| import OAI.AlgebraicGeometry.StableCoordinate.CoactionBaseNaturality |
| import OAI.AlgebraicGeometry.StableCoordinate.FrameCoassociativity |
| import OAI.AlgebraicGeometry.StableCoordinate.ObstructionDefinitions |
| import OAI.AlgebraicGeometry.StableCoordinate.BundleCharts |
|
|
| namespace OAI |
|
|
| /-! |
| # The action law for a normalized lift to the determinant algebra |
|
|
| The polynomial family constructed from a normalized determinant frame is not |
| assumed to be an action. This file proves its additive group law from that of |
| the base family, using the uniqueness of normalized frames. |
| -/ |
|
|
| noncomputable section |
| namespace StableCoordinate.FrameLiftCoassoc |
|
|
| open Polynomial |
| open CoactionLND |
|
|
| variable {k B A : Type*} [CommRing k] [CommRing B] [CommRing A] |
| [Algebra k B] [Algebra k A] |
|
|
| / |
| def iterateMap (β : A →ₐ[k] A[X]) : A →ₐ[k] A[X][X] := |
| (Polynomial.mapAlgHom β).comp β |
|
|
| / |
| def sumParameterMap (β : A →ₐ[k] A[X]) : A →ₐ[k] A[X][X] := |
| ((Polynomial.taylorAlgHom (X : A[X])).restrictScalars k).comp |
| ((Polynomial.mapAlgHom (Polynomial.CAlgHom : A →ₐ[k] A[X])).comp β) |
|
|
| @[simp] theorem iterateMap_apply (β : A →ₐ[k] A[X]) (r : A) : |
| iterateMap β r = (β r).map β.toRingHom := rfl |
|
|
| @[simp] theorem sumParameterMap_apply (β : A →ₐ[k] A[X]) (r : A) : |
| sumParameterMap β r = |
| taylor (X : A[X]) ((β r).map (C : A →+* A[X])) := rfl |
|
|
| / |
| quotient generators, and not merely on their values at complex points. -/ |
| theorem coassoc_of_generators |
| (β : Obstruction.C k →ₐ[k] (Obstruction.C k)[X]) |
| (h : ∀ i : Fin 5, |
| (β (Obstruction.cX k i)).map β.toRingHom = |
| taylor (X : (Obstruction.C k)[X]) |
| ((β (Obstruction.cX k i)).map (C : Obstruction.C k →+* (Obstruction.C k)[X]))) : |
| ∀ r, (β r).map β.toRingHom = |
| taylor (X : (Obstruction.C k)[X]) ((β r).map (C : Obstruction.C k →+* (Obstruction.C k)[X])) := by |
| have heq : iterateMap β = sumParameterMap β := by |
| apply Ideal.Quotient.algHom_ext |
| apply MvPolynomial.algHom_ext |
| intro i |
| change iterateMap β (Obstruction.cX k i) = sumParameterMap β (Obstruction.cX k i) |
| exact h i |
| intro r |
| exact AlgHom.congr_fun heq r |
|
|
| / |
| additive action automatically satisfies the complete bivariate action law. -/ |
| theorem coassoc_of_normalized_extension |
| [IsReduced (Obstruction.C k)] |
| (α : PolynomialAdditiveAction k (Obstruction.B k)) |
| (β : Obstruction.C k →ₐ[k] (Obstruction.C k)[X]) |
| (hidentity : ∀ r, (β r).coeff 0 = r) |
| (hbase : ∀ b, β (BundleCharts.baseMap k b) = |
| (α.toAlgHom b).map (BundleCharts.baseMap k).toRingHom) : |
| ∀ r, (β r).map β.toRingHom = |
| taylor (X : (Obstruction.C k)[X]) |
| ((β r).map (C : Obstruction.C k →+* (Obstruction.C k)[X])) := by |
| let F : FrameLift.Frame (Obstruction.C k) := |
| ⟨Obstruction.cX k 0, Obstruction.cX k 1, |
| Obstruction.cX k 2, Obstruction.cX k 3, |
| sub_eq_zero.mp (Obstruction.c_relation k)⟩ |
