fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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@[simps U V f t t', simps -isSimp J]
gluing : Scheme.GlueData.{u} where
J := 𝒰.I₀
U i := pullback (𝒰.f i ≫ f) g
V := fun ⟨i, j⟩ => v 𝒰 f g i j
f _ _ := pullback.fst _ _
f_id _ := inferInstance
f_open := inferInstance
t i j := t 𝒰 f g i j
t_id i := t_id 𝒰 f g i
t' i j k := t' 𝒰 f g i j k
t_fac i j k := by
apply pullback.hom_ext
on_goal 1 => apply pullback.hom_ext
all_goals
simp only [t'_snd_fst_fst, t'_snd_fst_snd, t'_snd_snd, t_fst_fst, t_fst_snd, t_snd,
Category.assoc]
cocycle i j k := cocycle 𝒰 f g i j k
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluing | Given `Uᵢ ×[Z] Y`, this is the glued fibred product `X ×[Z] Y`. |
gluing_ι (j : 𝒰.I₀) :
(gluing 𝒰 f g).ι j = Multicoequalizer.π (gluing 𝒰 f g).diagram j := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluing_ι | null |
p1 : (gluing 𝒰 f g).glued ⟶ X := by
apply Multicoequalizer.desc (gluing 𝒰 f g).diagram _ fun i ↦ pullback.fst _ _ ≫ 𝒰.f i
simp [t_fst_fst_assoc, ← pullback.condition] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | p1 | The first projection from the glued scheme into `X`. |
p2 : (gluing 𝒰 f g).glued ⟶ Y := by
apply Multicoequalizer.desc _ _ fun i ↦ pullback.snd _ _
simp [t_fst_snd] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | p2 | The second projection from the glued scheme into `Y`. |
p_comm : p1 𝒰 f g ≫ f = p2 𝒰 f g ≫ g := by
apply Multicoequalizer.hom_ext
simp [p1, p2, pullback.condition]
variable (s : PullbackCone f g) | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | p_comm | null |
gluedLiftPullbackMap (i j : 𝒰.I₀) :
pullback ((𝒰.pullback₁ s.fst).f i) ((𝒰.pullback₁ s.fst).f j) ⟶
(gluing 𝒰 f g).V ⟨i, j⟩ := by
refine (pullbackRightPullbackFstIso _ _ _).hom ≫ ?_
refine pullback.map _ _ _ _ ?_ (𝟙 _) (𝟙 _) ?_ ?_
· exact (pullbackSymmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (Category.id_comp _).symm s.condition
· simpa using pullback.condition
· simp only [Category.comp_id, Category.id_comp]
@[reassoc] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLiftPullbackMap | (Implementation)
The canonical map `(s.X ×[X] Uᵢ) ×[s.X] (s.X ×[X] Uⱼ) ⟶ (Uᵢ ×[Z] Y) ×[X] Uⱼ`
This is used in `gluedLift`. |
gluedLiftPullbackMap_fst (i j : 𝒰.I₀) :
gluedLiftPullbackMap 𝒰 f g s i j ≫ pullback.fst _ _ =
pullback.fst _ _ ≫
(pullbackSymmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (Category.id_comp _).symm s.condition := by
simp [gluedLiftPullbackMap]
@[reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLiftPullbackMap_fst | null |
gluedLiftPullbackMap_snd (i j : 𝒰.I₀) :
gluedLiftPullbackMap 𝒰 f g s i j ≫ pullback.snd _ _ = pullback.snd _ _ ≫ pullback.snd _ _ := by
simp [gluedLiftPullbackMap] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLiftPullbackMap_snd | null |
gluedLift : s.pt ⟶ (gluing 𝒰 f g).glued := by
fapply Cover.glueMorphisms (𝒰.pullback₁ s.fst)
· exact fun i ↦ (pullbackSymmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (Category.id_comp _).symm s.condition ≫ (gluing 𝒰 f g).ι i
intro i j
rw [← gluedLiftPullbackMap_fst_assoc, ← gluing_f, ← (gluing 𝒰 f g).glue_condition i j,
gluing_t, gluing_f]
simp_rw [← Category.assoc]
congr 1
apply pullback.hom_ext <;> simp_rw [Category.assoc]
· rw [t_fst_fst, gluedLiftPullbackMap_snd]
congr 1
rw [← Iso.inv_comp_eq, pullbackSymmetry_inv_comp_snd, pullback.lift_fst, Category.comp_id]
· rw [t_fst_snd, gluedLiftPullbackMap_fst_assoc, pullback.lift_snd, pullback.lift_snd]
simp_rw [pullbackSymmetry_hom_comp_snd_assoc]
exact pullback.condition_assoc _ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLift | The lifted map `s.X ⟶ (gluing 𝒰 f g).glued` in order to show that `(gluing 𝒰 f g).glued` is
indeed the pullback.
Given a pullback cone `s`, we have the maps `s.fst ⁻¹' Uᵢ ⟶ Uᵢ` and
`s.fst ⁻¹' Uᵢ ⟶ s.X ⟶ Y` that we may lift to a map `s.fst ⁻¹' Uᵢ ⟶ Uᵢ ×[Z] Y`.
to glue these into a map `s.X ⟶ Uᵢ ×[Z] Y`, we need to show that the maps agree on
`(s.fst ⁻¹' Uᵢ) ×[s.X] (s.fst ⁻¹' Uⱼ) ⟶ Uᵢ ×[Z] Y`. This is achieved by showing that both of these
maps factors through `gluedLiftPullbackMap`. |
gluedLift_p1 : gluedLift 𝒰 f g s ≫ p1 𝒰 f g = s.fst := by
rw [← cancel_epi (Cover.fromGlued <| 𝒰.pullback₁ s.fst)]
apply Multicoequalizer.hom_ext
intro b
simp_rw [Cover.fromGlued, Multicoequalizer.π_desc_assoc, gluedLift, ← Category.assoc]
simp_rw [Cover.ι_glueMorphisms (𝒰.pullback₁ s.fst)]
simp [p1, pullback.condition] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLift_p1 | null |
gluedLift_p2 : gluedLift 𝒰 f g s ≫ p2 𝒰 f g = s.snd := by
rw [← cancel_epi (Cover.fromGlued <| 𝒰.pullback₁ s.fst)]
apply Multicoequalizer.hom_ext
intro b
simp_rw [Cover.fromGlued, Multicoequalizer.π_desc_assoc, gluedLift, ← Category.assoc]
simp_rw [(Cover.ι_glueMorphisms <| 𝒰.pullback₁ s.fst)]
simp [p2] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedLift_p2 | null |
pullbackFstιToV (i j : 𝒰.I₀) :
pullback (pullback.fst (p1 𝒰 f g) (𝒰.f i)) ((gluing 𝒰 f g).ι j) ⟶
v 𝒰 f g j i :=
(pullbackSymmetry _ _ ≪≫ pullbackRightPullbackFstIso (p1 𝒰 f g) (𝒰.f i) _).hom ≫
(pullback.congrHom (Multicoequalizer.π_desc ..) rfl).hom
@[simp, reassoc] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackFstιToV | (Implementation)
The canonical map `(W ×[X] Uᵢ) ×[W] (Uⱼ ×[Z] Y) ⟶ (Uⱼ ×[Z] Y) ×[X] Uᵢ = V j i` where `W` is
the glued fibred product.
