AlgebraicGeometry.Scheme.exists_compatible_open_extension_fromSpecStalk_of_affineNeighborhood
An affine chart with finitely many irreducible components suffices for compatible extension from the original scheme stalk.
Source context 4 commands
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
universe u
open CategoryTheory Limits TopologicalSpace
namespace AlgebraicGeometry.SchemeLean source · full declaration
theorem exists_compatible_open_extension_fromSpecStalk_of_affineNeighborhood
{X Y : Scheme.{u}} (f : Y ⟶ X) [Etale f] [IsSeparated f]
(U : X.Opens) (s : U.toScheme ⟶ Y) (hs : s ≫ f = U.ι)
(x : X) (A : X.Opens) (hA : IsAffineOpen A) (hxA : x ∈ A)
(hfin : (irreducibleComponents A.toScheme).Finite)
(a : Spec (X.presheaf.stalk x) ⟶ Y) (ha : a ≫ f = X.fromSpecStalk x)
(hcompat : pullback.snd (X.fromSpecStalk x) U.ι ≫ s =
pullback.fst (X.fromSpecStalk x) U.ι ≫ a) :
∃ (W : X.Opens), x ∈ W ∧ ∃ b : W.toScheme ⟶ Y,
b ≫ f = W.ι ∧
X.homOfLE (show U ⊓ W ≤ U from inf_le_left) ≫ s =
X.homOfLE (show U ⊓ W ≤ W from inf_le_right) ≫ b := by
have : IsAffine A.toScheme := hA
let xA : A.toScheme := ⟨x, hxA⟩
let q : Spec (X.presheaf.stalk x) ⟶ X := X.fromSpecStalk x
let qA : Spec (A.toScheme.presheaf.stalk xA) ⟶ A.toScheme :=
A.toScheme.fromSpecStalk xA
let r : Spec (A.toScheme.presheaf.stalk xA) ⟶ Spec (X.presheaf.stalk x) :=
Spec.map (A.ι.stalkMap xA)
have hr : r ≫ q = qA ≫ A.ι := Scheme.SpecMap_stalkMap_fromSpecStalk A.ι
let fA : pullback f A.ι ⟶ A.toScheme := pullback.snd f A.ι
let aA : Spec (A.toScheme.presheaf.stalk xA) ⟶ pullback f A.ι :=
pullback.lift (r ≫ a) qA (by rw [Category.assoc, ha]; exact hr)
have haA : aA ≫ fA = qA := pullback.lift_snd _ _ _
let UA : A.toScheme.Opens := A.ι ⁻¹ᵁ U
let uU : UA.toScheme ⟶ U.toScheme := A.ι ∣_ U
have huU : uU ≫ U.ι = UA.ι ≫ A.ι := morphismRestrict_ι A.ι U
let sA : UA.toScheme ⟶ pullback f A.ι :=
pullback.lift (uU ≫ s) UA.ι (by rw [Category.assoc, hs]; exact huU)
have hsA : sA ≫ fA = UA.ι := pullback.lift_snd _ _ _
have hkcond : (pullback.fst qA UA.ι ≫ r) ≫ q =
(pullback.snd qA UA.ι ≫ uU) ≫ U.ι := by
rw [Category.assoc, hr, ← Category.assoc, pullback.condition,
Category.assoc, ← huU, ← Category.assoc]
let k : pullback qA UA.ι ⟶ pullback q U.ι := pullback.lift
(pullback.fst qA UA.ι ≫ r) (pullback.snd qA UA.ι ≫ uU) hkcond
have hAcompat : pullback.snd qA UA.ι ≫ sA = pullback.fst qA UA.ι ≫ aA := by
apply pullback.hom_ext
· have hkfst : k ≫ pullback.fst q U.ι = pullback.fst qA UA.ι ≫ r :=
pullback.lift_fst _ _ _
have hksnd : k ≫ pullback.snd q U.ι = pullback.snd qA UA.ι ≫ uU :=
pullback.lift_snd _ _ _
have h : k ≫ (pullback.snd q U.ι ≫ s) =
k ≫ (pullback.fst q U.ι ≫ a) :=
