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import OAI.Algebra.FiniteTensor.MovingCoordinates
import OAI.Algebra.FiniteTensor.RelativeDerivatives
namespace OAI
/-! Normal-cycle residues, restricted gradients and nonzero exterior pairings. -/
noncomputable section
open scoped BigOperators
namespace PD4Tensor.Forms
noncomputable section
open scoped TensorProduct BigOperators
variable (K A : Type*) [Field K] [CommRing A] [Algebra K A]
variable (σ : Type*) [Fintype σ] [DecidableEq σ]
variable (der : σ → A →ₗ[K] A) (p : ℕ)
theorem normalCycle_product (cs : List A) :
normalCycle K A σ der p cs =
scalar K A σ (cs.map (fun c => c^(p-1))).prod *
(cs.map (fun c => differential K A σ der (scalar K A σ c))).prod := by
induction cs with
| nil => simp [normalCycle]
| cons c cs ih =>
change (scalar K A σ (c^(p-1)) * differential K A σ der (scalar K A σ c)) *
normalCycle K A σ der p cs = _
rw [ih]
simp only [List.map_cons, List.prod_cons, scalar_mul]
rw [mul_assoc, ← mul_assoc (differential K A σ der (scalar K A σ c)),
← scalar_commute, mul_assoc, ← mul_assoc]
variable [Invertible (2 : K)]
theorem topCoefficient_normalCycle (N : ℕ) (der : Fin N → A →ₗ[K] A)
(c : Fin N → A) :
topCoefficient K A N (normalCycle K A (Fin N) der p (List.ofFn c)) =
(∏ i, c i^(p-1)) * Matrix.det (fun i j => der j (c i)) := by
rw [normalCycle_product]
have hscalar (a : A) (ω : Ω K A (Fin N)) : scalar K A (Fin N) a * ω = a • ω := by
induction ω using TensorProduct.inductionOn with
| add x y hx hy => simp [mul_add, smul_add, hx, hy]
| tmul b e => simp [scalar, Algebra.TensorProduct.tmul_mul_tmul, TensorProduct.smul_tmul']
rw [hscalar, map_smul, smul_eq_mul]
simp only [List.map_ofFn, Function.comp_def, List.prod_ofFn, differential_scalar]
rw [topCoefficient_oneForm_product]
end
end PD4Tensor.Forms
namespace PD4Tensor.FrobeniusTruncation
noncomputable section
open scoped TensorProduct BigOperators
variable (K : Type*) [Field K] (N p : ℕ) [CharP K p] [Fact p.Prime]
[Invertible (2 : K)]
theorem exterior_normal_residue (hp : 0 < p) (f : Fin N → MvPolynomial (Fin N) K)
(hf : ∀ i, MvPolynomial.constantCoeff (f i)=0) :
residue K (Fin N) p hp
(Forms.topCoefficient K (Ring K (Fin N) p) N
(Forms.normalCycle K (Ring K (Fin N) p) (Fin N) (derivative K (Fin N) p) p
(List.ofFn (fun i => Ideal.Quotient.mk (ideal K (Fin N) p) (f i))))) =
(tangentMatrix K (Fin N) f).det^p := by
rw [Forms.topCoefficient_normalCycle]
simp only [derivative_mk]
have hj : Matrix.det (fun i j => Ideal.Quotient.mk (ideal K (Fin N) p)
(MvPolynomial.pderiv j (f i))) =
Ideal.Quotient.mk (ideal K (Fin N) p) (jacobianDet K (Fin N) f) :=
(RingHom.map_det (Ideal.Quotient.mk (ideal K (Fin N) p)) _).symm
rw [hj]
exact normal_residue K (Fin N) p hp f hf
theorem exterior_normal_residue_ne_zero (hp : 0 < p) (f : Fin N → MvPolynomial (Fin N) K)
(hf : ∀ i, MvPolynomial.constantCoeff (f i)=0)
(hdet : (tangentMatrix K (Fin N) f).det ≠ 0) :
residue K (Fin N) p hp
(Forms.topCoefficient K (Ring K (Fin N) p) N
(Forms.normalCycle K (Ring K (Fin N) p) (Fin N) (derivative K (Fin N) p) p
(List.ofFn (fun i => Ideal.Quotient.mk (ideal K (Fin N) p) (f i))))) ≠ 0 := by
rw [exterior_normal_residue K N p hp f hf]
exact pow_ne_zero _ hdet
end
end PD4Tensor.FrobeniusTruncation
namespace PD4Tensor.Forms
noncomputable section
open scoped TensorProduct BigOperators
variable {K A B σ : Type*} [Field K] [CommRing A] [Algebra K A]
[CommRing B] [Algebra K B] [Fintype σ] [DecidableEq σ]
theorem restricted_gradient_boundary (r : A →ₐ[K] B) (hr : Function.Surjective r)
(v : σ → A) (ω : Ω K A σ) (hω : oneForm K A σ v * ω=0)
(hann : ∀ a, r a=0 → scalar K A σ a * ω=0)
(a : A) (ha : r a∈Ideal.span (Set.range (fun i => r (v i)))) :
scalar K A σ a * ω∈LinearMap.range (LinearMap.mulLeft K (oneForm K A σ v)) := by
obtain ⟨b,hb⟩ := Ideal.mem_span_range_iff_exists_fun.mp ha
choose c hc using fun i => hr (b i)
let a' : A := ∑ i, c i * v i
have he : r (a-a')=0 := by
simp only [a',map_sub,map_sum,map_mul,hc]
rw [hb,sub_self]
have heq : scalar K A σ a * ω=scalar K A σ a' * ω := by
have h := hann (a-a') he
simpa only [map_sub,scalar,TensorProduct.sub_tmul,sub_mul,sub_eq_zero] using h
rw [heq]
apply gradient_ideal_boundary v ω hω a'
exact Ideal.mem_span_range_iff_exists_fun.mpr ⟨c,rfl⟩
end
end PD4Tensor.Forms
namespace PD4Tensor.FrobeniusTruncation
noncomputable section
variable (K : Type*) [Field K] (N p : ℕ) [CharP K p] [Fact p.Prime]
[Invertible (2 : K)]
theorem inverse_normal_residue_ne_zero
(q : Ring K (Fin N) p ≃ₐ[K] Ring K (Fin N) p) :
residue K (Fin N) p (Fact.out : p.Prime).pos
(Forms.topCoefficient K (Ring K (Fin N) p) N
(Forms.normalCycle K (Ring K (Fin N) p) (Fin N) (derivative K (Fin N) p) p
(List.ofFn (fun i => q (coord K (Fin N) p i)))))≠0 := by
choose f hf using fun i => Ideal.Quotient.mk_surjective (q (coord K (Fin N) p i))
have hh := inverse_representatives K (Fin N) p q f hf
have hr := exterior_normal_residue_ne_zero K N p (Fact.out : p.Prime).pos f hh.1 hh.2
simpa only [hf] using hr
end
end PD4Tensor.FrobeniusTruncation
namespace PD4Tensor.Forms
noncomputable section
variable (K A σ : Type*) [Field K] [CommRing A] [Algebra K A] [Fintype σ] [DecidableEq σ]
(der : σ → A →ₗ[K] A) (p : ℕ)
@[simp] theorem normalCycle_append (cs ds : List A) :
normalCycle K A σ der p (cs++ds)=normalCycle K A σ der p cs * normalCycle K A σ der p ds := by
simp [normalCycle,List.prod_append]
end
end PD4Tensor.Forms
end
end OAI
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