This is used in `lift_comp_ι`. |
pullbackFstιToV_fst (i j : 𝒰.I₀) :
pullbackFstιToV 𝒰 f g i j ≫ pullback.fst _ _ = pullback.snd _ _ := by
simp [pullbackFstιToV, p1]
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackFstιToV_fst | null |
pullbackFstιToV_snd (i j : 𝒰.I₀) :
pullbackFstιToV 𝒰 f g i j ≫ pullback.snd _ _ = pullback.fst _ _ ≫ pullback.snd _ _ := by
simp [pullbackFstιToV, p1] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackFstιToV_snd | null |
lift_comp_ι (i : 𝒰.I₀) :
pullback.lift (pullback.snd _ _) (pullback.fst _ _ ≫ p2 𝒰 f g)
(by rw [← pullback.condition_assoc, Category.assoc, p_comm]) ≫
(gluing 𝒰 f g).ι i =
(pullback.fst _ _ : pullback (p1 𝒰 f g) (𝒰.f i) ⟶ _) := by
apply Cover.hom_ext ((gluing 𝒰 f g).openCover.pullback₁ (pullback.fst _ _))
intro j
dsimp only [Precoverage.ZeroHypercover.pullback₁_toPreZeroHypercover,
PreZeroHypercover.pullback₁_X, PreZeroHypercover.pullback₁_f]
trans pullbackFstιToV 𝒰 f g i j ≫ fV 𝒰 f g j i ≫ (gluing 𝒰 f g).ι _
· rw [← show _ = fV 𝒰 f g j i ≫ _ from (gluing 𝒰 f g).glue_condition j i]
simp_rw [← Category.assoc]
congr 1
rw [gluing_f, gluing_t]
apply pullback.hom_ext <;> simp_rw [Category.assoc]
· simp_rw [t_fst_fst, pullback.lift_fst, pullbackFstιToV_snd, GlueData.openCover_f]
· simp_rw [t_fst_snd, pullback.lift_snd, pullbackFstιToV_fst_assoc, pullback.condition_assoc,
GlueData.openCover_f, p2]
simp
· rw [pullback.condition, ← Category.assoc]
simp_rw [pullbackFstιToV_fst, GlueData.openCover_f] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | lift_comp_ι | We show that the map `W ×[X] Uᵢ ⟶ Uᵢ ×[Z] Y ⟶ W` is the first projection, where the
first map is given by the lift of `W ×[X] Uᵢ ⟶ Uᵢ` and `W ×[X] Uᵢ ⟶ W ⟶ Y`.
It suffices to show that the two map agrees when restricted onto `Uⱼ ×[Z] Y`. In this case,
both maps factor through `V j i` via `pullback_fst_ι_to_V` |
pullbackP1Iso (i : 𝒰.I₀) : pullback (p1 𝒰 f g) (𝒰.f i) ≅ pullback (𝒰.f i ≫ f) g := by
fconstructor
· exact
pullback.lift (pullback.snd _ _) (pullback.fst _ _ ≫ p2 𝒰 f g)
(by rw [← pullback.condition_assoc, Category.assoc, p_comm])
· exact pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst _ _)
(by rw [gluing_ι, p1, Multicoequalizer.π_desc])
· apply pullback.hom_ext
· simpa using lift_comp_ι 𝒰 f g i
· simp_rw [Category.assoc, pullback.lift_snd, pullback.lift_fst, Category.id_comp]
· apply pullback.hom_ext
· simp_rw [Category.assoc, pullback.lift_fst, pullback.lift_snd, Category.id_comp]
· simp [p2]
@[simp, reassoc] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso | The canonical isomorphism between `W ×[X] Uᵢ` and `Uᵢ ×[X] Y`. That is, the preimage of `Uᵢ` in
`W` along `p1` is indeed `Uᵢ ×[X] Y`. |
pullbackP1Iso_hom_fst (i : 𝒰.I₀) :
(pullbackP1Iso 𝒰 f g i).hom ≫ pullback.fst _ _ = pullback.snd _ _ := by
simp_rw [pullbackP1Iso, pullback.lift_fst]
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso_hom_fst | null |
pullbackP1Iso_hom_snd (i : 𝒰.I₀) :
(pullbackP1Iso 𝒰 f g i).hom ≫ pullback.snd _ _ = pullback.fst _ _ ≫ p2 𝒰 f g := by
simp_rw [pullbackP1Iso, pullback.lift_snd]
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso_hom_snd | null |
pullbackP1Iso_inv_fst (i : 𝒰.I₀) :
(pullbackP1Iso 𝒰 f g i).inv ≫ pullback.fst _ _ = (gluing 𝒰 f g).ι i := by
simp_rw [pullbackP1Iso, pullback.lift_fst]
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso_inv_fst | null |
pullbackP1Iso_inv_snd (i : 𝒰.I₀) :
(pullbackP1Iso 𝒰 f g i).inv ≫ pullback.snd _ _ = pullback.fst _ _ := by
simp_rw [pullbackP1Iso, pullback.lift_snd]
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso_inv_snd | null |
pullbackP1Iso_hom_ι (i : 𝒰.I₀) :
(pullbackP1Iso 𝒰 f g i).hom ≫ Multicoequalizer.π (gluing 𝒰 f g).diagram i =
pullback.fst _ _ := by
rw [← gluing_ι, ← pullbackP1Iso_inv_fst, Iso.hom_inv_id_assoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackP1Iso_hom_ι | null |
gluedIsLimit : IsLimit (PullbackCone.mk _ _ (p_comm 𝒰 f g)) := by
apply PullbackCone.isLimitAux'
intro s
refine ⟨gluedLift 𝒰 f g s, gluedLift_p1 𝒰 f g s, gluedLift_p2 𝒰 f g s, ?_⟩
intro m h₁ h₂
simp_rw [PullbackCone.mk_pt, PullbackCone.mk_π_app] at h₁ h₂
apply Cover.hom_ext <| 𝒰.pullback₁ s.fst
intro i
rw [gluedLift, (Cover.ι_glueMorphisms <| 𝒰.pullback₁ s.fst)]
dsimp only [Precoverage.ZeroHypercover.pullback₁_toPreZeroHypercover,
PreZeroHypercover.pullback₁_X, PullbackCone.mk_pt, PreZeroHypercover.pullback₁_f, gluing_ι]
rw [← cancel_epi
(pullbackRightPullbackFstIso (p1 𝒰 f g) (𝒰.f i) m ≪≫ pullback.congrHom h₁ rfl).hom,
Iso.trans_hom, Category.assoc, pullback.congrHom_hom, pullback.lift_fst_assoc,
Category.comp_id, pullbackRightPullbackFstIso_hom_fst_assoc, pullback.condition]
conv_lhs => rhs; rw [← pullbackP1Iso_hom_ι]
simp_rw [← Category.assoc]
congr 1
apply pullback.hom_ext
· simp_rw [Category.assoc, pullbackP1Iso_hom_fst, pullback.lift_fst, Category.comp_id,
pullbackSymmetry_hom_comp_fst, pullback.lift_snd, Category.comp_id,
pullbackRightPullbackFstIso_hom_snd]
· simp_rw [Category.assoc, pullbackP1Iso_hom_snd, pullback.lift_snd,
pullbackSymmetry_hom_comp_snd_assoc, pullback.lift_fst_assoc, Category.comp_id,
pullbackRightPullbackFstIso_hom_fst_assoc, ← pullback.condition_assoc, h₂]
include 𝒰 in | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | gluedIsLimit | The glued scheme (`(gluing 𝒰 f g).glued`) is indeed the pullback of `f` and `g`. |
hasPullback_of_cover : HasPullback f g :=
⟨⟨⟨_, gluedIsLimit 𝒰 f g⟩⟩⟩ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | hasPullback_of_cover | null |
affine_hasPullback {A B C : CommRingCat}
(f : Spec A ⟶ Spec C)
(g : Spec B ⟶ Spec C) : HasPullback f g := by
rw [← Scheme.Spec.map_preimage f, ← Scheme.Spec.map_preimage g]
exact ⟨⟨⟨_, isLimitOfHasPullbackOfPreservesLimit
Scheme.Spec (Scheme.Spec.preimage f) (Scheme.Spec.preimage g)⟩⟩⟩ | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | affine_hasPullback | null |
