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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
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