
| · simpa [AlgHom.comp_apply, baseMap_bX] using AlgHom.congr_fun h (Obstruction.bX k 3) |
| · rintro ⟨⟨hx, hy, hz⟩, hu⟩ |
| apply quadric_algHom_ext |
| intro i |
| fin_cases i |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hx |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hy |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hz |
| · simpa [AlgHom.comp_apply, baseMap_bX] using hu |
|
|
| /-- Normalized polynomial maps of the determinant quotient are unique over |
| the same base map. This is an actual equality of algebra homomorphisms. -/ |
| theorem normalized_quotientMap_unique {R σ : Type*} [CommRing R] [Algebra k R] |
| [IsReduced R] {φ ψ : Obstruction.C k →ₐ[k] MvPolynomial σ R} |
| (hbase : φ.comp (sourceBaseMap k) = ψ.comp (sourceBaseMap k)) |
| (hzero : (frameOfHom φ).map MvPolynomial.constantCoeff = |
| (frameOfHom ψ).map MvPolynomial.constantCoeff) : φ = ψ := by |
| obtain ⟨hF, hu⟩ := comp_baseMap_eq_iff.mp hbase |
| exact determinant_algHom_ext_frame (normalized_polynomial_unique hF hzero) hu |
|
|
| end StableCoordinate.FrameLift |
|
|
| end |
|
|
| end OAI |
|
|