affine_affine_hasPullback {B C : CommRingCat} {X : Scheme}
(f : X ⟶ Spec C) (g : Spec B ⟶ Spec C) :
HasPullback f g :=
hasPullback_of_cover X.affineCover f g | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | affine_affine_hasPullback | null |
base_affine_hasPullback {C : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec C)
(g : Y ⟶ Spec C) : HasPullback f g :=
@hasPullback_symmetry _ _ _ _ _ _ _
(@hasPullback_of_cover _ _ _ Y.affineCover g f fun _ =>
@hasPullback_symmetry _ _ _ _ _ _ _ <| affine_affine_hasPullback _ _) | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | base_affine_hasPullback | null |
left_affine_comp_pullback_hasPullback {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z)
(i : Z.affineCover.I₀) : HasPullback ((Z.affineCover.pullback₁ f).f i ≫ f) g := by
simpa [pullback.condition] using
hasPullback_assoc_symm f (Z.affineCover.f i) (Z.affineCover.f i) g | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | left_affine_comp_pullback_hasPullback | null |
isAffine_of_isAffine_isAffine_isAffine {X Y Z : Scheme}
(f : X ⟶ Z) (g : Y ⟶ Z) [IsAffine X] [IsAffine Y] [IsAffine Z] :
IsAffine (pullback f g) :=
.of_isIso
(pullback.map f g (Spec.map (Γ.map f.op)) (Spec.map (Γ.map g.op))
X.toSpecΓ Y.toSpecΓ Z.toSpecΓ
(Scheme.toSpecΓ_naturality f) (Scheme.toSpecΓ_naturality g) ≫
(PreservesPullback.iso Scheme.Spec _ _).inv) | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | isAffine_of_isAffine_isAffine_isAffine | null |
_root_.AlgebraicGeometry.Scheme.isEmpty_pullback
{X Y S : Scheme.{u}} (f : X ⟶ S) (g : Y ⟶ S)
(H : Disjoint (Set.range f.base) (Set.range g.base)) : IsEmpty ↑(Limits.pullback f g) :=
isEmpty_of_commSq (IsPullback.of_hasPullback f g).toCommSq H | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | _root_.AlgebraicGeometry.Scheme.isEmpty_pullback | null |
@[simps! I₀ X f]
openCoverOfLeft (𝒰 : OpenCover X) (f : X ⟶ Z) (g : Y ⟶ Z) : OpenCover (pullback f g) := by
fapply
((gluing 𝒰 f g).openCover.pushforwardIso
(limit.isoLimitCone ⟨_, gluedIsLimit 𝒰 f g⟩).inv).copy
𝒰.I₀ (fun i => pullback (𝒰.f i ≫ f) g)
(fun i => pullback.map _ _ _ _ (𝒰.f i) (𝟙 _) (𝟙 _) (Category.comp_id _) (by simp))
(Equiv.refl 𝒰.I₀) fun _ => Iso.refl _
rintro (i : 𝒰.I₀)
simp_rw [Cover.pushforwardIso_I₀, Cover.pushforwardIso_f, GlueData.openCover_f,
GlueData.openCover_I₀, gluing_J]
exact pullback.hom_ext (by simp [p1]) (by simp [p2]) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | openCoverOfLeft | Given an open cover `{ Xᵢ }` of `X`, then `X ×[Z] Y` is covered by `Xᵢ ×[Z] Y`. |
@[simps! I₀ X f]
openCoverOfRight (𝒰 : OpenCover Y) (f : X ⟶ Z) (g : Y ⟶ Z) : OpenCover (pullback f g) := by
fapply
((openCoverOfLeft 𝒰 g f).pushforwardIso (pullbackSymmetry _ _).hom).copy 𝒰.I₀
(fun i => pullback f (𝒰.f i ≫ g))
(fun i => pullback.map _ _ _ _ (𝟙 _) (𝒰.f i) (𝟙 _) (by simp) (Category.comp_id _))
(Equiv.refl _) fun i => pullbackSymmetry _ _
intro i
dsimp
apply pullback.hom_ext <;> simp | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | openCoverOfRight | Given an open cover `{ Yᵢ }` of `Y`, then `X ×[Z] Y` is covered by `X ×[Z] Yᵢ`. |
@[simps! I₀ X f]
openCoverOfLeftRight (𝒰X : X.OpenCover) (𝒰Y : Y.OpenCover) (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback f g).OpenCover := by
fapply
Cover.copy ((openCoverOfLeft 𝒰X f g).bind fun x => openCoverOfRight 𝒰Y (𝒰X.f x ≫ f) g)
(𝒰X.I₀ × 𝒰Y.I₀) (fun ij => pullback (𝒰X.f ij.1 ≫ f) (𝒰Y.f ij.2 ≫ g))
(fun ij =>
pullback.map _ _ _ _ (𝒰X.f ij.1) (𝒰Y.f ij.2) (𝟙 _) (Category.comp_id _)
(Category.comp_id _))
(Equiv.sigmaEquivProd _ _).symm fun _ => Iso.refl _
rintro ⟨i, j⟩
apply pullback.hom_ext <;> simp | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | openCoverOfLeftRight | Given an open cover `{ Xᵢ }` of `X` and an open cover `{ Yⱼ }` of `Y`, then
`X ×[Z] Y` is covered by `Xᵢ ×[Z] Yⱼ`. |
@[simps! f]
openCoverOfBase' (𝒰 : OpenCover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : OpenCover (pullback f g) := by
apply (openCoverOfLeft (𝒰.pullback₁ f) f g).bind
intro i
haveI := ((IsPullback.of_hasPullback (pullback.snd g (𝒰.f i))
(pullback.snd f (𝒰.f i))).paste_horiz (IsPullback.of_hasPullback _ _)).flip
refine
@coverOfIsIso _ _ _ _ _
(f := (pullbackSymmetry (pullback.snd f (𝒰.f i)) (pullback.snd g (𝒰.f i))).hom ≫
(limit.isoLimitCone ⟨_, this.isLimit⟩).inv ≫
pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) ?_ ?_) inferInstance
· simp [← pullback.condition]
· simp only [Category.comp_id, Category.id_comp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | openCoverOfBase' | (Implementation). Use `openCoverOfBase` instead. |
@[simps! I₀ X f]
openCoverOfBase (𝒰 : OpenCover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : OpenCover (pullback f g) := by
apply
(openCoverOfBase'.{u, u} 𝒰 f g).copy 𝒰.I₀
(fun i =>
pullback (pullback.snd _ _ : pullback f (𝒰.f i) ⟶ _)
(pullback.snd _ _ : pullback g (𝒰.f i) ⟶ _))
(fun i =>
pullback.map _ _ _ _ (pullback.fst _ _) (pullback.fst _ _) (𝒰.f i)
pullback.condition.symm pullback.condition.symm)
((Equiv.prodPUnit 𝒰.I₀).symm.trans (Equiv.sigmaEquivProd 𝒰.I₀ PUnit).symm) fun _ => Iso.refl _
intro i
rw [Iso.refl_hom, Category.id_comp, openCoverOfBase'_f]
ext : 1 <;>
· simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, Equiv.trans_apply,
Equiv.prodPUnit_symm_apply, Category.assoc, limit.lift_π_assoc, cospan_left, Category.comp_id,
limit.isoLimitCone_inv_π_assoc, PullbackCone.π_app_left, IsPullback.cone_fst,
pullbackSymmetry_hom_comp_snd_assoc, limit.isoLimitCone_inv_π,
PullbackCone.π_app_right, IsPullback.cone_snd, pullbackSymmetry_hom_comp_fst_assoc]
rfl
variable (f : X ⟶ Y) (𝒰 : Y.OpenCover) (𝒱 : ∀ i, ((𝒰.pullback₁ f).X i).OpenCover) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | openCoverOfBase | Given an open cover `{ Zᵢ }` of `Z`, then `X ×[Z] Y` is covered by `Xᵢ ×[Zᵢ] Yᵢ`, where
