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 ⌀ |
|---|---|---|---|---|---|---|
noncomputable
RationalMap.equivFunctionFieldOver [X.Over S] [Y.Over S] [IsIntegral X]
[LocallyOfFiniteType (Y ↘ S)] :
{ f : Spec X.functionField ⟶ Y // f.IsOver S } ≃ { f : X ⤏ Y // f.IsOver S } :=
((Equiv.subtypeEquivProp (by simp only [Hom.isOver_iff]; rfl)).trans
(RationalMap.equivFunctionField (X ↘ S) (Y ↘ S))).trans
(Equiv.subtypeEquivProp (by ext f; rw [RationalMap.isOver_iff])) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.equivFunctionFieldOver | 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`. |
RationalMap.domain (f : X ⤏ Y) : X.Opens :=
sSup { PartialMap.domain g | (g) (_ : g.toRationalMap = f) } | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.domain | The domain of definition of a rational map. |
PartialMap.le_domain_toRationalMap (f : X.PartialMap Y) :
f.domain ≤ f.toRationalMap.domain :=
le_sSup ⟨f, rfl, rfl⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.le_domain_toRationalMap | null |
RationalMap.mem_domain {f : X ⤏ Y} {x} :
x ∈ f.domain ↔ ∃ g : X.PartialMap Y, x ∈ g.domain ∧ g.toRationalMap = f :=
TopologicalSpace.Opens.mem_sSup.trans (by simp [@and_comm (x ∈ _)]) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.mem_domain | null |
RationalMap.dense_domain (f : X ⤏ Y) : Dense (X := X) f.domain :=
f.inductionOn (fun g ↦ g.dense_domain.mono g.le_domain_toRationalMap) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.dense_domain | null |
noncomputable
RationalMap.openCoverDomain (f : X ⤏ Y) : f.domain.toScheme.OpenCover where
I₀ := { PartialMap.domain g | (g) (_ : g.toRationalMap = f) }
X U := U.1.toScheme
f U := X.homOfLE (le_sSup U.2)
mem₀ := by
rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, inferInstance⟩
use ⟨_, (TopologicalSpace.Opens.mem_sSup.mp x.2).choose_spec.1⟩
exact ⟨⟨x.1, (TopologicalSpace.Opens.mem_sSup.mp x.2).choose_spec.2⟩, Subtype.ext (by simp)⟩ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.openCoverDomain | The open cover of the domain of `f : X ⤏ Y`,
consisting of all the domains of the partial maps in the equivalence class. |
noncomputable
RationalMap.toPartialMap [IsReduced X] [Y.IsSeparated] (f : X ⤏ Y) : X.PartialMap Y := by
refine ⟨f.domain, f.dense_domain, f.openCoverDomain.glueMorphisms
(fun x ↦ (X.isoOfEq x.2.choose_spec.2).inv ≫ x.2.choose.hom) ?_⟩
intro x y
let g (x : f.openCoverDomain.I₀) := x.2.choose
have hg₁ (x) : (g x).toRationalMap = f := x.2.choose_spec.1
have hg₂ (x) : (g x).domain = x.1 := x.2.choose_spec.2
refine (cancel_epi (isPullback_opens_inf_le (le_sSup x.2) (le_sSup y.2)).isoPullback.hom).mp ?_
simp only [openCoverDomain, IsPullback.isoPullback_hom_fst_assoc,
IsPullback.isoPullback_hom_snd_assoc]
change _ ≫ _ ≫ (g x).hom = _ ≫ _ ≫ (g y).hom
simp_rw [← cancel_epi (X.isoOfEq congr($(hg₂ x) ⊓ $(hg₂ y))).hom, ← Category.assoc]
convert (PartialMap.equiv_iff_of_isSeparated (S := ⊤_ _) (f := g x) (g := g y)).mp ?_ using 1
· dsimp; congr 1; simp [g, ← cancel_mono (Opens.ι _)]
· dsimp; congr 1; simp [g, ← cancel_mono (Opens.ι _)]
· rw [← PartialMap.toRationalMap_eq_iff, hg₁, hg₁] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.toPartialMap | If `f : X ⤏ Y` is a rational map from a reduced scheme to a separated scheme,
then `f` can be represented as a partial map on its domain of definition. |
PartialMap.toPartialMap_toRationalMap_restrict [IsReduced X] [Y.IsSeparated]
(f : X.PartialMap Y) : (f.toRationalMap.toPartialMap.restrict _ f.dense_domain
