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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IsLocallyNoetherian (X : Scheme) : Prop where
component_noetherian : ∀ (U : X.affineOpens),
IsNoetherianRing Γ(X, U) := by infer_instance | class | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | IsLocallyNoetherian | A scheme `X` is locally Noetherian if `𝒪ₓ(U)` is Noetherian for all affine `U`. |
isNoetherianRing_of_away : IsNoetherianRing R := by
apply monotone_stabilizes_iff_noetherian.mp
intro I
let floc s := algebraMap R (Away (M := R) s)
let suitableN s :=
{ n : ℕ | ∀ m : ℕ, n ≤ m → (Ideal.map (floc s) (I n)) = (Ideal.map (floc s) (I m)) }
let minN s := sInf (suitableN s)
have hSuit : ∀ s :... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isNoetherianRing_of_away | Let `R` be a ring, and `f i` a finite collection of elements of `R` generating the unit ideal.
If the localization of `R` at each `f i` is Noetherian, so is `R`.
We follow the proof given in [Har77], Proposition II.3.2 |
isLocallyNoetherian_of_affine_cover {ι} {S : ι → X.affineOpens}
(hS : (⨆ i, S i : X.Opens) = ⊤)
(hS' : ∀ i, IsNoetherianRing Γ(X, S i)) : IsLocallyNoetherian X := by
refine ⟨fun U => ?_⟩
induction U using of_affine_open_cover S hS with
| basicOpen U f hN =>
have := U.prop.isLocalization_basicOpen f
... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isLocallyNoetherian_of_affine_cover | If a scheme `X` has a cover by affine opens whose sections are Noetherian rings,
then `X` is locally Noetherian. |
isLocallyNoetherian_iff_of_iSup_eq_top {ι} {S : ι → X.affineOpens}
(hS : (⨆ i, S i : X.Opens) = ⊤) :
IsLocallyNoetherian X ↔ ∀ i, IsNoetherianRing Γ(X, S i) :=
⟨fun _ i => IsLocallyNoetherian.component_noetherian (S i),
isLocallyNoetherian_of_affine_cover hS⟩ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isLocallyNoetherian_iff_of_iSup_eq_top | A scheme is locally Noetherian if and only if it is covered by affine opens whose sections
are Noetherian rings.
See [Har77], Proposition II.3.2. |
isLocallyNoetherian_iff_of_affine_openCover (𝒰 : Scheme.OpenCover.{v, u} X)
[∀ i, IsAffine (𝒰.X i)] :
IsLocallyNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing Γ(𝒰.X i, ⊤) := by
constructor
· intro h i
let U := Scheme.Hom.opensRange (𝒰.f i)
have := h.component_noetherian ⟨U, isAffineOpen_opensRang... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isLocallyNoetherian_iff_of_affine_openCover | A version of `isLocallyNoetherian_iff_of_iSup_eq_top` using `Scheme.OpenCover`. |
isLocallyNoetherian_of_isOpenImmersion {Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f]
[IsLocallyNoetherian Y] : IsLocallyNoetherian X := by
refine ⟨fun U => ?_⟩
let V : Y.affineOpens := ⟨f ''ᵁ U, IsAffineOpen.image_of_isOpenImmersion U.prop _⟩
suffices Γ(X, U) ≅ Γ(Y, V) by
convert isNoetherianRing_of_ringEq... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isLocallyNoetherian_of_isOpenImmersion | null |
isLocallyNoetherian_iff_openCover (𝒰 : Scheme.OpenCover X) :
IsLocallyNoetherian X ↔ ∀ (i : 𝒰.I₀), IsLocallyNoetherian (𝒰.X i) := by
constructor
· intro h i
exact isLocallyNoetherian_of_isOpenImmersion (𝒰.f i)
· rw [isLocallyNoetherian_iff_of_affine_openCover (𝒰 := 𝒰.affineRefinement.openCover)]
... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isLocallyNoetherian_iff_openCover | If `𝒰` is an open cover of a scheme `X`, then `X` is locally Noetherian if and only if
`𝒰.X i` are all locally Noetherian. |