`Xᵢ = X ×[Z] Zᵢ` and `Yᵢ = Y ×[Z] Zᵢ` is the preimage of `Zᵢ` in `X` and `Y`. |
noncomputable
diagonalCover : (pullback.diagonalObj f).OpenCover :=
(openCoverOfBase 𝒰 f f).bind
fun i ↦ openCoverOfLeftRight (𝒱 i) (𝒱 i) (𝒰.pullbackHom _ _) (𝒰.pullbackHom _ _) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | diagonalCover | Given `𝒰 i` covering `Y` and `𝒱 i j` covering `𝒰 i`, this is the open cover
`𝒱 i j₁ ×[𝒰 i] 𝒱 i j₂` ranging over all `i`, `j₁`, `j₂`. |
noncomputable
diagonalCoverDiagonalRange : (pullback.diagonalObj f).Opens :=
⨆ i : Σ i, (𝒱 i).I₀, ((diagonalCover f 𝒰 𝒱).f ⟨i.1, i.2, i.2⟩).opensRange | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | diagonalCoverDiagonalRange | The image of `𝒱 i j₁ ×[𝒰 i] 𝒱 i j₂` in `diagonalCover` with `j₁ = j₂` |
diagonalCover_map (I) : (diagonalCover f 𝒰 𝒱).f I =
pullback.map _ _ _ _
((𝒱 I.fst).f _ ≫ pullback.fst _ _) ((𝒱 I.fst).f _ ≫ pullback.fst _ _) (𝒰.f _)
(by simp)
(by simp) := by
cases I
ext1 <;> simp [diagonalCover, Cover.pullbackHom] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | diagonalCover_map | null |
noncomputable
diagonalRestrictIsoDiagonal (i j) :
Arrow.mk (pullback.diagonal f ∣_ ((diagonalCover f 𝒰 𝒱).f ⟨i, j, j⟩).opensRange) ≅
Arrow.mk (pullback.diagonal ((𝒱 i).f j ≫ pullback.snd _ _)) := by
refine (morphismRestrictOpensRange _ _).trans ?_
refine Arrow.isoMk ?_ (Iso.refl _) ?_
· exact pullback.congrHom rfl (diagonalCover_map _ _ _ _) ≪≫
pullbackDiagonalMapIso _ _ _ _ ≪≫ (asIso (pullback.diagonal _)).symm
have H : pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
pullback.snd _ _ = pullback.snd _ _ ≫ pullback.fst _ _ := by
rw [← cancel_mono ((𝒱 i).f _)]
apply pullback.hom_ext
· trans pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
(diagonalCover f 𝒰 𝒱).f _ ≫ pullback.snd _ _
· simp [diagonalCover_map]
symm
trans pullback.snd (pullback.diagonal f) ((diagonalCover f 𝒰 𝒱).f ⟨i, (j, j)⟩) ≫
(diagonalCover f 𝒰 𝒱).f _ ≫ pullback.fst _ _
· simp [diagonalCover_map]
· rw [← pullback.condition_assoc, ← pullback.condition_assoc]
simp
· simp [pullback.condition, Cover.pullbackHom]
dsimp [Cover.pullbackHom] at H ⊢
apply pullback.hom_ext
· simp only [Category.assoc, pullback.diagonal_fst, Category.comp_id]
simp only [← Category.assoc, IsIso.comp_inv_eq]
apply pullback.hom_ext <;> simp [H]
· simp only [Category.assoc, pullback.diagonal_snd, Category.comp_id]
simp only [← Category.assoc, IsIso.comp_inv_eq]
apply pullback.hom_ext <;> simp [H] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | diagonalRestrictIsoDiagonal | The restriction of the diagonal `X ⟶ X ×ₛ X` to `𝒱 i j ×[𝒰 i] 𝒱 i j` is the diagonal
`𝒱 i j ⟶ 𝒱 i j ×[𝒰 i] 𝒱 i j`. |
Scheme.pullback_map_isOpenImmersion {X Y S X' Y' S' : Scheme}
(f : X ⟶ S) (g : Y ⟶ S) (f' : X' ⟶ S') (g' : Y' ⟶ S')
(i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f') (e₂ : g ≫ i₃ = i₂ ≫ g')
[IsOpenImmersion i₁] [IsOpenImmersion i₂] [Mono i₃] :
IsOpenImmersion (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂) := by
rw [pullback_map_eq_pullbackFstFstIso_inv]
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | Scheme.pullback_map_isOpenImmersion | null |
noncomputable
pullbackSpecIso :
pullback (Spec.map (CommRingCat.ofHom (algebraMap R S)))
(Spec.map (CommRingCat.ofHom (algebraMap R T))) ≅ Spec(S ⊗[R] T) :=
letI H := IsLimit.equivIsoLimit (PullbackCone.eta _)
(PushoutCocone.isColimitEquivIsLimitOp _ (CommRingCat.pushoutCoconeIsColimit R S T))
limit.isoLimitCone ⟨_, isLimitPullbackConeMapOfIsLimit Scheme.Spec _ H⟩ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackSpecIso | The isomorphism between the fibred product of two schemes `Spec S` and `Spec T`
over a scheme `Spec R` and the `Spec` of the tensor product `S ⊗[R] T`. |
@[reassoc (attr := simp)]
pullbackSpecIso_inv_fst :
(pullbackSpecIso R S T).inv ≫ pullback.fst _ _ = Spec.map (ofHom includeLeftRingHom) :=
limit.isoLimitCone_inv_π _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackSpecIso_inv_fst | The composition of the inverse of the isomorphism `pullbackSpecIso R S T` (from the pullback of
`Spec S ⟶ Spec R` and `Spec T ⟶ Spec R` to `Spec (S ⊗[R] T)`) with the first projection is
the morphism `Spec (S ⊗[R] T) ⟶ Spec S` obtained by applying `Spec.map` to the ring morphism
`s ↦ s ⊗ₜ[R] 1`. |
@[reassoc (attr := simp)]
pullbackSpecIso_inv_snd :
(pullbackSpecIso R S T).inv ≫ pullback.snd _ _ =
Spec.map (ofHom (R := T) (S := S ⊗[R] T) (toRingHom includeRight)) :=
limit.isoLimitCone_inv_π _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackSpecIso_inv_snd | The composition of the inverse of the isomorphism `pullbackSpecIso R S T` (from the pullback of
`Spec S ⟶ Spec R` and `Spec T ⟶ Spec R` to `Spec (S ⊗[R] T)`) with the second projection is
the morphism `Spec (S ⊗[R] T) ⟶ Spec T` obtained by applying `Spec.map` to the ring morphism
`t ↦ 1 ⊗ₜ[R] t`. |
@[reassoc (attr := simp)]
pullbackSpecIso_hom_fst :
(pullbackSpecIso R S T).hom ≫ Spec.map (ofHom includeLeftRingHom) = pullback.fst _ _ := by
rw [← pullbackSpecIso_inv_fst, Iso.hom_inv_id_assoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackSpecIso_hom_fst | The composition of the isomorphism `pullbackSpecIso R S T` (from the pullback of
`Spec S ⟶ Spec R` and `Spec T ⟶ Spec R` to `Spec (S ⊗[R] T)`) with the morphism
`Spec (S ⊗[R] T) ⟶ Spec S` obtained by applying `Spec.map` to the ring morphism `s ↦ s ⊗ₜ[R] 1`
is the first projection. |
@[reassoc (attr := simp)]
pullbackSpecIso_hom_snd :
(pullbackSpecIso R S T).hom ≫ Spec.map (ofHom (toRingHom includeRight)) = pullback.snd _ _ := by
rw [← pullbackSpecIso_inv_snd, Iso.hom_inv_id_assoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | pullbackSpecIso_hom_snd | The composition of the isomorphism `pullbackSpecIso R S T` (from the pullback of
