| import Mathlib |
| import OAI.Algebra.FiniteTensor.FiniteObstruction |
|
|
| namespace OAI |
|
|
| /-! Complementary graph matrices and tangent invertibility. -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace PD4Tensor.FiniteCoordinates |
| noncomputable section |
| variable (R υ σ τ : Type*) [CommRing R] (p : ℕ) (e : υ ≃ σ ⊕ τ) |
|
|
| |
| def planeRestriction (hp : 0<p) : |
| Ring R υ (fun _ => p) →ₐ[R] Ring R σ (fun _ => p) := |
| (uniformLeft R σ τ p hp).comp (reindexEquiv R υ (σ ⊕ τ) p e).toAlgHom |
|
|
| @[simp] theorem planeRestriction_coord (hp : 0<p) (i : υ) : |
| planeRestriction R υ σ τ p e hp (coord R υ (fun _ => p) i)= |
| Sum.elim (coord R σ (fun _ => p)) (fun _ => 0) (e i) := by |
| simp only [planeRestriction,AlgHom.comp_apply,AlgEquiv.coe_toAlgHom, |
| reindexEquiv_coord,uniformLeft_coord] |
|
|
| |
| variable [Fintype σ] [Fintype τ] |
|
|
| def combinedGraph (H : υ → MvPowerSeries σ R) (G : υ → MvPowerSeries τ R) : |
| υ → MvPowerSeries υ R := |
| fun i => MvPowerSeries.rename (fun j => e.symm (Sum.inl j)) (H i) + |
| MvPowerSeries.rename (fun j => e.symm (Sum.inr j)) (G i) |
|
|
| |
|
|
| theorem combinedGraph_eval_left [Fintype υ] (hp : 0<p) |
| (H : υ → MvPowerSeries σ R) (G : υ → MvPowerSeries τ R) |
| (hG : ∀ i, MvPowerSeries.constantCoeff (G i)=0) (i : υ) : |
| planeRestriction R υ σ τ p e hp |
| (nilEval R (coord R υ (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) |
| (combinedGraph R υ σ τ e H G i)) = |
| nilEval R (coord R σ (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) (H i) := by |
| have hh := nilEval_map (planeRestriction R υ σ τ p e hp) |
| (coord R υ (fun _ => p)) (fun j => (show IsNilpotent (coord R υ (fun _ => p) j) from |
| ⟨p,coord_pow _ _ _ j⟩)) (combinedGraph R υ σ τ e H G i) |
| rw [hh] |
| simp only [combinedGraph,map_add,nilEval_rename,Function.comp_def, |
| planeRestriction_coord,Equiv.apply_symm_apply,Sum.elim_inl,Sum.elim_inr] |
| rw [nilEval_zero,hG,map_zero,add_zero] |
|
|
| end |
| end PD4Tensor.FiniteCoordinates |
|
|
| namespace PD4Tensor |
| noncomputable section |
| open scoped BigOperators |
| variable {K υ σ τ : Type*} [Field K] [Fintype υ] |
| [Fintype σ] [Fintype τ] |
|
|
| def joinedMatrix (e : υ ≃ σ ⊕ τ) (L : Matrix υ σ K) (R : Matrix υ τ K) : Matrix υ υ K := |
| fun i j => Sum.elim (L i) (R i) (e j) |
|
|
| theorem joinedMatrix_mulVec (e : υ ≃ σ ⊕ τ) (L : Matrix υ σ K) (R : Matrix υ τ K) |
| (v : υ → K) : |
| (joinedMatrix e L R).mulVec v= |
| L.mulVec (fun j => v (e.symm (Sum.inl j))) + R.mulVec (fun j => v (e.symm (Sum.inr j))) := by |
| ext i |
| simp only [Matrix.mulVec,dotProduct,joinedMatrix,Pi.add_apply] |
| rw [←e.symm.sum_comp] |
| simp only [Equiv.apply_symm_apply,Fintype.sum_sum_type,Sum.elim_inl,Sum.elim_inr] |
|
|
| |
|
|
| theorem joinedMatrix_det_ne_zero [DecidableEq υ] |
| (e : υ ≃ σ ⊕ τ) (L : Matrix υ σ K) (R : Matrix υ τ K) |
| (hL : Function.Injective L.mulVec) (hR : Function.Injective R.mulVec) |
| (hcomp : IsCompl (Matrix.mulVecLin L).range (Matrix.mulVecLin R).range) : |
| (joinedMatrix e L R).det≠0 := by |
| suffices hi : Function.Injective (joinedMatrix e L R).mulVec by |
| exact ((Matrix.isUnit_iff_isUnit_det _).mp |
| (Matrix.mulVec_injective_iff_isUnit.mp hi)).ne_zero |
| change Function.Injective (Matrix.mulVecLin (joinedMatrix e L R)) |
| apply LinearMap.ker_eq_bot.mp |
| apply LinearMap.ker_eq_bot'.mpr |
| intro v hv |
| change (joinedMatrix e L R).mulVec v=0 at hv |
| rw [joinedMatrix_mulVec] at hv |
| have hzero : L.mulVec (fun j => v (e.symm (Sum.inl j)))=0 := by |
| apply (Submodule.disjoint_def.mp hcomp.disjoint) |
| · exact ⟨fun j => v (e.symm (Sum.inl j)),rfl⟩ |
| · refine ⟨-(fun j => v (e.symm (Sum.inr j))),?_⟩ |
| rw [map_neg] |
| exact (eq_neg_of_add_eq_zero_left hv).symm |
| have hzL : (fun j => v (e.symm (Sum.inl j)))=0 := |
| hL (by simpa using hzero) |
| have hzR : (fun j => v (e.symm (Sum.inr j)))=0 := hR (by simpa [hzero] using hv) |
| ext i |