f.le_domain_toRationalMap).hom = f.hom := by
dsimp [RationalMap.toPartialMap]
refine (f.toRationalMap.openCoverDomain.ι_glueMorphisms _ _ ⟨_, f, rfl, rfl⟩).trans ?_
generalize_proofs _ _ H _
have : H.choose = f := (equiv_iff_of_domain_eq_of_isSeparated (S := ⊤_ _) H.choose_spec.2).mp
(toRationalMap_eq_iff.mp H.choose_spec.1)
exact ((ext_iff _ _).mp this.symm).choose_spec.symm
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | PartialMap.toPartialMap_toRationalMap_restrict | null |
RationalMap.toRationalMap_toPartialMap [IsReduced X] [Y.IsSeparated]
(f : X ⤏ Y) : f.toPartialMap.toRationalMap = f := by
obtain ⟨f, rfl⟩ := PartialMap.toRationalMap_surjective f
trans (f.toRationalMap.toPartialMap.restrict _
f.dense_domain f.le_domain_toRationalMap).toRationalMap
· simp
· congr 1
exact PartialMap.ext _ f rfl (by simpa using f.toPartialMap_toRationalMap_restrict) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.SpreadingOut",
"Mathlib.AlgebraicGeometry.FunctionField",
"Mathlib.AlgebraicGeometry.Morphisms.Separated"
] | Mathlib/AlgebraicGeometry/RationalMap.lean | RationalMap.toRationalMap_toPartialMap | null |
residueField (x : X) : CommRingCat :=
CommRingCat.of <| IsLocalRing.ResidueField (X.presheaf.stalk x) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueField | The residue field of `X` at a point `x` is the residue field of the stalk of `X`
at `x`. |
residue (X : Scheme.{u}) (x) : X.presheaf.stalk x ⟶ X.residueField x :=
CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk x)) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residue | The residue map from the stalk to the residue field. |
descResidueField {K : Type u} [Field K] {X : Scheme.{u}} {x : X}
(f : X.presheaf.stalk x ⟶ .of K) [IsLocalHom f.hom] :
X.residueField x ⟶ .of K :=
CommRingCat.ofHom (IsLocalRing.ResidueField.lift (S := K) f.hom)
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | descResidueField | See `AlgebraicGeometry.IsClosedImmersion.Spec_map_residue` for the stronger result that
`Spec.map (X.residue x)` is a closed immersion. -/
instance {X : Scheme.{u}} (x) : IsPreimmersion (Spec.map (X.residue x)) :=
IsPreimmersion.mk_Spec_map
(PrimeSpectrum.isClosedEmbedding_comap_of_surjective _ _
Ideal.Quotient.mk_surjective).isEmbedding
(RingHom.surjectiveOnStalks_of_surjective (Ideal.Quotient.mk_surjective))
@[simp]
lemma Spec_map_residue_apply {X : Scheme.{u}} (x : X) (s : Spec (X.residueField x)) :
(Spec.map (X.residue x)).base s = closedPoint (X.presheaf.stalk x) :=
IsLocalRing.PrimeSpectrum.comap_residue _ s
lemma residue_surjective (X : Scheme.{u}) (x) : Function.Surjective (X.residue x) :=
Ideal.Quotient.mk_surjective
instance (X : Scheme.{u}) (x) : Epi (X.residue x) :=
ConcreteCategory.epi_of_surjective _ (X.residue_surjective x)
/-- If `K` is a field and `f : 𝒪_{X, x} ⟶ K` is a ring map, then this is the induced
map `κ(x) ⟶ K`. |
residue_descResidueField {K : Type u} [Field K] {X : Scheme.{u}} {x}
(f : X.presheaf.stalk x ⟶ .of K) [IsLocalHom f.hom] :
X.residue x ≫ X.descResidueField f = f :=
CommRingCat.hom_ext <| RingHom.ext fun _ ↦ rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residue_descResidueField | null |
evaluation (U : X.Opens) (x : X) (hx : x ∈ U) : Γ(X, U) ⟶ X.residueField x :=
X.presheaf.germ U x hx ≫ X.residue _
@[reassoc] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | evaluation | If `U` is an open of `X` containing `x`, we have a canonical ring map from the sections
over `U` to the residue field of `x`.
