Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Projective.PolynomialCharts

namespace OAI

section
noncomputable section
                                             
section

namespace MaximalSeshadri.Projective
noncomputable section
open MvPolynomial
variable {R A σ : Type*} [CommRing R] [CommRing A] [Algebra R A]
  (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]

lemma aeval_mem_grade (x : σ → A) (hx : ∀ i, x i ∈ 𝒜 1)
    {p : MvPolynomial σ R} {d : ℕ} (hp : p.IsHomogeneous d) :
    aeval x p ∈ 𝒜 d := by
  classical
  rw [p.as_sum]
  simp only [map_sum, aeval_monomial]
  apply sum_mem
  intro m hm
  have h := SetLike.prod_pow_mem_graded 𝒜 (fun _ : σ => (1 : ℕ)) x
    (F := m.support) m (fun i _ => hx i)
  simp only [smul_eq_mul, mul_one] at h
  rw [← hp.degree_eq_sum_deg_support hm] at h
  simpa only [Algebra.smul_def, Finsupp.prod] using (𝒜 d).smul_mem (p.coeff m) h

attribute [local instance] MvPolynomial.gradedAlgebra

def gradedPolynomialEval (x : σ → A) (hx : ∀ i, x i ∈ 𝒜 1) :
    homogeneousSubmodule σ R →+*ᵍ 𝒜 where
  toRingHom := (aeval x).toRingHom
  map_mem := fun hp => aeval_mem_grade 𝒜 x hx hp

end

noncomputable section
open AlgebraicGeometry CategoryTheory HomogeneousIdeal MvPolynomial
universe u
variable {K σ : Type u} [CommRing K]
attribute [local instance] MvPolynomial.gradedAlgebra

def shiftedVariable (c : σ → K) : Option σ → MvPolynomial (Option σ) K
  | none => X none
  | some i => X (some i) + C (c i) * X none

lemma shiftedVariable_homogeneous (c : σ → K) (i : Option σ) :
    shiftedVariable c i ∈ PolyGrade K (Option σ) 1 := by
  cases i with
  | none => exact isHomogeneous_X _ _
  | some i =>
    exact (isHomogeneous_X _ _).add (by
      simpa using (isHomogeneous_C (Option σ) (c i)).mul (isHomogeneous_X K none))

def projectiveShiftHom (c : σ → K) :
    PolyGrade K (Option σ) →+*ᵍ PolyGrade K (Option σ) :=
  gradedPolynomialEval _ (shiftedVariable c) (shiftedVariable_homogeneous c)

@[simp] lemma projectiveShiftHom_X (c : σ → K) (i : Option σ) :
    projectiveShiftHom c (X i) = shiftedVariable c i := aeval_X _ _

@[simp] lemma projectiveShiftHom_C (c : σ → K) (a : K) :
    projectiveShiftHom c (C a) = C a := aeval_C _ _

lemma projectiveShiftHom_inverse (c : σ → K) :
    (projectiveShiftHom (fun i => -c i)).comp (projectiveShiftHom c) =
      GradedRingHom.id (PolyGrade K (Option σ)) := by
  have h : ((projectiveShiftHom (fun i => -c i)).comp
      (projectiveShiftHom c)).toRingHom = RingHom.id _ := by
    apply MvPolynomial.ringHom_ext
    · intro a
      simp
    · intro i
      cases i <;> simp [shiftedVariable]
  exact GradedRingHom.ext (RingHom.congr_fun h)

lemma projectiveShiftHom_surjective (c : σ → K) :
    Function.Surjective (projectiveShiftHom c) := by
  intro p
  refine ⟨projectiveShiftHom (fun i => -c i) p, ?_⟩
  have h := projectiveShiftHom_inverse (fun i => -c i)
  simp only [neg_neg] at h
  exact DFunLike.congr_fun h p

lemma shift_irrelevant (c : σ → K) :
    (PolyGrade K (Option σ))₊ ≤
      (PolyGrade K (Option σ))₊.map (projectiveShiftHom c) := by
  apply (HomogeneousIdeal.irrelevant_le _).mpr
  intro d hd b hb
  have eqb : projectiveShiftHom c (projectiveShiftHom (fun i => -c i) b) = b := by
    have hh := projectiveShiftHom_inverse (fun i => -c i)
    simp only [neg_neg] at hh
    exact DFunLike.congr_fun hh b
  rw [← eqb]
  exact Ideal.mem_map_of_mem _
    (HomogeneousIdeal.mem_irrelevant_of_mem _ hd ((projectiveShiftHom _).map_mem hb))

def projectiveShift (c : σ → K) :
    Proj (PolyGrade K (Option σ)) ≅ Proj (PolyGrade K (Option σ)) where
  hom := Proj.map (projectiveShiftHom c) (shift_irrelevant c)
  inv := Proj.map (projectiveShiftHom (fun i => -c i)) (shift_irrelevant _)
  hom_inv_id := by
    rw [← Proj.map_comp]
    have hh := projectiveShiftHom_inverse (fun i => -c i)
    simp only [neg_neg] at hh
    simpa only [hh] using (Proj.map_id (𝒜 := PolyGrade K (Option σ)))
  inv_hom_id := by
    rw [← Proj.map_comp]
    simpa only [projectiveShiftHom_inverse] using
      (Proj.map_id (𝒜 := PolyGrade K (Option σ)))

end
end MaximalSeshadri.Projective

end


end
end

end OAI

	import Mathlib
	import OAI.AlgebraicGeometry.Seshadri.Projective.PolynomialCharts

