File size: 5,455 Bytes
b797a5b | 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 | import Mathlib
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Topology
theorem generic_iff_no_proper_generalization {T : Type*} [TopologicalSpace T] [QuasiSober T]
(x : T) : x ∈ genericPoints T ↔ ∀ y : T, y ⤳ x → x ⤳ y := by
constructor
· intro hx y hy
apply specializes_iff_closure_subset.mpr
exact hx.2 isIrreducible_singleton.closure hy.closure_subset
· intro hx
let C : irreducibleComponents T :=
⟨irreducibleComponent x, irreducibleComponent_mem_irreducibleComponents x⟩
let y := (genericPoints.ofComponent C).val
have hy : y ⤳ x := (genericPoints.isGenericPoint_ofComponent C).specializes
mem_irreducibleComponent
have he : closure ({x} : Set T) = C.val := by
apply Set.Subset.antisymm
· exact closure_minimal (Set.singleton_subset_iff.mpr mem_irreducibleComponent)
(isClosed_of_mem_irreducibleComponents C.val C.property)
· rw [← (genericPoints.isGenericPoint_ofComponent C).def]
exact (hx y hy).closure_subset
change closure ({x} : Set T) ∈ irreducibleComponents T
rw [he]
exact C.property
theorem generic_embedding_iff {T S : Type*} [TopologicalSpace T] [QuasiSober T]
[TopologicalSpace S] [QuasiSober S] {f : S → T} (hf : IsEmbedding f)
(hstable : ∀ y : S, ∀ x : T, x ⤳ f y → x ∈ Set.range f) (y : S) :
y ∈ genericPoints S ↔ f y ∈ genericPoints T := by
rw [generic_iff_no_proper_generalization, generic_iff_no_proper_generalization]
constructor
· intro hy x hx
obtain ⟨z,rfl⟩ := hstable y x hx
exact (hy z (hf.isInducing.specializes_iff.mp hx)).map hf.continuous
· intro hy z hz
exact hf.isInducing.specializes_iff.mp (hy (f z) (hz.map hf.continuous))
def genericPointsEmbeddingEquiv {T S : Type*} [TopologicalSpace T] [QuasiSober T]
[TopologicalSpace S] [QuasiSober S] {f : S → T} (hf : IsEmbedding f)
(hstable : ∀ y : S, ∀ x : T, x ⤳ f y → x ∈ Set.range f) :
genericPoints S ≃ {x : genericPoints T // x.val ∈ Set.range f} := by
let g : genericPoints S → {x : genericPoints T // x.val ∈ Set.range f} := fun y =>
⟨⟨f y.val, (generic_embedding_iff hf hstable y.val).mp y.property⟩, ⟨y.val,rfl⟩⟩
apply Equiv.ofBijective g
constructor
· intro x y h
exact Subtype.ext (hf.injective (congrArg (fun z => z.val.val) h))
· intro x
obtain ⟨y,hy⟩ := x.property
have hgen : y ∈ genericPoints S :=
(generic_embedding_iff hf hstable y).mpr (hy ▸ x.val.property)
exact ⟨⟨y,hgen⟩, Subtype.ext (Subtype.ext hy)⟩
def minimalPrimesGenericEquiv (A : CommRingCat) :
minimalPrimes A ≃ genericPoints (Spec A) where
toFun P := ⟨⟨P.val,P.property.1.1⟩, by
change closure ({⟨P.val,P.property.1.1⟩} : Set (PrimeSpectrum A)) ∈
irreducibleComponents (PrimeSpectrum A)
rw [← PrimeSpectrum.vanishingIdeal_mem_minimalPrimes,
PrimeSpectrum.vanishingIdeal_singleton]
exact P.property⟩
invFun y := ⟨y.val.asIdeal, by
rw [← PrimeSpectrum.vanishingIdeal_singleton y.val]
exact PrimeSpectrum.vanishingIdeal_mem_minimalPrimes.mpr y.property⟩
left_inv P := rfl
right_inv y := by rfl
theorem fromSpecStalk_generizing (X : Scheme.{0}) (x : X)
(y : Spec (X.presheaf.stalk x)) (z : X) (hz : z ⤳ X.fromSpecStalk x y) :
z ∈ Set.range (X.fromSpecStalk x) := by
rw [Scheme.range_fromSpecStalk]
have hy : X.fromSpecStalk x y ⤳ x := by
have hmem : X.fromSpecStalk x y ∈ Set.range (X.fromSpecStalk x) := ⟨y,rfl⟩
rw [Scheme.range_fromSpecStalk] at hmem
exact hmem
exact hz.trans hy
def genericPointsThroughEquiv (X : Scheme.{0}) (x : X) :
{y : genericPoints X // y.val ∈ Set.range (X.fromSpecStalk x)} ≃
{C : irreducibleComponents X // x ∈ C.val} where
toFun y := ⟨genericPoints.component y.val, by
have hmem : y.val.val ⤳ x :=
(congrArg (fun S : Set X => y.val.val ∈ S)
(Scheme.range_fromSpecStalk (X := X) (x := x))).mp y.property
exact hmem.mem_closure⟩
invFun C := ⟨genericPoints.ofComponent C.val, by
rw [Scheme.range_fromSpecStalk]
exact (genericPoints.isGenericPoint_ofComponent C.val).specializes C.property⟩
left_inv y := by
apply Subtype.ext
exact genericPoints.ofComponent_component y.val
right_inv C := by
apply Subtype.ext
exact genericPoints.component_ofComponent C.val
def stalkMinimalPrimesEquivComponentsThrough (X : Scheme.{0}) (x : X) :
minimalPrimes (X.presheaf.stalk x) ≃
{C : irreducibleComponents X // x ∈ C.val} :=
(minimalPrimesGenericEquiv (X.presheaf.stalk x)).trans
((genericPointsEmbeddingEquiv (X.fromSpecStalk x).isEmbedding
(fromSpecStalk_generizing X x)).trans (genericPointsThroughEquiv X x))
theorem stalkMinimalPrimesEquivComponentsThrough_point (X : Scheme.{0}) (x : X)
(P : minimalPrimes (X.presheaf.stalk x)) :
X.fromSpecStalk x (⟨P.val,P.property.1.1⟩ : Spec (X.presheaf.stalk x)) =
(genericPoints.ofComponent
(stalkMinimalPrimesEquivComponentsThrough X x P).val).val := by
symm
exact congrArg Subtype.val (genericPoints.ofComponent_component
((genericPointsEmbeddingEquiv (X.fromSpecStalk x).isEmbedding
(fromSpecStalk_generizing X x))
(minimalPrimesGenericEquiv (X.presheaf.stalk x) P)).val)
end
end PiExponent.CurveCycle
end OAI
|