If we interpret sections over `U` as functions of `X` defined on `U`, then this ring map
corresponds to evaluation at `x`. |
germ_residue (x hx) : X.presheaf.germ U x hx ≫ X.residue x = X.evaluation U x hx := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | germ_residue | null |
Γevaluation (x : X) : Γ(X, ⊤) ⟶ X.residueField x :=
X.evaluation ⊤ x trivial
@[simp] | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Γevaluation | The global evaluation map from `Γ(X, ⊤)` to the residue field at `x`. |
evaluation_eq_zero_iff_notMem_basicOpen (x : X) (hx : x ∈ U) (f : Γ(X, U)) :
X.evaluation U x hx f = 0 ↔ x ∉ X.basicOpen f :=
X.toLocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen ⟨x, hx⟩ f
@[deprecated (since := "2025-05-23")]
alias evaluation_eq_zero_iff_not_mem_basicOpen := evaluation_eq_zero_iff_notMem_basicOpen | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | evaluation_eq_zero_iff_notMem_basicOpen | null |
evaluation_ne_zero_iff_mem_basicOpen (x : X) (hx : x ∈ U) (f : Γ(X, U)) :
X.evaluation U x hx f ≠ 0 ↔ x ∈ X.basicOpen f := by
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | evaluation_ne_zero_iff_mem_basicOpen | null |
basicOpen_eq_bot_iff_forall_evaluation_eq_zero (f : X.presheaf.obj (op U)) :
X.basicOpen f = ⊥ ↔ ∀ (x : U), X.evaluation U x x.property f = 0 :=
X.toLocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero f
variable {X Y : Scheme.{u}} (f : X ⟶ Y) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | basicOpen_eq_bot_iff_forall_evaluation_eq_zero | null |
Hom.residueFieldMap (f : X.Hom Y) (x : X) :
Y.residueField (f.base x) ⟶ X.residueField x :=
CommRingCat.ofHom <| IsLocalRing.ResidueField.map (f.stalkMap x).hom
@[reassoc] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Hom.residueFieldMap | If `X ⟶ Y` is a morphism of locally ringed spaces and `x` a point of `X`, we obtain
a morphism of residue fields in the other direction. |
residue_residueFieldMap (x : X) :
Y.residue (f.base x) ≫ f.residueFieldMap x = f.stalkMap x ≫ X.residue x := by
simp [Hom.residueFieldMap]
rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residue_residueFieldMap | null |
residueFieldMap_id (x : X) :
Hom.residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x) :=
LocallyRingedSpace.residueFieldMap_id _
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldMap_id | null |
residueFieldMap_comp {Z : Scheme.{u}} (g : Y ⟶ Z) (x : X) :
(f ≫ g).residueFieldMap x = g.residueFieldMap (f.base x) ≫ f.residueFieldMap x :=
LocallyRingedSpace.residueFieldMap_comp _ _ _
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldMap_comp | null |
evaluation_naturality {V : Opens Y} (x : X) (hx : f.base x ∈ V) :
Y.evaluation V (f.base x) hx ≫ f.residueFieldMap x =
f.app V ≫ X.evaluation (f ⁻¹ᵁ V) x hx :=
LocallyRingedSpace.evaluation_naturality f.1 ⟨x, hx⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | evaluation_naturality | null |
evaluation_naturality_apply {V : Opens Y} (x : X) (hx : f.base x ∈ V) (s) :
f.residueFieldMap x (Y.evaluation V (f.base x) hx s) =
X.evaluation (f ⁻¹ᵁ V) x hx (f.app V s) :=
LocallyRingedSpace.evaluation_naturality_apply f.1 ⟨x, hx⟩ s
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | evaluation_naturality_apply | null |
Γevaluation_naturality (x : X) :
Y.Γevaluation (f.base x) ≫ f.residueFieldMap x =
f.c.app (op ⊤) ≫ X.Γevaluation x :=
LocallyRingedSpace.Γevaluation_naturality f.toLRSHom x | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Γevaluation_naturality | null |
Γevaluation_naturality_apply (x : X) (a : Y.presheaf.obj (op ⊤)) :
f.residueFieldMap x (Y.Γevaluation (f.base x) a) =
X.Γevaluation x (f.c.app (op ⊤) a) :=
LocallyRingedSpace.Γevaluation_naturality_apply f.toLRSHom x a | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Γevaluation_naturality_apply | null |
residueFieldCongr {x y : X} (h : x = y) :
X.residueField x ≅ X.residueField y :=
eqToIso (by subst h; rfl)
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr | The isomorphism between residue fields of equal points. |
residueFieldCongr_refl {x : X} :
X.residueFieldCongr (refl x) = Iso.refl _ := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_refl | null |