	namespace OAI

	section
	noncomputable section

	section

	namespace MaximalSeshadri.Projective
	noncomputable section
	open MvPolynomial
	variable {R A σ : Type*} [CommRing R] [CommRing A] [Algebra R A]
	(𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]

	lemma aeval_mem_grade (x : σ → A) (hx : ∀ i, x i ∈ 𝒜 1)
	{p : MvPolynomial σ R} {d : ℕ} (hp : p.IsHomogeneous d) :
	aeval x p ∈ 𝒜 d := by
	classical
	rw [p.as_sum]
	simp only [map_sum, aeval_monomial]
	apply sum_mem
	intro m hm
	have h := SetLike.prod_pow_mem_graded 𝒜 (fun _ : σ => (1 : ℕ)) x
	(F := m.support) m (fun i _ => hx i)
	simp only [smul_eq_mul, mul_one] at h
	rw [← hp.degree_eq_sum_deg_support hm] at h
	simpa only [Algebra.smul_def, Finsupp.prod] using (𝒜 d).smul_mem (p.coeff m) h

	attribute [local instance] MvPolynomial.gradedAlgebra

	def gradedPolynomialEval (x : σ → A) (hx : ∀ i, x i ∈ 𝒜 1) :
	homogeneousSubmodule σ R →+*ᵍ 𝒜 where
	toRingHom := (aeval x).toRingHom
	map_mem := fun hp => aeval_mem_grade 𝒜 x hx hp

	end

	noncomputable section
	open AlgebraicGeometry CategoryTheory HomogeneousIdeal MvPolynomial
	universe u
	variable {K σ : Type u} [CommRing K]
	attribute [local instance] MvPolynomial.gradedAlgebra

	def shiftedVariable (c : σ → K) : Option σ → MvPolynomial (Option σ) K
	\| none => X none
	\| some i => X (some i) + C (c i) * X none

	lemma shiftedVariable_homogeneous (c : σ → K) (i : Option σ) :
	shiftedVariable c i ∈ PolyGrade K (Option σ) 1 := by
	cases i with
	\| none => exact isHomogeneous_X _ _
	\| some i =>
	exact (isHomogeneous_X _ _).add (by
	simpa using (isHomogeneous_C (Option σ) (c i)).mul (isHomogeneous_X K none))

	def projectiveShiftHom (c : σ → K) :
	PolyGrade K (Option σ) →+*ᵍ PolyGrade K (Option σ) :=
	gradedPolynomialEval _ (shiftedVariable c) (shiftedVariable_homogeneous c)

	@[simp] lemma projectiveShiftHom_X (c : σ → K) (i : Option σ) :
	projectiveShiftHom c (X i) = shiftedVariable c i := aeval_X _ _

	@[simp] lemma projectiveShiftHom_C (c : σ → K) (a : K) :
	projectiveShiftHom c (C a) = C a := aeval_C _ _

	lemma projectiveShiftHom_inverse (c : σ → K) :
	(projectiveShiftHom (fun i => -c i)).comp (projectiveShiftHom c) =
	GradedRingHom.id (PolyGrade K (Option σ)) := by
	have h : ((projectiveShiftHom (fun i => -c i)).comp
	(projectiveShiftHom c)).toRingHom = RingHom.id _ := by
	apply MvPolynomial.ringHom_ext
	· intro a
	simp
	· intro i
	cases i <;> simp [shiftedVariable]
	exact GradedRingHom.ext (RingHom.congr_fun h)

	lemma projectiveShiftHom_surjective (c : σ → K) :
	Function.Surjective (projectiveShiftHom c) := by
	intro p
	refine ⟨projectiveShiftHom (fun i => -c i) p, ?_⟩
	have h := projectiveShiftHom_inverse (fun i => -c i)
	simp only [neg_neg] at h
	exact DFunLike.congr_fun h p

	lemma shift_irrelevant (c : σ → K) :
	(PolyGrade K (Option σ))₊ ≤
	(PolyGrade K (Option σ))₊.map (projectiveShiftHom c) := by
	apply (HomogeneousIdeal.irrelevant_le _).mpr
	intro d hd b hb
	have eqb : projectiveShiftHom c (projectiveShiftHom (fun i => -c i) b) = b := by
	have hh := projectiveShiftHom_inverse (fun i => -c i)
	simp only [neg_neg] at hh
	exact DFunLike.congr_fun hh b
	rw [← eqb]
	exact Ideal.mem_map_of_mem _
	(HomogeneousIdeal.mem_irrelevant_of_mem _ hd ((projectiveShiftHom _).map_mem hb))

	def projectiveShift (c : σ → K) :
	Proj (PolyGrade K (Option σ)) ≅ Proj (PolyGrade K (Option σ)) where
	hom := Proj.map (projectiveShiftHom c) (shift_irrelevant c)
	inv := Proj.map (projectiveShiftHom (fun i => -c i)) (shift_irrelevant _)
	hom_inv_id := by
	rw [← Proj.map_comp]
	have hh := projectiveShiftHom_inverse (fun i => -c i)
	simp only [neg_neg] at hh
	simpa only [hh] using (Proj.map_id (𝒜 := PolyGrade K (Option σ)))
	inv_hom_id := by
	rw [← Proj.map_comp]
	simpa only [projectiveShiftHom_inverse] using
	(Proj.map_id (𝒜 := PolyGrade K (Option σ)))

	end
	end MaximalSeshadri.Projective

	end


	end
	end

	end OAI