@[mk_iff]
IsNoetherian (X : Scheme) : Prop extends IsLocallyNoetherian X, CompactSpace X | class | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | IsNoetherian | If `R` is a Noetherian ring, `Spec R` is a Noetherian topological space. -/
instance {R : CommRingCat} [IsNoetherianRing R] :
NoetherianSpace (Spec R) := by
convert PrimeSpectrum.instNoetherianSpace (R := R)
lemma noetherianSpace_of_isAffine [IsAffine X] [IsNoetherianRing Γ(X, ⊤)] :
NoetherianSpace X :=
(n... |
isNoetherian_iff_of_finite_iSup_eq_top {ι} [Finite ι] {S : ι → X.affineOpens}
(hS : (⨆ i, S i : X.Opens) = ⊤) :
IsNoetherian X ↔ ∀ i, IsNoetherianRing Γ(X, S i) := by
constructor
· intro h i
apply (isLocallyNoetherian_iff_of_iSup_eq_top hS).mp
exact h.toIsLocallyNoetherian
· intro h
convert Is... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isNoetherian_iff_of_finite_iSup_eq_top | A scheme is Noetherian if and only if it is covered by finitely many affine opens whose
sections are Noetherian rings. |
isNoetherian_iff_of_finite_affine_openCover {𝒰 : Scheme.OpenCover.{v, u} X}
[Finite 𝒰.I₀] [∀ i, IsAffine (𝒰.X i)] :
IsNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing Γ(𝒰.X i, ⊤) := by
constructor
· intro h i
apply (isLocallyNoetherian_iff_of_affine_openCover _).mp
exact h.toIsLocallyNoetherian
... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isNoetherian_iff_of_finite_affine_openCover | A version of `isNoetherian_iff_of_finite_iSup_eq_top` using `Scheme.OpenCover`. |
isNoetherian_Spec {R : CommRingCat} :
IsNoetherian (Spec R) ↔ IsNoetherianRing R :=
⟨fun _ => inferInstance,
fun _ => inferInstance⟩ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | isNoetherian_Spec | A Noetherian scheme has a Noetherian underlying topological space. -/
@[stacks 01OZ]
instance (priority := 100) IsNoetherian.noetherianSpace [IsNoetherian X] :
NoetherianSpace X := by
apply TopologicalSpace.noetherian_univ_iff.mp
let 𝒰 := X.affineCover.finiteSubcover
rw [← 𝒰.iUnion_range]
suffices ∀ i : �... |
@[stacks 0BA8]
finite_irreducibleComponents_of_isNoetherian [IsNoetherian X] :
(irreducibleComponents X).Finite := NoetherianSpace.finite_irreducibleComponents | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated",
"Mathlib.RingTheory.Localization.Submodule",
"Mathlib.RingTheory.Spectrum.Prime.Noetherian"
] | Mathlib/AlgebraicGeometry/Noetherian.lean | finite_irreducibleComponents_of_isNoetherian | A Noetherian scheme has a finite number of irreducible components. |
IsOpenImmersion : MorphismProperty (Scheme.{u}) :=
fun _ _ f ↦ LocallyRingedSpace.IsOpenImmersion f.toLRSHom | abbrev | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | IsOpenImmersion | A morphism of Schemes is an open immersion if it is an open immersion as a morphism
of LocallyRingedSpaces |
IsOpenImmersion.comp {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)
[IsOpenImmersion f] [IsOpenImmersion g] : IsOpenImmersion (f ≫ g) :=
LocallyRingedSpace.IsOpenImmersion.comp f.toLRSHom g.toLRSHom | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | IsOpenImmersion.comp | null |
protected scheme (X : LocallyRingedSpace.{u})
(h :
∀ x : X,
∃ (R : CommRingCat) (f : Spec.toLocallyRingedSpace.obj (op R) ⟶ X),
(x ∈ Set.range f.base :) ∧ LocallyRingedSpace.IsOpenImmersion f) :
Scheme where
toLocallyRingedSpace := X
local_affine := by
intro x
obtain ⟨R, f, h... | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | scheme | To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. |