residueFieldCongr_symm {x y : X} (e : x = y) :
(X.residueFieldCongr e).symm = X.residueFieldCongr e.symm := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_symm | null |
residueFieldCongr_inv {x y : X} (e : x = y) :
(X.residueFieldCongr e).inv = (X.residueFieldCongr e.symm).hom := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_inv | null |
residueFieldCongr_trans {x y z : X} (e : x = y) (e' : y = z) :
X.residueFieldCongr e ≪≫ X.residueFieldCongr e' = X.residueFieldCongr (e.trans e') := by
subst e e'
rfl
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_trans | null |
residueFieldCongr_trans_hom (X : Scheme) {x y z : X} (e : x = y) (e' : y = z) :
(X.residueFieldCongr e).hom ≫ (X.residueFieldCongr e').hom =
(X.residueFieldCongr (e.trans e')).hom := by
subst e e'
rfl
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_trans_hom | null |
residue_residueFieldCongr (X : Scheme) {x y : X} (h : x = y) :
X.residue x ≫ (X.residueFieldCongr h).hom =
(X.presheaf.stalkCongr (.of_eq h)).hom ≫ X.residue y := by
subst h
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residue_residueFieldCongr | null |
Hom.residueFieldMap_congr {f g : X ⟶ Y} (e : f = g) (x : X) :
f.residueFieldMap x = (Y.residueFieldCongr (by subst e; rfl)).hom ≫ g.residueFieldMap x := by
subst e; simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Hom.residueFieldMap_congr | null |
fromSpecResidueField (X : Scheme) (x : X) :
Spec (X.residueField x) ⟶ X :=
Spec.map (X.residue x) ≫ X.fromSpecStalk x | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | fromSpecResidueField | The canonical map `Spec κ(x) ⟶ X`. |
@[reassoc (attr := simp)]
residueFieldCongr_fromSpecResidueField {x y : X} (h : x = y) :
Spec.map (X.residueFieldCongr h).hom ≫ X.fromSpecResidueField _ =
X.fromSpecResidueField _ := by
subst h; simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | residueFieldCongr_fromSpecResidueField | null |
@[reassoc (attr := simp)]
Hom.Spec_map_residueFieldMap_fromSpecResidueField (x : X) :
Spec.map (f.residueFieldMap x) ≫ Y.fromSpecResidueField _ =
X.fromSpecResidueField x ≫ f := by
dsimp only [fromSpecResidueField]
rw [Category.assoc, ← Spec_map_stalkMap_fromSpecStalk, ← Spec.map_comp_assoc,
← Spec.map_comp_assoc]
rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | Hom.Spec_map_residueFieldMap_fromSpecResidueField | null |
@[simp]
fromSpecResidueField_apply (x : X.carrier) (s : Spec (X.residueField x)) :
(X.fromSpecResidueField x).base s = x := by
simp [fromSpecResidueField] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | fromSpecResidueField_apply | null |
range_fromSpecResidueField (x : X.carrier) :
Set.range (X.fromSpecResidueField x).base = {x} := by
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | range_fromSpecResidueField | null |
descResidueField_fromSpecResidueField {K : Type*} [Field K] (X : Scheme) {x}
(f : X.presheaf.stalk x ⟶ .of K) [IsLocalHom f.hom] :
Spec.map (X.descResidueField f) ≫
X.fromSpecResidueField x = Spec.map f ≫ X.fromSpecStalk x := by
simp [fromSpecResidueField, ← Spec.map_comp_assoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | descResidueField_fromSpecResidueField | null |
descResidueField_stalkClosedPointTo_fromSpecResidueField
(K : Type u) [Field K] (X : Scheme.{u}) (f : Spec(K) ⟶ X) :
Spec.map (@descResidueField (CommRingCat.of K) _ X _ (Scheme.stalkClosedPointTo f)
_) ≫
X.fromSpecResidueField (f.base (closedPoint K)) = f := by
rw [X.descResidueField_fromSpecResidueField]
rw [Scheme.Spec_stalkClosedPointTo_fromSpecStalk] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | descResidueField_stalkClosedPointTo_fromSpecResidueField | null |
SpecToEquivOfField_eq_iff {K : Type*} [Field K] {X : Scheme}
{f₁ f₂ : Σ x : X.carrier, X.residueField x ⟶ .of K} :
f₁ = f₂ ↔ ∃ e : f₁.1 = f₂.1, f₁.2 = (X.residueFieldCongr e).hom ≫ f₂.2 := by
constructor
· rintro rfl
simp
· obtain ⟨f, _⟩ := f₁
obtain ⟨g, _⟩ := f₂
rintro ⟨(rfl : f = g), h⟩
simpa | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | SpecToEquivOfField_eq_iff | A helper lemma to work with `AlgebraicGeometry.Scheme.SpecToEquivOfField`. |
SpecToEquivOfField (K : Type u) [Field K] (X : Scheme.{u}) :
(Spec(K) ⟶ X) ≃ Σ x, X.residueField x ⟶ .of K where
