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7f5fab8 | 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 | import Mathlib
import OAI.AlgebraicGeometry.RelativeTriviality.Cech
namespace OAI
namespace CampanaPeternellSix.ActualCech
noncomputable section
open AlgebraicGeometry CategoryTheory Limits Opposite
open RelativeLineCharts
open scoped Polynomial LaurentPolynomial
universe u
lemma laurent_parts (R : Type u) [CommRing R] (f : R[T;T⁻¹]) :
∃ p q : R[X], f = p.toLaurent + LaurentPolynomial.invert q.toLaurent := by
induction f using LaurentPolynomial.induction_on' with
| add f g hf hg =>
obtain ⟨p,q,h⟩ := hf
obtain ⟨p',q',h'⟩ := hg
exact ⟨p+p',q+q', by simp only [map_add,h,h']; abel⟩
| C_mul_T n a =>
cases n with
| ofNat n =>
exact ⟨Polynomial.C a * Polynomial.X^n,0, by simp⟩
| negSucc n =>
exact ⟨0,Polynomial.C a * Polynomial.X^(n+1), by
simp only [map_zero, zero_add, map_mul, Polynomial.toLaurent_C,
map_mul, map_pow, Polynomial.toLaurent_X, LaurentPolynomial.invert_C,
LaurentPolynomial.invert_T, LaurentPolynomial.T_pow]
congr 2
omega⟩
variable (X : Scheme.{u}) (I : Type u) [Finite I]
def freeProductIso : (SheafOfModules.free I : X.Modules) ≅
∏ᶜ (fun _ : I => SheafOfModules.unit X.ringCatSheaf) := by
classical
letI := Fintype.ofFinite I
let f : I → X.Modules := fun _ => SheafOfModules.unit X.ringCatSheaf
letI : HasBiproduct f := HasBiproduct.of_hasProduct f
exact (biproduct.isoCoproduct f).symm ≪≫ biproduct.isoProduct f
def freeProjection (i : I) : (SheafOfModules.free I : X.Modules) ⟶
SheafOfModules.unit X.ringCatSheaf := (freeProductIso X I).hom ≫ Pi.π _ i
local instance freeSectionsModule (U : X.Opens) :
Module Γ(X,U) Γ((SheafOfModules.free I : X.Modules), U) :=
((SheafOfModules.free I : X.Modules).val.obj (op U)).isModule
def freeSectionEquiv (U : X.Opens) :
Γ((SheafOfModules.free I : X.Modules), U) ≃ₗ[Γ(X,U)] (I → Γ(X,U)) := by
classical
letI := Fintype.ofFinite I
let F := SheafOfModules.evaluation X.ringCatSheaf (op U)
let f : I → X.Modules := fun _ => SheafOfModules.unit X.ringCatSheaf
exact (F.mapIso (freeProductIso X I) ≪≫ asIso (piComparison F f) ≪≫
ModuleCat.piIsoPi (F.obj ∘ f)).toLinearEquiv
lemma freeSectionEquiv_apply (U : X.Opens)
(m : Γ((SheafOfModules.free I : X.Modules), U)) (i : I) :
freeSectionEquiv X I U m i = (freeProjection X I i).val.app (op U) m := by
classical
let := Fintype.ofFinite I
let F := SheafOfModules.evaluation X.ringCatSheaf (op U)
let f : I → X.Modules := fun _ => SheafOfModules.unit X.ringCatSheaf
have h : (F.map (freeProductIso X I).hom ≫ piComparison F f ≫
(ModuleCat.piIsoPi (F.obj ∘ f)).hom) ≫ ModuleCat.ofHom (LinearMap.proj i) =
F.map ((freeProductIso X I).hom ≫ Pi.π f i) := by
simp only [Category.assoc, ModuleCat.piIsoPi_hom_ker_subtype]
erw [piComparison_comp_π, ← CategoryTheory.Functor.map_comp]
rfl
exact ConcreteCategory.congr_hom h m
lemma freeSectionEquiv_restrict {U V : X.Opens} (h : U ≤ V)
(m : Γ((SheafOfModules.free I : X.Modules), V)) (i : I) :
freeSectionEquiv X I U ((SheafOfModules.free I : X.Modules).val.map
(homOfLE h).op m) i =
X.presheaf.map (homOfLE h).op (freeSectionEquiv X I V m i) := by
erw [freeSectionEquiv_apply, freeSectionEquiv_apply]
exact ConcreteCategory.congr_hom ((freeProjection X I i).val.naturality (homOfLE h).op) m
variable (R : Type u) [CommRing R]
lemma free_ranges_top :
leftImage R (SheafOfModules.free I) ⊔ rightImage R (SheafOfModules.free I) = ⊤ := by
classical
let E : (scheme R).Modules := SheafOfModules.free I
apply top_unique
intro m hm
change Γ(E,leftOpen R ⊓ rightOpen R) at m
let c : I → R[T;T⁻¹] := fun i => overlapSectionEquiv R
(freeSectionEquiv (scheme R) I (leftOpen R ⊓ rightOpen R) m i)
choose p q hpq using fun i => laurent_parts R (c i)
let a : Γ(E,leftOpen R) := (freeSectionEquiv (scheme R) I (leftOpen R)).symm
(fun i => (leftSectionEquiv R).symm (p i))
let b : Γ(E,rightOpen R) := (freeSectionEquiv (scheme R) I (rightOpen R)).symm
(fun i => (rightSectionEquiv R).symm (q i))
let ra : Γ(E,leftOpen R ⊓ rightOpen R) := E.val.map (homOfLE (show leftOpen R ⊓ rightOpen R ≤ leftOpen R from inf_le_left)).op a
let rb : Γ(E,leftOpen R ⊓ rightOpen R) := E.val.map (homOfLE (show leftOpen R ⊓ rightOpen R ≤ rightOpen R from inf_le_right)).op b
have hsum : ra + rb = m := by
apply (freeSectionEquiv (scheme R) I (leftOpen R ⊓ rightOpen R)).injective
ext i
rw [map_add, Pi.add_apply]
dsimp only [ra, rb]
rw [freeSectionEquiv_restrict, freeSectionEquiv_restrict]
change _ = freeSectionEquiv (scheme R) I (leftOpen R ⊓ rightOpen R) m i
apply (overlapSectionEquiv R).injective
rw [map_add, (standard_section_restrictions R).1, (standard_section_restrictions R).2]
simp only [a,b,LinearEquiv.apply_symm_apply, RingEquiv.apply_symm_apply]
exact (hpq i).symm
have ha : ra ∈ leftImage R E := ⟨a,rfl⟩
have hb : rb ∈ rightImage R E := ⟨b,rfl⟩
exact hsum ▸ Submodule.add_mem_sup ha hb
theorem subsingleton_free_quotient :
Subsingleton (quotient R (SheafOfModules.free I)) :=
Submodule.Quotient.subsingleton_iff.mpr (free_ranges_top I R)
end
end CampanaPeternellSix.ActualCech
end OAI
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