congrArg (fun z : pullback q U.ι ⟶ Y => k ≫ z) hcompat
have h' : (pullback.snd qA UA.ι ≫ uU) ≫ s =
(pullback.fst qA UA.ι ≫ r) ≫ a := by
rw [← Category.assoc, hksnd, ← Category.assoc, hkfst] at h
exact h
rw [Category.assoc, pullback.lift_fst, Category.assoc, pullback.lift_fst]
simpa only [Category.assoc] using h'
· rw [Category.assoc, pullback.lift_snd, Category.assoc, pullback.lift_snd]
exact pullback.condition.symm
obtain ⟨B, hxB, bB, _, hbBf, hbBU⟩ :=
Scheme.exists_compatible_open_extension_fromSpecStalk hfin fA UA sA hsA
xA aA haA hAcompat
let W : X.Opens := A.ι ''ᵁ B
have hxW : x ∈ W := ⟨xA, hxB, rfl⟩
let e : B.toScheme ≅ W.toScheme := A.ι.isoImage B
let bW : W.toScheme ⟶ Y := e.inv ≫ bB ≫ pullback.fst f A.ι
have hbWf : bW ≫ f = W.ι := by
change (e.inv ≫ bB ≫ pullback.fst f A.ι) ≫ f = W.ι
rw [Category.assoc, Category.assoc, pullback.condition,
← Category.assoc bB, hbBf]
exact A.ι.isoImage_inv_ι B
let OA : A.toScheme.Opens := UA ⊓ B
let O : X.Opens := U ⊓ W
have hrange : Set.range (OA.ι ≫ A.ι) = Set.range O.ι := by
rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp,
Scheme.Opens.range_ι, Scheme.Opens.range_ι]
change A.ι '' (A.ι ⁻¹' (U : Set X) ∩ (B : Set A.toScheme)) =
(U : Set X) ∩ A.ι '' (B : Set A.toScheme)
exact Set.image_preimage_inter A.ι B U
let eO : OA.toScheme ≅ O.toScheme :=
IsOpenImmersion.isoOfRangeEq (OA.ι ≫ A.ι) O.ι hrange
have heO : eO.hom ≫ O.ι = OA.ι ≫ A.ι :=
IsOpenImmersion.isoOfRangeEq_hom_fac (OA.ι ≫ A.ι) O.ι hrange
let dUA : OA.toScheme ⟶ UA.toScheme := A.toScheme.homOfLE inf_le_left
let dB : OA.toScheme ⟶ B.toScheme := A.toScheme.homOfLE inf_le_right
let dU : O.toScheme ⟶ U.toScheme := X.homOfLE inf_le_left
let dW : O.toScheme ⟶ W.toScheme := X.homOfLE inf_le_right
have heOU : eO.hom ≫ dU = dUA ≫ uU := by
apply (cancel_mono U.ι).mp
rw [Category.assoc, Scheme.homOfLE_ι, heO, Category.assoc, huU,
← Category.assoc, Scheme.homOfLE_ι]
have heOW : eO.hom ≫ dW = dB ≫ e.hom := by
apply (cancel_mono W.ι).mp
rw [Category.assoc, Scheme.homOfLE_ι, heO, Category.assoc,
Scheme.Hom.isoImage_hom_ι, ← Category.assoc, Scheme.homOfLE_ι]
have heb : e.hom ≫ bW = bB ≫ pullback.fst f A.ι := by
change e.hom ≫ (e.inv ≫ bB ≫ pullback.fst f A.ι) = _
rw [← Category.assoc, e.hom_inv_id, Category.id_comp]
have hoverlap : dUA ≫ (uU ≫ s) = dB ≫ (bB ≫ pullback.fst f A.ι) := by
have h := congrArg (fun z : OA.toScheme ⟶ pullback f A.ι =>
z ≫ pullback.fst f A.ι) hbBU
simpa only [sA, Category.assoc, pullback.lift_fst] using h
refine ⟨W, hxW, bW, hbWf, ?_⟩
apply (cancel_epi eO.hom).mp
change eO.hom ≫ (dU ≫ s) = eO.hom ≫ (dW ≫ bW)
rw [← Category.assoc, heOU, Category.assoc,
← Category.assoc eO.hom, heOW, Category.assoc, heb]
exact hoverlap