toFun f :=
⟨_, X.descResidueField (Scheme.stalkClosedPointTo f)⟩
invFun xf := Spec.map xf.2 ≫ X.fromSpecResidueField xf.1
left_inv := Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField K X
right_inv f := by
rw [SpecToEquivOfField_eq_iff]
simp only [CommRingCat.coe_of, Scheme.comp_coeBase, TopCat.coe_comp, Function.comp_apply,
Scheme.fromSpecResidueField_apply, exists_true_left]
rw [← Spec.map_inj, Spec.map_comp, ← cancel_mono (X.fromSpecResidueField _)]
erw [Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField]
simp | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField"
] | Mathlib/AlgebraicGeometry/ResidueField.lean | SpecToEquivOfField | For a field `K` and a scheme `X`, the morphisms `Spec K ⟶ X` bijectively correspond
to pairs of points `x` of `X` and embeddings `κ(x) ⟶ K`. |
@[coe]
toScheme {X : Scheme.{u}} (U : X.Opens) : Scheme.{u} :=
X.restrict U.isOpenEmbedding | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | toScheme | Open subset of a scheme as a scheme. |
ι : ↑U ⟶ X := X.ofRestrict _
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι | The restriction of a scheme to an open subset. |
ι_base_apply (x : U) : U.ι.base x = x.val := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_base_apply | null |
toScheme_carrier : (U : Type u) = (U : Set X) := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | toScheme_carrier | null |
toScheme_presheaf_obj (V) : Γ(U, V) = Γ(X, U.ι ''ᵁ V) := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | toScheme_presheaf_obj | null |
toScheme_presheaf_map {V W} (i : V ⟶ W) :
U.toScheme.presheaf.map i = X.presheaf.map (U.ι.opensFunctor.map i.unop).op := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | toScheme_presheaf_map | null |
ι_app (V) : U.ι.app V = X.presheaf.map
(homOfLE (x := U.ι ''ᵁ U.ι ⁻¹ᵁ V) (Set.image_preimage_subset _ _)).op :=
rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_app | null |
ι_appTop :
U.ι.appTop = X.presheaf.map (homOfLE (x := U.ι ''ᵁ ⊤) le_top).op :=
rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_appTop | null |
ι_appLE (V W e) :
U.ι.appLE V W e =
X.presheaf.map (homOfLE (x := U.ι ''ᵁ W) (Set.image_subset_iff.mpr ‹_›)).op := by
simp only [Hom.appLE, ι_app, toScheme_presheaf_map, Quiver.Hom.unop_op,
Hom.opensFunctor_map_homOfLE, ← Functor.map_comp]
rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_appLE | null |
ι_appIso (V) : U.ι.appIso V = Iso.refl _ :=
X.ofRestrict_appIso _ _
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_appIso | null |
opensRange_ι : U.ι.opensRange = U :=
Opens.ext Subtype.range_val
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | opensRange_ι | null |
range_ι : Set.range U.ι.base = U :=
Subtype.range_val | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | range_ι | null |
ι_image_top : U.ι ''ᵁ ⊤ = U :=
U.isOpenEmbedding_obj_top | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_image_top | null |
ι_image_le (W : U.toScheme.Opens) : U.ι ''ᵁ W ≤ U := by
simp_rw [← U.ι_image_top]
exact U.ι.image_le_image_of_le le_top
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_image_le | null |
ι_preimage_self : U.ι ⁻¹ᵁ U = ⊤ :=
Opens.inclusion'_map_eq_top _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_preimage_self | null |
ι_appLE_isIso :
IsIso (U.ι.appLE U ⊤ U.ι_preimage_self.ge) := by
simp only [ι, ofRestrict_appLE]
change IsIso (X.presheaf.map (eqToIso U.ι_image_top).hom.op)
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_appLE_isIso | null |
ι_app_self : U.ι.app U = X.presheaf.map (eqToHom (X := U.ι ''ᵁ _) (by simp)).op := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ι_app_self | null |
eq_presheaf_map_eqToHom {V W : Opens U} (e : U.ι ''ᵁ V = U.ι ''ᵁ W) :
X.presheaf.map (eqToHom e).op =
U.toScheme.presheaf.map (eqToHom <| U.isOpenEmbedding.functor_obj_injective e).op := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | eq_presheaf_map_eqToHom | null |
nonempty_iff : Nonempty U.toScheme ↔ (U : Set X).Nonempty := by
simp only [toScheme_carrier, SetLike.coe_sort_coe, nonempty_subtype]
rfl
attribute [-simp] eqToHom_op in | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | nonempty_iff | null |
@[simps!]