IsOpenImmersion.isOpen_range {X Y : Scheme.{u}} (f : X ⟶ Y) [H : IsOpenImmersion f] :
IsOpen (Set.range f.base) :=
H.base_open.isOpen_range | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | IsOpenImmersion.isOpen_range | null |
isOpenEmbedding : IsOpenEmbedding f.base :=
H.base_open | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isOpenEmbedding | null |
@[simps]
opensRange : Y.Opens :=
⟨_, f.isOpenEmbedding.isOpen_range⟩ | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | opensRange | The image of an open immersion as an open set. |
opensFunctor : X.Opens ⥤ Y.Opens :=
LocallyRingedSpace.IsOpenImmersion.opensFunctor f.toLRSHom | abbrev | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | opensFunctor | The functor `opens X ⥤ opens Y` associated with an open immersion `f : X ⟶ Y`. |
appIso (U) : Γ(Y, f ''ᵁ U) ≅ Γ(X, U) :=
(asIso <| LocallyRingedSpace.IsOpenImmersion.invApp f.toLRSHom U).symm
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso | `f ''ᵁ U` is notation for the image (as an open set) of `U` under an open immersion `f`. -/
scoped[AlgebraicGeometry] notation3:90 f:91 " ''ᵁ " U:90 => (Scheme.Hom.opensFunctor f).obj U
lemma image_le_image_of_le {U V : X.Opens} (e : U ≤ V) : f ''ᵁ U ≤ f ''ᵁ V := by
rintro a ⟨u, hu, rfl⟩
exact Set.mem_image_of_mem... |
appIso_inv_naturality {U V : X.Opens} (i : op U ⟶ op V) :
X.presheaf.map i ≫ (f.appIso V).inv =
(f.appIso U).inv ≫ Y.presheaf.map (f.opensFunctor.op.map i) :=
PresheafedSpace.IsOpenImmersion.inv_naturality _ _ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso_inv_naturality | null |
appIso_hom (U) :
(f.appIso U).hom = f.app (f ''ᵁ U) ≫ X.presheaf.map
(eqToHom (preimage_image_eq f U).symm).op :=
(PresheafedSpace.IsOpenImmersion.inv_invApp f.toPshHom U).trans (by rw [eqToHom_op]) | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso_hom | null |
appIso_hom' (U) :
(f.appIso U).hom = f.appLE (f ''ᵁ U) U (preimage_image_eq f U).ge :=
f.appIso_hom U
@[reassoc (attr := simp)] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso_hom' | null |
app_appIso_inv (U) :
f.app U ≫ (f.appIso (f ⁻¹ᵁ U)).inv =
Y.presheaf.map (homOfLE (Set.image_preimage_subset f.base U.1)).op :=
PresheafedSpace.IsOpenImmersion.app_invApp _ _ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | app_appIso_inv | null |
@[reassoc]
app_invApp' (U) (hU : U ≤ f.opensRange) :
f.app U ≫ (f.appIso (f ⁻¹ᵁ U)).inv =
Y.presheaf.map (eqToHom (Opens.ext <| by simpa [Set.image_preimage_eq_inter_range])).op :=
PresheafedSpace.IsOpenImmersion.app_invApp _ _
@[reassoc (attr := simp), elementwise nosimp] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | app_invApp' | A variant of `app_invApp` that gives an `eqToHom` instead of `homOfLE`. |
appIso_inv_app (U) :
(f.appIso U).inv ≫ f.app (f ''ᵁ U) = X.presheaf.map (eqToHom (preimage_image_eq f U)).op :=
(PresheafedSpace.IsOpenImmersion.invApp_app _ _).trans (by rw [eqToHom_op])
@[reassoc (attr := simp), elementwise nosimp] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso_inv_app | null |
appLE_appIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] {U : Y.Opens}
{V : X.Opens} (e : V ≤ f ⁻¹ᵁ U) :
f.appLE U V e ≫ (f.appIso V).inv =
Y.presheaf.map (homOfLE <| (f.image_le_image_of_le e).trans
(f.image_preimage_eq_opensRange_inter U ▸ inf_le_right)).op := by
simp only [appL... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appLE_appIso_inv | null |