`Spec S ⟶ Spec R` and `Spec T ⟶ Spec R` to `Spec (S ⊗[R] T)`) with the morphism
`Spec (S ⊗[R] T) ⟶ Spec T` obtained by applying `Spec.map` to the ring morphism `t ↦ 1 ⊗ₜ[R] t`
is the second projection. |
isPullback_Spec_map_isPushout {A B C P : CommRingCat} (f : A ⟶ B) (g : A ⟶ C)
(inl : B ⟶ P) (inr : C ⟶ P) (h : IsPushout f g inl inr) :
IsPullback (Spec.map inl) (Spec.map inr) (Spec.map f) (Spec.map g) :=
IsPullback.map Scheme.Spec h.op.flip | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | isPullback_Spec_map_isPushout | null |
isPullback_Spec_map_pushout {A B C : CommRingCat} (f : A ⟶ B) (g : A ⟶ C) :
IsPullback (Spec.map (pushout.inl f g))
(Spec.map (pushout.inr f g)) (Spec.map f) (Spec.map g) := by
apply isPullback_Spec_map_isPushout
exact IsPushout.of_hasPushout f g | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | isPullback_Spec_map_pushout | null |
diagonal_Spec_map :
pullback.diagonal (Spec.map (CommRingCat.ofHom (algebraMap R S))) =
Spec.map (CommRingCat.ofHom (Algebra.TensorProduct.lmul' R : S ⊗[R] S →ₐ[R] S).toRingHom) ≫
(pullbackSpecIso R S S).inv := by
ext1 <;> simp only [pullback.diagonal_fst, pullback.diagonal_snd, ← Spec.map_comp, ← Spec.map_id,
AlgHom.toRingHom_eq_coe, Category.assoc, pullbackSpecIso_inv_fst, pullbackSpecIso_inv_snd]
· congr 1; ext x; change x = Algebra.TensorProduct.lmul' R (S := S) (x ⊗ₜ[R] 1); simp
· congr 1; ext x; change x = Algebra.TensorProduct.lmul' R (S := S) (1 ⊗ₜ[R] x); simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Gluing",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.CategoryTheory.Limits.Shapes.Diagonal",
"Mathlib.CategoryTheory.Monoidal.Cartesian.Over"
] | Mathlib/AlgebraicGeometry/Pullbacks.lean | diagonal_Spec_map | null |
PartialMap (X Y : Scheme.{u}) where
/-- The domain of definition of a partial map. -/
domain : X.Opens
dense_domain : Dense (domain : Set X)
/-- The underlying morphism of a partial map. -/
hom : ↑domain ⟶ Y
variable (S) in | structure | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap | A partial map from `X` to `Y` (`X.PartialMap Y`) is a morphism into `Y`
defined on a dense open subscheme of `X`. |
PartialMap.IsOver [X.Over S] [Y.Over S] (f : X.PartialMap Y) :=
f.hom.IsOver S | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.IsOver | A partial map is a `S`-map if the underlying morphism is. |
ext_iff (f g : X.PartialMap Y) :
f = g ↔ ∃ e : f.domain = g.domain, f.hom = (X.isoOfEq e).hom ≫ g.hom := by
constructor
· rintro rfl
simp only [exists_true_left, Scheme.isoOfEq_rfl, Iso.refl_hom, Category.id_comp]
· obtain ⟨U, hU, f⟩ := f
obtain ⟨V, hV, g⟩ := g
rintro ⟨rfl : U = V, e⟩
congr 1
simpa using e
@[ext] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | ext_iff | null |
ext (f g : X.PartialMap Y) (e : f.domain = g.domain)
(H : f.hom = (X.isoOfEq e).hom ≫ g.hom) : f = g := by
rw [ext_iff]
exact ⟨e, H⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | ext | null |
@[simps hom domain]
noncomputable
restrict (f : X.PartialMap Y) (U : X.Opens)
(hU : Dense (U : Set X)) (hU' : U ≤ f.domain) : X.PartialMap Y where
domain := U
dense_domain := hU
hom := X.homOfLE hU' ≫ f.hom
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict | The restriction of a partial map to a smaller domain. |
restrict_id (f : X.PartialMap Y) : f.restrict f.domain f.dense_domain le_rfl = f := by
ext1 <;> simp [restrict_domain] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict_id | null |
restrict_id_hom (f : X.PartialMap Y) :
(f.restrict f.domain f.dense_domain le_rfl).hom = f.hom := by
simp
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict_id_hom | null |
restrict_restrict (f : X.PartialMap Y)
(U : X.Opens) (hU : Dense (U : Set X)) (hU' : U ≤ f.domain)
(V : X.Opens) (hV : Dense (V : Set X)) (hV' : V ≤ U) :
(f.restrict U hU hU').restrict V hV hV' = f.restrict V hV (hV'.trans hU') := by
ext1 <;> simp [restrict_domain] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict_restrict | null |
restrict_restrict_hom (f : X.PartialMap Y)
(U : X.Opens) (hU : Dense (U : Set X)) (hU' : U ≤ f.domain)
(V : X.Opens) (hV : Dense (V : Set X)) (hV' : V ≤ U) :
((f.restrict U hU hU').restrict V hV hV').hom = (f.restrict V hV (hV'.trans hU')).hom := by
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict_restrict_hom | null |
@[simps]
compHom (f : X.PartialMap Y) (g : Y ⟶ Z) : X.PartialMap Z where
domain := f.domain
dense_domain := f.dense_domain
hom := f.hom ≫ g | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | compHom | The composition of a partial map and a morphism on the right. |
@[simps]
_root_.AlgebraicGeometry.Scheme.Hom.toPartialMap (f : X.Hom Y) :
X.PartialMap Y := ⟨⊤, dense_univ, X.topIso.hom ≫ f⟩ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | _root_.AlgebraicGeometry.Scheme.Hom.toPartialMap | A scheme morphism as a partial map. |
isOver_iff [X.Over S] [Y.Over S] {f : X.PartialMap Y} :
f.IsOver S ↔ (f.compHom (Y ↘ S)).hom = f.domain.ι ≫ X ↘ S := by
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | isOver_iff | null |
isOver_iff_eq_restrict [X.Over S] [Y.Over S] {f : X.PartialMap Y} :
f.IsOver S ↔ f.compHom (Y ↘ S) = (X ↘ S).toPartialMap.restrict _ f.dense_domain (by simp) := by
simp [PartialMap.ext_iff] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | isOver_iff_eq_restrict | null |
noncomputable
fromSpecStalkOfMem (f : X.PartialMap Y) {x} (hx : x ∈ f.domain) :
Spec (X.presheaf.stalk x) ⟶ Y :=
f.domain.fromSpecStalkOfMem x hx ≫ f.hom | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromSpecStalkOfMem | If `x` is in the domain of a partial map `f`, then `f` restricts to a map from `Spec 𝒪_x`. |
noncomputable
fromFunctionField [IrreducibleSpace X] (f : X.PartialMap Y) :
Spec X.functionField ⟶ Y :=
f.fromSpecStalkOfMem