topIso : Γ(U, ⊤) ≅ Γ(X, U) :=
X.presheaf.mapIso (eqToIso U.ι_image_top.symm).op | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | topIso | The global sections of the restriction is isomorphic to the sections on the open set. |
stalkIso {X : Scheme.{u}} (U : X.Opens) (x : U) :
U.toScheme.presheaf.stalk x ≅ X.presheaf.stalk x.1 :=
X.restrictStalkIso (Opens.isOpenEmbedding _) _
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | stalkIso | The stalks of an open subscheme are isomorphic to the stalks of the original scheme. |
germ_stalkIso_hom {X : Scheme.{u}} (U : X.Opens)
{V : U.toScheme.Opens} (x : U) (hx : x ∈ V) :
U.toScheme.presheaf.germ V x hx ≫ (U.stalkIso x).hom =
X.presheaf.germ (U.ι ''ᵁ V) x.1 ⟨x, hx, rfl⟩ :=
PresheafedSpace.restrictStalkIso_hom_eq_germ _ U.isOpenEmbedding _ _ _
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | germ_stalkIso_hom | null |
germ_stalkIso_inv {X : Scheme.{u}} (U : X.Opens) (V : U.toScheme.Opens) (x : U)
(hx : x ∈ V) : X.presheaf.germ (U.ι ''ᵁ V) x ⟨x, hx, rfl⟩ ≫
(U.stalkIso x).inv = U.toScheme.presheaf.germ V x hx :=
PresheafedSpace.restrictStalkIso_inv_eq_germ X.toPresheafedSpace U.isOpenEmbedding V x hx | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | germ_stalkIso_inv | null |
stalkIso_inv {X : Scheme.{u}} (U : X.Opens) (x : U) :
(U.stalkIso x).inv = U.ι.stalkMap x := by
rw [← Category.comp_id (U.stalkIso x).inv, Iso.inv_comp_eq]
apply TopCat.Presheaf.stalk_hom_ext
intro W hxW
simp only [Category.comp_id, U.germ_stalkIso_hom_assoc]
convert (Scheme.stalkMap_germ U.ι (U.ι ''ᵁ W) x ⟨_, hxW, rfl⟩).symm
refine (U.toScheme.presheaf.germ_res (homOfLE ?_) _ _).symm
exact (Set.preimage_image_eq _ Subtype.val_injective).le | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | stalkIso_inv | null |
@[simps! I₀ X f]
Scheme.openCoverOfIsOpenCover {s : Type*} (X : Scheme.{u}) (U : s → X.Opens)
(hU : TopologicalSpace.IsOpenCover U) : X.OpenCover where
I₀ := s
X i := U i
f i := (U i).ι
mem₀ := by
rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, inferInstance⟩
have hx : x ∈ ⨆ i, U i := hU.symm ▸ show x ∈ (⊤ : X.Opens) by trivial
rw [Opens.mem_iSup] at hx
obtain ⟨i, hi⟩ := hx
use i
simpa
@[deprecated (since := "2025-09-30")]
noncomputable alias Scheme.openCoverOfISupEqTop := Scheme.openCoverOfIsOpenCover | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.openCoverOfIsOpenCover | If `U` is a family of open sets that covers `X`, then `X.restrict U` forms an `X.open_cover`. |
@[simps!]
opensRestrict :
Scheme.Opens U ≃ { V : X.Opens // V ≤ U } :=
(IsOpenImmersion.opensEquiv (U.ι)).trans (Equiv.subtypeEquivProp (by simp)) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | opensRestrict | The open sets of an open subscheme corresponds to the open sets containing in the subset. |
ΓRestrictAlgebra {X : Scheme.{u}} (U : X.Opens) :
Algebra (Γ(X, ⊤)) Γ(U, ⊤) :=
U.ι.appTop.hom.toAlgebra | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | ΓRestrictAlgebra | null |
Scheme.map_basicOpen (r : Γ(U, ⊤)) :
U.ι ''ᵁ U.toScheme.basicOpen r = X.basicOpen
(X.presheaf.map (eqToHom U.isOpenEmbedding_obj_top.symm).op r) := by
refine (Scheme.image_basicOpen (X.ofRestrict U.isOpenEmbedding) r).trans ?_
rw [← Scheme.basicOpen_res_eq _ _ (eqToHom U.isOpenEmbedding_obj_top).op]
rw [← CommRingCat.comp_apply, ← CategoryTheory.Functor.map_comp, ← op_comp, eqToHom_trans,
eqToHom_refl, op_id, CategoryTheory.Functor.map_id]
congr
exact PresheafedSpace.IsOpenImmersion.ofRestrict_invApp _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.map_basicOpen | null |
Scheme.Opens.ι_image_basicOpen (r : Γ(U, ⊤)) :
U.ι ''ᵁ U.toScheme.basicOpen r = X.basicOpen r := by
rw [Scheme.map_basicOpen, Scheme.basicOpen_res_eq] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Opens.ι_image_basicOpen | null |
Scheme.map_basicOpen_map (r : Γ(X, U)) :