| obtain ⟨j,rfl⟩ := e.symm.surjective i |
| cases j with |
| | inl j => exact congr_fun hzL j |
| | inr j => exact congr_fun hzR j |
| |
| end |
| end PD4Tensor |
| |
| namespace PD4Tensor |
| noncomputable section |
| variable {R σ τ : Type*} [CommRing R] |
| |
| @[simp] theorem coeff_single_rename_embedding (e : σ ↪ τ) (f : MvPowerSeries σ R) (i : σ) : |
| MvPowerSeries.coeff (Finsupp.single (e i) 1) (MvPowerSeries.rename e f)= |
| MvPowerSeries.coeff (Finsupp.single i 1) f := by |
| simpa only [Finsupp.embDomain_single] using |
| (MvPowerSeries.coeff_embDomain_rename e f (Finsupp.single i 1)) |
| |
| theorem coeff_single_rename_outside [Fintype σ] (e : σ → τ) (f : MvPowerSeries σ R) |
| (i : τ) (hi : i∉Set.range e) : |
| MvPowerSeries.coeff (Finsupp.single i 1) (MvPowerSeries.rename e f)=0 := by |
| classical |
| apply MvPowerSeries.coeff_rename_eq_zero |
| rintro ⟨a,ha⟩ |
| apply hi |
| apply Finsupp.mem_range_of_mapDomain_ne_zero (x:=a) |
| rw [ha,Finsupp.single_eq_same] |
| exact one_ne_zero |
| |
| end |
| end PD4Tensor |
| |
| namespace PD4Tensor.FiniteCoordinates |
| noncomputable section |
| variable (R υ σ τ : Type*) [CommRing R] |
| [Fintype σ] [Fintype τ] |
| (e : υ ≃ σ ⊕ τ) |
| |
| @[simp] theorem combinedGraph_constantCoeff |
| (H : υ → MvPowerSeries σ R) (G : υ → MvPowerSeries τ R) (i : υ) : |
| MvPowerSeries.constantCoeff (combinedGraph R υ σ τ e H G i)= |
| MvPowerSeries.constantCoeff (H i)+MvPowerSeries.constantCoeff (G i) := by |
| simp [combinedGraph] |
| |
| theorem combinedGraph_coeff (H : υ → MvPowerSeries σ R) (G : υ → MvPowerSeries τ R) |
| (i j : υ) : |
| MvPowerSeries.coeff (Finsupp.single j 1) (combinedGraph R υ σ τ e H G i)= |
| Sum.elim (fun k => MvPowerSeries.coeff (Finsupp.single k 1) (H i)) |
| (fun k => MvPowerSeries.coeff (Finsupp.single k 1) (G i)) (e j) := by |
| let el : σ ↪ υ := ⟨fun j => e.symm (Sum.inl j),e.symm.injective.comp Sum.inl_injective⟩ |
| let er : τ ↪ υ := ⟨fun j => e.symm (Sum.inr j),e.symm.injective.comp Sum.inr_injective⟩ |
| obtain ⟨j,rfl⟩ := e.symm.surjective j |
| cases j with |
| | inl j => |
| change MvPowerSeries.coeff (Finsupp.single (el j) 1) |
| (MvPowerSeries.rename el (H i)+MvPowerSeries.rename er (G i))=_ |
| rw [map_add,coeff_single_rename_embedding] |
| rw [coeff_single_rename_outside er _ (el j) (by |
| rintro ⟨k,hk⟩ |
| have hh := e.symm.injective hk |
| cases hh)] |
| simp |
| | inr j => |
| change MvPowerSeries.coeff (Finsupp.single (er j) 1) |
| (MvPowerSeries.rename el (H i)+MvPowerSeries.rename er (G i))=_ |
| rw [map_add,coeff_single_rename_embedding] |
| rw [coeff_single_rename_outside el _ (er j) (by |
| rintro ⟨k,hk⟩ |
| have hh := e.symm.injective hk |
| cases hh)] |
| simp |
| |
| variable {K : Type*} [Field K] [Fintype υ] [DecidableEq υ] |
| |
| theorem combinedGraph_central_tangent |
| (a : R →+* K) (H : υ → MvPowerSeries σ R) (G : υ → MvPowerSeries τ R) |
| (hH : Function.Injective (Matrix.mulVecLin |
| (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (H i))))) |
| (hG : Function.Injective (Matrix.mulVecLin |
| (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (G i))))) |
| (hcomp : IsCompl (Matrix.mulVecLin |
| (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (H i)))).range |
| (Matrix.mulVecLin (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (G i)))).range) : |
| Matrix.det (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) |
| (combinedGraph R υ σ τ e H G i)))≠0 := by |
| have heq : (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) |
| (combinedGraph R υ σ τ e H G i)))= |
| joinedMatrix e (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (H i))) |
| (fun i j => a (MvPowerSeries.coeff (Finsupp.single j 1) (G i))) := by |
| funext i j |
| rw [combinedGraph_coeff] |
| unfold joinedMatrix |
| cases e j <;> rfl |
| rw [heq] |
| exact joinedMatrix_det_ne_zero e _ _ hH hG hcomp |
| |
| end |
| end PD4Tensor.FiniteCoordinates |
| end |
| |
| end OAI |
| |