((genericPoint_specializes _).mem_open f.domain.2 f.dense_domain.nonempty.choose_spec) | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromFunctionField | A partial map restricts to a map from `Spec K(X)`. |
fromSpecStalkOfMem_restrict (f : X.PartialMap Y)
{U : X.Opens} (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) {x} (hx : x ∈ U) :
(f.restrict U hU hU').fromSpecStalkOfMem hx = f.fromSpecStalkOfMem (hU' hx) := by
dsimp only [fromSpecStalkOfMem, restrict, Scheme.Opens.fromSpecStalkOfMem]
have e : ⟨x, hU' hx⟩ = (X.homOfLE hU').base ⟨x, hx⟩ := by
rw [Scheme.homOfLE_base]
rfl
rw [Category.assoc, ← Spec_map_stalkMap_fromSpecStalk_assoc,
← Spec_map_stalkSpecializes_fromSpecStalk (Inseparable.of_eq e).specializes,
← TopCat.Presheaf.stalkCongr_inv _ (Inseparable.of_eq e)]
simp only [← Category.assoc, ← Spec.map_comp]
congr 3
rw [Iso.eq_inv_comp, ← Category.assoc, IsIso.comp_inv_eq, IsIso.eq_inv_comp,
stalkMap_congr_hom _ _ (X.homOfLE_ι hU').symm]
simp only [TopCat.Presheaf.stalkCongr_hom]
rw [← stalkSpecializes_stalkMap_assoc, stalkMap_comp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromSpecStalkOfMem_restrict | null |
fromFunctionField_restrict (f : X.PartialMap Y) [IrreducibleSpace X]
{U : X.Opens} (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) :
(f.restrict U hU hU').fromFunctionField = f.fromFunctionField :=
fromSpecStalkOfMem_restrict f _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromFunctionField_restrict | null |
noncomputable
ofFromSpecStalk [IrreducibleSpace X] [LocallyOfFiniteType sY] {x : X} [X.IsGermInjectiveAt x]
(φ : Spec (X.presheaf.stalk x) ⟶ Y) (h : φ ≫ sY = X.fromSpecStalk x ≫ sX) : X.PartialMap Y where
hom := (spread_out_of_isGermInjective' sX sY φ h).choose_spec.choose_spec.choose
domain := (spread_out_of_isGermInjective' sX sY φ h).choose
dense_domain := (spread_out_of_isGermInjective' sX sY φ h).choose.2.dense
⟨_, (spread_out_of_isGermInjective' sX sY φ h).choose_spec.choose⟩ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | ofFromSpecStalk | Given `S`-schemes `X` and `Y` such that `Y` is locally of finite type and
`X` is irreducible germ-injective at `x` (e.g. when `X` is integral),
any `S`-morphism `Spec 𝒪ₓ ⟶ Y` spreads out to a partial map from `X` to `Y`. |
ofFromSpecStalk_comp [IrreducibleSpace X] [LocallyOfFiniteType sY]
{x : X} [X.IsGermInjectiveAt x] (φ : Spec (X.presheaf.stalk x) ⟶ Y)
(h : φ ≫ sY = X.fromSpecStalk x ≫ sX) :
(ofFromSpecStalk sX sY φ h).hom ≫ sY = (ofFromSpecStalk sX sY φ h).domain.ι ≫ sX :=
(spread_out_of_isGermInjective' sX sY φ h).choose_spec.choose_spec.choose_spec.2 | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | ofFromSpecStalk_comp | null |
mem_domain_ofFromSpecStalk [IrreducibleSpace X] [LocallyOfFiniteType sY]
{x : X} [X.IsGermInjectiveAt x] (φ : Spec (X.presheaf.stalk x) ⟶ Y)
(h : φ ≫ sY = X.fromSpecStalk x ≫ sX) : x ∈ (ofFromSpecStalk sX sY φ h).domain :=
(spread_out_of_isGermInjective' sX sY φ h).choose_spec.choose | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | mem_domain_ofFromSpecStalk | null |
fromSpecStalkOfMem_ofFromSpecStalk [IrreducibleSpace X] [LocallyOfFiniteType sY]
{x : X} [X.IsGermInjectiveAt x] (φ : Spec (X.presheaf.stalk x) ⟶ Y)
(h : φ ≫ sY = X.fromSpecStalk x ≫ sX) :
(ofFromSpecStalk sX sY φ h).fromSpecStalkOfMem (mem_domain_ofFromSpecStalk sX sY φ h) = φ :=
(spread_out_of_isGermInjective' sX sY φ h).choose_spec.choose_spec.choose_spec.1.symm
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromSpecStalkOfMem_ofFromSpecStalk | null |
fromSpecStalkOfMem_compHom (f : X.PartialMap Y) (g : Y ⟶ Z) (x) (hx) :
(f.compHom g).fromSpecStalkOfMem (x := x) hx = f.fromSpecStalkOfMem hx ≫ g := by
simp [fromSpecStalkOfMem]
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromSpecStalkOfMem_compHom | null |
fromSpecStalkOfMem_toPartialMap (f : X ⟶ Y) (x) :
f.toPartialMap.fromSpecStalkOfMem (x := x) trivial = X.fromSpecStalk x ≫ f := by
simp [fromSpecStalkOfMem] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | fromSpecStalkOfMem_toPartialMap | null |
protected noncomputable
equiv (f g : X.PartialMap Y) : Prop :=
∃ (W : X.Opens) (hW : Dense (W : Set X)) (hWl : W ≤ f.domain) (hWr : W ≤ g.domain),
(f.restrict W hW hWl).hom = (g.restrict W hW hWr).hom | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv | Two partial maps are equivalent if they are equal on a dense open subscheme. |
equivalence_rel : Equivalence (@Scheme.PartialMap.equiv X Y) where
refl f := ⟨f.domain, f.dense_domain, by simp⟩
symm {f g} := by
intro ⟨W, hW, hWl, hWr, e⟩
exact ⟨W, hW, hWr, hWl, e.symm⟩
trans {f g h} := by
intro ⟨W₁, hW₁, hW₁l, hW₁r, e₁⟩ ⟨W₂, hW₂, hW₂l, hW₂r, e₂⟩
refine ⟨W₁ ⊓ W₂, hW₁.inter_of_isOpen_left hW₂ W₁.2, inf_le_left.trans hW₁l,
inf_le_right.trans hW₂r, ?_⟩
dsimp at e₁ e₂
simp only [restrict_domain, restrict_hom, ← X.homOfLE_homOfLE (U := W₁ ⊓ W₂) inf_le_left hW₁l,
Category.assoc, e₁, ← X.homOfLE_homOfLE (U := W₁ ⊓ W₂) inf_le_right hW₂r, ← e₂]
simp only [homOfLE_homOfLE_assoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equivalence_rel | null |
restrict_equiv (f : X.PartialMap Y) (U : X.Opens)
(hU : Dense (U : Set X)) (hU' : U ≤ f.domain) : (f.restrict U hU hU').equiv f :=
⟨U, hU, le_rfl, hU', by simp⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | restrict_equiv | null |
equiv_of_fromSpecStalkOfMem_eq [IrreducibleSpace X]
{x : X} [X.IsGermInjectiveAt x] (f g : X.PartialMap Y)
(hxf : x ∈ f.domain) (hxg : x ∈ g.domain)
(H : f.fromSpecStalkOfMem hxf = g.fromSpecStalkOfMem hxg) : f.equiv g := by
have hdense : Dense ((f.domain ⊓ g.domain) : Set X) :=
f.dense_domain.inter_of_isOpen_left g.dense_domain f.domain.2
have := (isGermInjectiveAt_iff_of_isOpenImmersion (f := (f.domain ⊓ g.domain).ι)
(x := ⟨x, hxf, hxg⟩)).mp ‹_›