| have hcoord (i : Fin 4) : FrameLift.CoassociativeAt β.toRingHom |
| (BundleCharts.baseMap k (Obstruction.bX k i)) := |
| coassoc_on_image_of_extension α β (BundleCharts.baseMap k) hbase (Obstruction.bX k i) |
| have hx : BundleCharts.baseMap k (Obstruction.bX k 0) = |
| Obstruction.cX k 0 * Obstruction.cX k 2 := by |
| change BundleCharts.baseMap k (BundleCharts.qgen k 0) = BundleCharts.dgen k 0 * BundleCharts.dgen k 2 |
| exact BundleCharts.baseMap_qgen k 0 |
| have hy : BundleCharts.baseMap k (Obstruction.bX k 1) = |
| Obstruction.cX k 1 * Obstruction.cX k 3 := by |
| change BundleCharts.baseMap k (BundleCharts.qgen k 1) = BundleCharts.dgen k 1 * BundleCharts.dgen k 3 |
| exact BundleCharts.baseMap_qgen k 1 |
| have hz : BundleCharts.baseMap k (Obstruction.bX k 2) = |
| Obstruction.cX k 1 * Obstruction.cX k 2 := by |
| change BundleCharts.baseMap k (BundleCharts.qgen k 2) = BundleCharts.dgen k 1 * BundleCharts.dgen k 2 |
| exact BundleCharts.baseMap_qgen k 2 |
| have hu : BundleCharts.baseMap k (Obstruction.bX k 3) = |
| Obstruction.cX k 4 := by |
| change BundleCharts.baseMap k (BundleCharts.qgen k 3) = BundleCharts.dgen k 4 |
| exact BundleCharts.baseMap_qgen k 3 |
| have hab : FrameLift.CoassociativeAt β.toRingHom (F.a * F.b) := by |
| simpa only [hx] using hcoord 0 |
| have hdc : FrameLift.CoassociativeAt β.toRingHom (F.d * F.c) := by |
| simpa only [hy] using hcoord 1 |
| have hdb : FrameLift.CoassociativeAt β.toRingHom (F.d * F.b) := by |
| simpa only [hz] using hcoord 2 |
| have hF := FrameLift.coassociative_on_frame_entries β.toRingHom hidentity F |
| ⟨hab, hdc, hdb⟩ |
| apply coassoc_of_generators β |
| intro i |
| fin_cases i |
| · exact hF.1 |
| · exact hF.2.1 |
| · exact hF.2.2.1 |
| · exact hF.2.2.2 |
| · change FrameLift.CoassociativeAt β.toRingHom (Obstruction.cX k 4) |
| simpa only [hu] using hcoord 3 |
|
|
| / |
| The family is supplied by the separately proved determinant-frame construction; |
| its group law is proved here, not taken as an assumption. -/ |
| def actionOfNormalizedExtension |
| [IsReduced (Obstruction.C k)] |
| (α : PolynomialAdditiveAction k (Obstruction.B k)) |
| (β : Obstruction.C k →ₐ[k] (Obstruction.C k)[X]) |
| (hidentity : ∀ r, (β r).coeff 0 = r) |
| (hbase : ∀ b, β (BundleCharts.baseMap k b) = |
| (α.toAlgHom b).map (BundleCharts.baseMap k).toRingHom) : |
| PolynomialAdditiveAction k (Obstruction.C k) where |
| toAlgHom := β |
| identity := hidentity |
| coassoc := coassoc_of_normalized_extension α β hidentity hbase |
|
|
| @[simp] theorem actionOfNormalizedExtension_toAlgHom |
| [IsReduced (Obstruction.C k)] |
| (α : PolynomialAdditiveAction k (Obstruction.B k)) |
| (β : Obstruction.C k →ₐ[k] (Obstruction.C k)[X]) |
| (hidentity : ∀ r, (β r).coeff 0 = r) |
| (hbase : ∀ b, β (BundleCharts.baseMap k b) = |
| (α.toAlgHom b).map (BundleCharts.baseMap k).toRingHom) : |
| (actionOfNormalizedExtension α β hidentity hbase).toAlgHom = β := rfl |
|
|
| end StableCoordinate.FrameLiftCoassoc |
|
|
| end |
|
|
| end OAI |
|
|