U.ι ''ᵁ (U.toScheme.basicOpen <| U.topIso.inv r) = X.basicOpen r := by
simp only [Scheme.Opens.toScheme_presheaf_obj]
rw [Scheme.map_basicOpen, Scheme.basicOpen_res_eq, Scheme.Opens.topIso_inv,
Scheme.basicOpen_res_eq X] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.map_basicOpen_map | null |
protected noncomputable
Scheme.homOfLE (X : Scheme.{u}) {U V : X.Opens} (e : U ≤ V) : (U : Scheme.{u}) ⟶ V :=
IsOpenImmersion.lift V.ι U.ι (by simpa using e)
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE | If `U ≤ V`, then `U` is also a subscheme of `V`. |
Scheme.homOfLE_ι (X : Scheme.{u}) {U V : X.Opens} (e : U ≤ V) :
X.homOfLE e ≫ V.ι = U.ι :=
IsOpenImmersion.lift_fac _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_ι | null |
@[simp]
Scheme.homOfLE_rfl (X : Scheme.{u}) (U : X.Opens) : X.homOfLE (refl U) = 𝟙 _ := by
rw [← cancel_mono U.ι, Scheme.homOfLE_ι, Category.id_comp]
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_rfl | null |
Scheme.homOfLE_homOfLE (X : Scheme.{u}) {U V W : X.Opens} (e₁ : U ≤ V) (e₂ : V ≤ W) :
X.homOfLE e₁ ≫ X.homOfLE e₂ = X.homOfLE (e₁.trans e₂) := by
rw [← cancel_mono W.ι, Category.assoc, Scheme.homOfLE_ι, Scheme.homOfLE_ι, Scheme.homOfLE_ι] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_homOfLE | null |
Scheme.homOfLE_base {U V : X.Opens} (e : U ≤ V) :
(X.homOfLE e).base = (Opens.toTopCat _).map (homOfLE e) := by
ext a; refine Subtype.ext ?_ -- Porting note: `ext` did not pick up `Subtype.ext`
exact congr($(X.homOfLE_ι e).base a)
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_base | null |
Scheme.homOfLE_apply {U V : X.Opens} (e : U ≤ V) (x : U) :
((X.homOfLE e).base x).1 = x := by
rw [homOfLE_base]
rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_apply | null |
Scheme.ι_image_homOfLE_le_ι_image {U V : X.Opens} (e : U ≤ V) (W : Opens V) :
U.ι ''ᵁ (X.homOfLE e ⁻¹ᵁ W) ≤ V.ι ''ᵁ W := by
simp only [← SetLike.coe_subset_coe, IsOpenMap.coe_functor_obj, Set.image_subset_iff,
Scheme.homOfLE_base, Opens.map_coe]
rintro _ h
exact ⟨_, h, rfl⟩
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.ι_image_homOfLE_le_ι_image | null |
Scheme.homOfLE_app {U V : X.Opens} (e : U ≤ V) (W : Opens V) :
(X.homOfLE e).app W =
X.presheaf.map (homOfLE <| X.ι_image_homOfLE_le_ι_image e W).op := by
have e₁ := Scheme.congr_app (X.homOfLE_ι e) (V.ι ''ᵁ W)
have : V.ι ⁻¹ᵁ V.ι ''ᵁ W = W := W.map_functor_eq (U := V)
have e₂ := (X.homOfLE e).naturality (eqToIso this).hom.op
have e₃ := e₂.symm.trans e₁
dsimp at e₃ ⊢
rw [← IsIso.eq_comp_inv, ← Functor.map_inv, ← Functor.map_comp] at e₃
rw [e₃, ← Functor.map_comp]
congr 1 | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_app | null |
Scheme.homOfLE_appTop {U V : X.Opens} (e : U ≤ V) :
(X.homOfLE e).appTop =
X.presheaf.map (homOfLE <| X.ι_image_homOfLE_le_ι_image e ⊤).op :=
homOfLE_app .. | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.homOfLE_appTop | null |
Scheme.Opens.iSupOpenCover {J : Type*} {X : Scheme} (U : J → X.Opens) :
(⨆ i, U i).toScheme.OpenCover where
I₀ := J
X i := U i
f j := X.homOfLE (le_iSup _ _)
mem₀ := by
rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, inferInstance⟩
obtain ⟨i, hi⟩ := TopologicalSpace.Opens.mem_iSup.mp x.2
use i, ⟨x.1, hi⟩
apply Subtype.ext
simp
variable (X) in | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Opens.iSupOpenCover | The open cover of `⋃ Vᵢ` by `Vᵢ`. |
@[simps! obj_left obj_hom map_left]
Scheme.restrictFunctor : X.Opens ⥤ Over X where
obj U := Over.mk U.ι
map {U V} i := Over.homMk (X.homOfLE i.le) (by simp)
map_id U := by
ext1
exact Scheme.homOfLE_rfl _ _
map_comp {U V W} i j := by
ext1
exact (X.homOfLE_homOfLE i.le j.le).symm | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.restrictFunctor | The functor taking open subsets of `X` to open subschemes of `X`. |
@[simps!]
