File size: 4,240 Bytes
12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
|