have := spread_out_unique_of_isGermInjective' (X := (f.domain ⊓ g.domain).toScheme)
(X.homOfLE inf_le_left ≫ f.hom) (X.homOfLE inf_le_right ≫ g.hom) (x := ⟨x, hxf, hxg⟩) ?_
· obtain ⟨U, hxU, e⟩ := this
refine ⟨(f.domain ⊓ g.domain).ι ''ᵁ U, ((f.domain ⊓ g.domain).ι ''ᵁ U).2.dense
⟨_, ⟨_, hxU, rfl⟩⟩,
((Set.image_subset_range _ _).trans_eq (Subtype.range_val)).trans inf_le_left,
((Set.image_subset_range _ _).trans_eq (Subtype.range_val)).trans inf_le_right, ?_⟩
rw [← cancel_epi (Scheme.Hom.isoImage _ _).hom]
simp only [restrict_hom, ← Category.assoc] at e ⊢
convert e using 2 <;> rw [← cancel_mono (Scheme.Opens.ι _)] <;> simp
· rw [← f.fromSpecStalkOfMem_restrict hdense inf_le_left ⟨hxf, hxg⟩,
← g.fromSpecStalkOfMem_restrict hdense inf_le_right ⟨hxf, hxg⟩] at H
simpa only [fromSpecStalkOfMem, restrict_domain, Opens.fromSpecStalkOfMem, Spec.map_inv,
restrict_hom, Category.assoc, IsIso.eq_inv_comp, IsIso.hom_inv_id_assoc] using H | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv_of_fromSpecStalkOfMem_eq | null |
Opens.isDominant_ι {U : X.Opens} (hU : Dense (X := X) U) : IsDominant U.ι :=
⟨by simpa [DenseRange] using hU⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | Opens.isDominant_ι | null |
Opens.isDominant_homOfLE {U V : X.Opens} (hU : Dense (X := X) U) (hU' : U ≤ V) :
IsDominant (X.homOfLE hU') :=
have : IsDominant (X.homOfLE hU' ≫ Opens.ι _) := by simpa using Opens.isDominant_ι hU
IsDominant.of_comp_of_isOpenImmersion (g := Opens.ι _) _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | Opens.isDominant_homOfLE | null |
equiv_iff_of_isSeparated_of_le [X.Over S] [Y.Over S] [IsReduced X]
[IsSeparated (Y ↘ S)] {f g : X.PartialMap Y} [f.IsOver S] [g.IsOver S]
{W : X.Opens} (hW : Dense (X := X) W) (hWl : W ≤ f.domain) (hWr : W ≤ g.domain) : f.equiv g ↔
(f.restrict W hW hWl).hom = (g.restrict W hW hWr).hom := by
refine ⟨fun ⟨V, hV, hVl, hVr, e⟩ ↦ ?_, fun e ↦ ⟨_, _, _, _, e⟩⟩
have : IsDominant (X.homOfLE (inf_le_left : W ⊓ V ≤ W)) :=
Opens.isDominant_homOfLE (hW.inter_of_isOpen_left hV W.2) _
apply ext_of_isDominant_of_isSeparated' S (X.homOfLE (inf_le_left : W ⊓ V ≤ W))
simpa using congr(X.homOfLE (inf_le_right : W ⊓ V ≤ V) ≫ $e) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv_iff_of_isSeparated_of_le | Two partial maps from reduced schemes to separated schemes are equivalent if and only if
they are equal on **any** open dense subset. |
equiv_iff_of_isSeparated [X.Over S] [Y.Over S] [IsReduced X]
[IsSeparated (Y ↘ S)] {f g : X.PartialMap Y}
[f.IsOver S] [g.IsOver S] : f.equiv g ↔
(f.restrict _ (f.2.inter_of_isOpen_left g.2 f.domain.2) inf_le_left).hom =
(g.restrict _ (f.2.inter_of_isOpen_left g.2 f.domain.2) inf_le_right).hom :=
equiv_iff_of_isSeparated_of_le (S := S) _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv_iff_of_isSeparated | Two partial maps from reduced schemes to separated schemes are equivalent if and only if
they are equal on the intersection of the domains. |
equiv_iff_of_domain_eq_of_isSeparated [X.Over S] [Y.Over S] [IsReduced X]
[IsSeparated (Y ↘ S)] {f g : X.PartialMap Y} (hfg : f.domain = g.domain)
[f.IsOver S] [g.IsOver S] : f.equiv g ↔ f = g := by
rw [equiv_iff_of_isSeparated_of_le (S := S) f.dense_domain le_rfl hfg.le]
obtain ⟨Uf, _, f⟩ := f
obtain ⟨Ug, _, g⟩ := g
obtain rfl : Uf = Ug := hfg
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv_iff_of_domain_eq_of_isSeparated | Two partial maps from reduced schemes to separated schemes with the same domain are equivalent
if and only if they are equal. |
equiv_toPartialMap_iff_of_isSeparated [X.Over S] [Y.Over S] [IsReduced X]
[IsSeparated (Y ↘ S)] {f : X.PartialMap Y} {g : X ⟶ Y}
[f.IsOver S] [g.IsOver S] : f.equiv g.toPartialMap ↔
f.hom = f.domain.ι ≫ g := by
rw [equiv_iff_of_isSeparated (S := S), ← cancel_epi (X.isoOfEq (inf_top_eq f.domain)).hom]
simp
rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | equiv_toPartialMap_iff_of_isSeparated | A partial map from a reduced scheme to a separated scheme is equivalent to a morphism
if and only if it is equal to the restriction of the morphism. |
RationalMap (X Y : Scheme.{u}) : Type u :=
@Quotient (X.PartialMap Y) inferInstance | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap | A rational map from `X` to `Y` (`X ⤏ Y`) is an equivalence class of partial maps,
where two partial maps are equivalent if they are equal on a dense open subscheme. |
PartialMap.toRationalMap (f : X.PartialMap Y) : X ⤏ Y := Quotient.mk _ f | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.toRationalMap | The notation for rational maps. -/
scoped[AlgebraicGeometry] infix:10 " ⤏ " => Scheme.RationalMap
/-- A partial map as a rational map. |
Hom.toRationalMap (f : X.Hom Y) : X ⤏ Y := f.toPartialMap.toRationalMap
variable (S) in | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | Hom.toRationalMap | A scheme morphism as a rational map. |
RationalMap.IsOver [X.Over S] [Y.Over S] (f : X ⤏ Y) : Prop where
exists_partialMap_over : ∃ g : X.PartialMap Y, g.IsOver S ∧ g.toRationalMap = f | class | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.IsOver | A rational map is a `S`-map if some partial map in the equivalence class is a `S`-map. |
PartialMap.toRationalMap_surjective : Function.Surjective (@toRationalMap X Y) :=
Quotient.exists_rep | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.toRationalMap_surjective | null |
RationalMap.exists_rep (f : X ⤏ Y) : ∃ g : X.PartialMap Y, g.toRationalMap = f :=
Quotient.exists_rep f | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.exists_rep | null |
PartialMap.toRationalMap_eq_iff {f g : X.PartialMap Y} :
f.toRationalMap = g.toRationalMap ↔ f.equiv g :=
Quotient.eq
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.toRationalMap_eq_iff | null |
PartialMap.restrict_toRationalMap (f : X.PartialMap Y) (U : X.Opens)