appIso_inv_appLE {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] {U V : X.Opens}
(e : V ≤ f ⁻¹ᵁ f ''ᵁ U) :
(f.appIso U).inv ≫ f.appLE (f ''ᵁ U) V e =
X.presheaf.map (homOfLE (by rwa [preimage_image_eq] at e)).op := by
simp only [appLE, appIso_inv_app_assoc, eqToHom_op]
rw [← Functor.map_comp]
r... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | appIso_inv_appLE | null |
@[simps]
IsOpenImmersion.opensEquiv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] :
X.Opens ≃ { U : Y.Opens // U ≤ f.opensRange } where
toFun U := ⟨f ''ᵁ U, Set.image_subset_range _ _⟩
invFun U := f ⁻¹ᵁ U
left_inv _ := Opens.ext (Set.preimage_image_eq _ f.isOpenEmbedding.injective)
right_inv U := Subty... | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | IsOpenImmersion.opensEquiv | The open sets of an open subscheme corresponds to the open sets containing in the image. |
basic_open_isOpenImmersion {R : CommRingCat.{u}} (f : R) :
IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away f)))) := by
apply SheafedSpace.IsOpenImmersion.of_stalk_iso (H := ?_)
· exact (PrimeSpectrum.localization_away_isOpenEmbedding (Localization.Away f) f :)
· intro x
exact... | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | basic_open_isOpenImmersion | null |
_root_.AlgebraicGeometry.IsOpenImmersion.of_isLocalization {R S} [CommRing R] [CommRing S]
[Algebra R S] (f : R) [IsLocalization.Away f S] :
IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap R S))) := by
have e := (IsLocalization.algEquiv (.powers f) S
(Localization.Away f)).symm.toAlgHom.comp_alge... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | _root_.AlgebraicGeometry.IsOpenImmersion.of_isLocalization | null |
exists_affine_mem_range_and_range_subset
{X : Scheme.{u}} {x : X} {U : X.Opens} (hxU : x ∈ U) :
∃ (R : CommRingCat) (f : Spec R ⟶ X),
IsOpenImmersion f ∧ x ∈ Set.range f.base ∧ Set.range f.base ⊆ U := by
obtain ⟨⟨V, hxV⟩, R, ⟨e⟩⟩ := X.2 x
have : e.hom.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.base ≫ V.inclusi... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | exists_affine_mem_range_and_range_subset | null |
toScheme : Scheme := by
apply LocallyRingedSpace.IsOpenImmersion.scheme (toLocallyRingedSpace _ f)
intro x
obtain ⟨R, i, _, h₁, h₂⟩ :=
Scheme.exists_affine_mem_range_and_range_subset (U := ⟨_, H.base_open.isOpen_range⟩) ⟨x, rfl⟩
refine ⟨R, LocallyRingedSpace.IsOpenImmersion.lift (toLocallyRingedSpaceHom _ f... | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | toScheme | If `X ⟶ Y` is an open immersion, and `Y` is a scheme, then so is `X`. |
toScheme_toLocallyRingedSpace :
(toScheme Y f).toLocallyRingedSpace = toLocallyRingedSpace Y.1 f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | toScheme_toLocallyRingedSpace | null |
toSchemeHom : toScheme Y f ⟶ Y :=
⟨toLocallyRingedSpaceHom _ f⟩
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | toSchemeHom | If `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a Scheme, we can
upgrade it into a morphism of Schemes. |
toSchemeHom_toPshHom : (toSchemeHom Y f).toPshHom = f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | toSchemeHom_toPshHom | null |
toSchemeHom_isOpenImmersion : AlgebraicGeometry.IsOpenImmersion (toSchemeHom Y f) :=
H | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | toSchemeHom_isOpenImmersion | null |
scheme_eq_of_locallyRingedSpace_eq {X Y : Scheme.{u}}
(H : X.toLocallyRingedSpace = Y.toLocallyRingedSpace) : X = Y := by