Scheme.restrictFunctorΓ : X.restrictFunctor.op ⋙ (Over.forget X).op ⋙ Scheme.Γ ≅ X.presheaf :=
NatIso.ofComponents
(fun U => X.presheaf.mapIso ((eqToIso (unop U).isOpenEmbedding_obj_top).symm.op :))
(by
intro U V i
dsimp
rw [X.homOfLE_appTop, ← Functor.map_comp, ← Functor.map_comp]
congr 1) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.restrictFunctorΓ | The functor that restricts to open subschemes and then takes global section is
isomorphic to the structure sheaf. |
noncomputable
Scheme.restrictRestrictComm (X : Scheme.{u}) (U V : X.Opens) :
(U.ι ⁻¹ᵁ V).toScheme ≅ V.ι ⁻¹ᵁ U :=
IsOpenImmersion.isoOfRangeEq (Opens.ι _ ≫ U.ι) (Opens.ι _ ≫ V.ι) <| by
simp only [comp_coeBase, TopCat.coe_comp, Set.range_comp, Opens.range_ι, Opens.map_coe,
Set.image_preimage_eq_inter_range, Set.inter_comm (U : Set X)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.restrictRestrictComm | `X ∣_ U ∣_ V` is isomorphic to `X ∣_ V ∣_ U` |
noncomputable
Scheme.Hom.isoImage
{X Y : Scheme.{u}} (f : X.Hom Y) [IsOpenImmersion f] (U : X.Opens) :
U.toScheme ≅ f ''ᵁ U :=
IsOpenImmersion.isoOfRangeEq (Opens.ι _ ≫ f) (Opens.ι _) (by simp [Set.range_comp])
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoImage | If `f : X ⟶ Y` is an open immersion, then for any `U : X.Opens`,
we have the isomorphism `U ≅ f ''ᵁ U`. |
Scheme.Hom.isoImage_hom_ι
{X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) :
(f.isoImage U).hom ≫ (f ''ᵁ U).ι = U.ι ≫ f :=
IsOpenImmersion.isoOfRangeEq_hom_fac _ _ _
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoImage_hom_ι | null |
Scheme.Hom.isoImage_inv_ι
{X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) :
(f.isoImage U).inv ≫ U.ι ≫ f = (f ''ᵁ U).ι :=
IsOpenImmersion.isoOfRangeEq_inv_fac _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoImage_inv_ι | null |
Scheme.Hom.isoOpensRange {X Y : Scheme.{u}} (f : X.Hom Y) [IsOpenImmersion f] :
X ≅ f.opensRange :=
IsOpenImmersion.isoOfRangeEq f f.opensRange.ι (by simp)
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoOpensRange | If `f : X ⟶ Y` is an open immersion, then `X` is isomorphic to its image in `Y`. |
Scheme.Hom.isoOpensRange_hom_ι {X Y : Scheme.{u}} (f : X.Hom Y) [IsOpenImmersion f] :
f.isoOpensRange.hom ≫ f.opensRange.ι = f := by
simp [isoOpensRange]
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoOpensRange_hom_ι | null |
Scheme.Hom.isoOpensRange_inv_comp {X Y : Scheme.{u}} (f : X.Hom Y) [IsOpenImmersion f] :
f.isoOpensRange.inv ≫ f = f.opensRange.ι := by
simp [isoOpensRange] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.Hom.isoOpensRange_inv_comp | null |
@[simps hom]
Scheme.topIso (X : Scheme) : ↑(⊤ : X.Opens) ≅ X where
hom := Scheme.Opens.ι _
inv := ⟨X.restrictTopIso.inv⟩
hom_inv_id := Hom.ext' X.restrictTopIso.hom_inv_id
inv_hom_id := Hom.ext' X.restrictTopIso.inv_hom_id
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.topIso | `(⊤ : X.Opens)` as a scheme is isomorphic to `X`. |
Scheme.toIso_inv_ι (X : Scheme.{u}) : X.topIso.inv ≫ Opens.ι _ = 𝟙 _ :=
X.topIso.inv_hom_id
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.toIso_inv_ι | null |
Scheme.ι_toIso_inv (X : Scheme.{u}) : Opens.ι _ ≫ X.topIso.inv = 𝟙 _ :=
X.topIso.hom_inv_id | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.ι_toIso_inv | null |
noncomputable
Scheme.isoOfEq (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(U : Scheme.{u}) ≅ V :=
IsOpenImmersion.isoOfRangeEq U.ι V.ι (by rw [e])
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.isoOfEq | If `U = V`, then `X ∣_ U` is isomorphic to `X ∣_ V`. |
Scheme.isoOfEq_hom_ι (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(X.isoOfEq e).hom ≫ V.ι = U.ι :=
IsOpenImmersion.isoOfRangeEq_hom_fac _ _ _
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.isoOfEq_hom_ι | null |
Scheme.isoOfEq_inv_ι (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(X.isoOfEq e).inv ≫ U.ι = V.ι :=
IsOpenImmersion.isoOfRangeEq_inv_fac _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.isoOfEq_inv_ι | null |
Scheme.isoOfEq_hom (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(X.isoOfEq e).hom = X.homOfLE e.le := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.Over"
] | Mathlib/AlgebraicGeometry/Restrict.lean | Scheme.isoOfEq_hom | null |
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