(hU : Dense (U : Set X)) (hU' : U ≤ f.domain) :
(f.restrict U hU hU').toRationalMap = f.toRationalMap :=
toRationalMap_eq_iff.mpr (f.restrict_equiv U hU hU') | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.restrict_toRationalMap | null |
RationalMap.exists_partialMap_over [X.Over S] [Y.Over S] (f : X ⤏ Y) [f.IsOver S] :
∃ g : X.PartialMap Y, g.IsOver S ∧ g.toRationalMap = f :=
IsOver.exists_partialMap_over | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.exists_partialMap_over | null |
RationalMap.compHom (f : X ⤏ Y) (g : Y ⟶ Z) : X ⤏ Z := by
refine Quotient.map (PartialMap.compHom · g) ?_ f
intro f₁ f₂ ⟨W, hW, hWl, hWr, e⟩
refine ⟨W, hW, hWl, hWr, ?_⟩
simp only [PartialMap.restrict_domain, PartialMap.restrict_hom, PartialMap.compHom_domain,
PartialMap.compHom_hom] at e ⊢
rw [reassoc_of% e]
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.compHom | The composition of a rational map and a morphism on the right. |
RationalMap.compHom_toRationalMap (f : X.PartialMap Y) (g : Y ⟶ Z) :
(f.compHom g).toRationalMap = f.toRationalMap.compHom g := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.compHom_toRationalMap | null |
PartialMap.exists_restrict_isOver [X.Over S] [Y.Over S] (f : X.PartialMap Y)
[f.toRationalMap.IsOver S] : ∃ U hU hU', (f.restrict U hU hU').IsOver S := by
obtain ⟨f', hf₁, hf₂⟩ := RationalMap.IsOver.exists_partialMap_over (S := S) (f := f.toRationalMap)
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp hf₂
exact ⟨U, hU, hUr, by rw [IsOver, ← e]; infer_instance⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.exists_restrict_isOver | null |
RationalMap.isOver_iff [X.Over S] [Y.Over S] {f : X ⤏ Y} :
f.IsOver S ↔ f.compHom (Y ↘ S) = (X ↘ S).toRationalMap := by
constructor
· intro h
obtain ⟨g, hg, e⟩ := f.exists_partialMap_over S
rw [← e, Hom.toRationalMap, ← compHom_toRationalMap, PartialMap.isOver_iff_eq_restrict.mp hg,
PartialMap.restrict_toRationalMap]
· intro e
obtain ⟨f, rfl⟩ := PartialMap.toRationalMap_surjective f
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp e
exact ⟨⟨f.restrict U hU hUl, by simpa using e, by simp⟩⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.isOver_iff | null |
PartialMap.isOver_toRationalMap_iff_of_isSeparated [X.Over S] [Y.Over S] [IsReduced X]
[S.IsSeparated] {f : X.PartialMap Y} :
f.toRationalMap.IsOver S ↔ f.IsOver S := by
refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩
obtain ⟨U, hU, hU', H⟩ := f.exists_restrict_isOver (S := S)
rw [isOver_iff]
have : IsDominant (X.homOfLE hU') := Opens.isDominant_homOfLE hU _
exact ext_of_isDominant (ι := X.homOfLE hU') (by simpa using H.1) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.isOver_toRationalMap_iff_of_isSeparated | null |
noncomputable
RationalMap.fromFunctionField [IrreducibleSpace X] (f : X ⤏ Y) :
Spec X.functionField ⟶ Y := by
refine Quotient.lift PartialMap.fromFunctionField ?_ f
intro f g ⟨W, hW, hWl, hWr, e⟩
have : f.restrict W hW hWl = g.restrict W hW hWr := by ext1; rfl; rw [e]; simp
rw [← f.fromFunctionField_restrict hW hWl, this, g.fromFunctionField_restrict]
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.fromFunctionField | A rational map restricts to a map from `Spec K(X)`. |
RationalMap.fromFunctionField_toRationalMap [IrreducibleSpace X] (f : X.PartialMap Y) :
f.toRationalMap.fromFunctionField = f.fromFunctionField := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.fromFunctionField_toRationalMap | null |
noncomputable
RationalMap.ofFunctionField [IsIntegral X] [LocallyOfFiniteType sY]
(f : Spec X.functionField ⟶ Y) (h : f ≫ sY = X.fromSpecStalk _ ≫ sX) : X ⤏ Y :=
(PartialMap.ofFromSpecStalk sX sY f h).toRationalMap | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.ofFunctionField | Given `S`-schemes `X` and `Y` such that `Y` is locally of finite type and `X` is integral,
any `S`-morphism `Spec K(X) ⟶ Y` spreads out to a rational map from `X` to `Y`. |
RationalMap.fromFunctionField_ofFunctionField [IsIntegral X] [LocallyOfFiniteType sY]
(f : Spec X.functionField ⟶ Y) (h : f ≫ sY = X.fromSpecStalk _ ≫ sX) :
(ofFunctionField sX sY f h).fromFunctionField = f :=
PartialMap.fromSpecStalkOfMem_ofFromSpecStalk sX sY _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.fromFunctionField_ofFunctionField | null |
RationalMap.eq_of_fromFunctionField_eq [IsIntegral X] (f g : X.RationalMap Y)
(H : f.fromFunctionField = g.fromFunctionField) : f = g := by
obtain ⟨f, rfl⟩ := f.exists_rep
obtain ⟨g, rfl⟩ := g.exists_rep
refine PartialMap.toRationalMap_eq_iff.mpr ?_
exact PartialMap.equiv_of_fromSpecStalkOfMem_eq _ _ _ _ H | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.eq_of_fromFunctionField_eq | null |
noncomputable
RationalMap.equivFunctionField [IsIntegral X] [LocallyOfFiniteType sY] :
{ f : Spec X.functionField ⟶ Y // f ≫ sY = X.fromSpecStalk _ ≫ sX } ≃
{ f : X ⤏ Y // f.compHom sY = sX.toRationalMap } where
toFun f := ⟨.ofFunctionField sX sY f f.2, PartialMap.toRationalMap_eq_iff.mpr
⟨_, PartialMap.dense_domain _, le_rfl, le_top, by simp [PartialMap.ofFromSpecStalk_comp]⟩⟩
invFun f := ⟨f.1.fromFunctionField, by
obtain ⟨f, hf⟩ := f
obtain ⟨f, rfl⟩ := f.exists_rep
simpa [fromFunctionField_toRationalMap] using congr(RationalMap.fromFunctionField $hf)⟩
left_inv f := Subtype.ext (RationalMap.fromFunctionField_ofFunctionField _ _ _ _)
right_inv f := Subtype.ext (RationalMap.eq_of_fromFunctionField_eq
(ofFunctionField sX sY f.1.fromFunctionField _) f
(RationalMap.fromFunctionField_ofFunctionField _ _ _ _)) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.equivFunctionField | Given `S`-schemes `X` and `Y` such that `Y` is locally of finite type and `X` is integral,
`S`-morphisms `Spec K(X) ⟶ Y` correspond bijectively to `S`-rational maps from `X` to `Y`. |
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