cases X; cases Y; congr | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | scheme_eq_of_locallyRingedSpace_eq | null |
scheme_toScheme {X Y : Scheme.{u}} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] :
toScheme Y f.toPshHom = X := by
apply scheme_eq_of_locallyRingedSpace_eq
exact locallyRingedSpace_toLocallyRingedSpace f.toLRSHom | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | scheme_toScheme | null |
@[simps! -isSimp carrier, simps! presheaf_obj]
Scheme.restrict : Scheme :=
{ PresheafedSpace.IsOpenImmersion.toScheme X (X.toPresheafedSpace.ofRestrict h) with
toPresheafedSpace := X.toPresheafedSpace.restrict h } | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.restrict | The restriction of a Scheme along an open embedding. |
Scheme.restrict_toPresheafedSpace :
(X.restrict h).toPresheafedSpace = X.toPresheafedSpace.restrict h := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.restrict_toPresheafedSpace | null |
@[simps! toLRSHom_base, simps! -isSimp toLRSHom_c_app]
Scheme.ofRestrict : X.restrict h ⟶ X :=
⟨X.toLocallyRingedSpace.ofRestrict h⟩
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.ofRestrict | The canonical map from the restriction to the subspace. |
Scheme.ofRestrict_app (V) :
(X.ofRestrict h).app V = X.presheaf.map (h.isOpenMap.adjunction.counit.app V).op :=
rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.ofRestrict_app | null |
IsOpenImmersion.ofRestrict : IsOpenImmersion (X.ofRestrict h) :=
show PresheafedSpace.IsOpenImmersion (X.toPresheafedSpace.ofRestrict h) by infer_instance
@[simp] | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | IsOpenImmersion.ofRestrict | null |
Scheme.ofRestrict_appLE (V W e) :
(X.ofRestrict h).appLE V W e = X.presheaf.map
(homOfLE (show X.ofRestrict h ''ᵁ _ ≤ _ by exact Set.image_subset_iff.mpr e)).op := by
dsimp [Hom.appLE]
exact (X.presheaf.map_comp _ _).symm
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.ofRestrict_appLE | null |
Scheme.ofRestrict_appIso (U) :
(X.ofRestrict h).appIso U = Iso.refl _ := by
ext1
simp only [restrict_presheaf_obj, Hom.appIso_hom', ofRestrict_appLE, homOfLE_refl, op_id,
CategoryTheory.Functor.map_id, Iso.refl_hom]
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.ofRestrict_appIso | null |
Scheme.restrict_presheaf_map (V W) (i : V ⟶ W) :
(X.restrict h).presheaf.map i = X.presheaf.map (homOfLE (show X.ofRestrict h ''ᵁ W.unop ≤
X.ofRestrict h ''ᵁ V.unop from Set.image_mono i.unop.le)).op := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | Scheme.restrict_presheaf_map | null |
to_iso {X Y : Scheme.{u}} (f : X ⟶ Y) [h : IsOpenImmersion f] [Epi f.base] : IsIso f :=
@isIso_of_reflects_iso _ _ _ _ _ _ f
(Scheme.forgetToLocallyRingedSpace ⋙
LocallyRingedSpace.forgetToSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace)
(@PresheafedSpace.IsOpenImmersion.to_iso _ _ _ _ f.toPshHom h ... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | to_iso | null |
of_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) (hf : IsOpenEmbedding f.base)
[∀ x, IsIso (f.stalkMap x)] : IsOpenImmersion f :=
haveI (x : X) : IsIso (f.toShHom.stalkMap x) := inferInstanceAs <| IsIso (f.stalkMap x)
SheafedSpace.IsOpenImmersion.of_stalk_iso f.toShHom hf | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | of_stalk_iso | null |
stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (x : X) :
IsIso (f.stalkMap x) :=
inferInstanceAs <| IsIso (f.toLRSHom.stalkMap x) | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | stalk_iso | null |
of_comp {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) [IsOpenImmersion g]
[IsOpenImmersion (f ≫ g)] : IsOpenImmersion f :=
haveI (x : X) : IsIso (f.stalkMap x) :=
haveI : IsIso (g.stalkMap (f.base x) ≫ f.stalkMap x) := by
rw [← Scheme.stalkMap_comp]
infer_instance
IsIso.of_isIso_comp_left (f :=... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | of_comp | null |
iff_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) :
IsOpenImmersion f ↔ IsOpenEmbedding f.base ∧ ∀ x, IsIso (f.stalkMap x) :=
⟨fun H => ⟨H.1, fun x ↦ inferInstanceAs <| IsIso (f.toPshHom.stalkMap x)⟩,
fun ⟨h, _⟩ => IsOpenImmersion.of_stalk_iso f h⟩ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | iff_stalk_iso | null |
_root_.AlgebraicGeometry.isIso_iff_isOpenImmersion {X Y : Scheme.{u}} (f : X ⟶ Y) :
IsIso f ↔ IsOpenImmersion f ∧ Epi f.base :=
⟨fun _ => ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ => IsOpenImmersion.to_iso f⟩ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | _root_.AlgebraicGeometry.isIso_iff_isOpenImmersion | null |
_root_.AlgebraicGeometry.isIso_iff_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) :
IsIso f ↔ IsIso f.base ∧ ∀ x, IsIso (f.stalkMap x) := by
rw [isIso_iff_isOpenImmersion, IsOpenImmersion.iff_stalk_iso, and_comm, ← and_assoc]
refine and_congr ⟨?_, ?_⟩ Iff.rfl
· rintro ⟨h₁, h₂⟩
convert_to
IsIso
(To... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | _root_.AlgebraicGeometry.isIso_iff_stalk_iso | null |
isoRestrict : X ≅ (Z.restrict H.base_open :) :=
Scheme.fullyFaithfulForgetToLocallyRingedSpace.preimageIso
(LocallyRingedSpace.IsOpenImmersion.isoRestrict f.toLRSHom)
local notation "forget" => Scheme.forgetToLocallyRingedSpace | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isoRestrict | An open immersion induces an isomorphism from the domain onto the image |
mono : Mono f :=
Scheme.forgetToLocallyRingedSpace.mono_of_mono_map
(show Mono f.toLRSHom by infer_instance) | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | mono | null |
le_monomorphisms :
IsOpenImmersion ≤ MorphismProperty.monomorphisms Scheme.{u} := fun _ _ _ _ ↦
MorphismProperty.monomorphisms.infer_property _ | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | le_monomorphisms | null |
forget_map_isOpenImmersion : LocallyRingedSpace.IsOpenImmersion ((forget).map f) :=
⟨H.base_open, H.c_iso⟩ | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forget_map_isOpenImmersion | null |
hasLimit_cospan_forget_of_left :
HasLimit (cospan f g ⋙ Scheme.forgetToLocallyRingedSpace) := by
rw [hasLimit_iff_of_iso (diagramIsoCospan _)]
exact inferInstanceAs (HasLimit (cospan ((forget).map f) ((forget).map g)))
open CategoryTheory.Limits.WalkingCospan | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasLimit_cospan_forget_of_left | null |
hasLimit_cospan_forget_of_left' :
HasLimit (cospan ((cospan f g ⋙ forget).map Hom.inl) ((cospan f g ⋙ forget).map Hom.inr)) :=
show HasLimit (cospan ((forget).map f) ((forget).map g)) from inferInstance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasLimit_cospan_forget_of_left' | null |
hasLimit_cospan_forget_of_right : HasLimit (cospan g f ⋙ forget) := by
rw [hasLimit_iff_of_iso (diagramIsoCospan _)]
exact inferInstanceAs (HasLimit (cospan ((forget).map g) ((forget).map f))) | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasLimit_cospan_forget_of_right | null |
hasLimit_cospan_forget_of_right' :
HasLimit (cospan ((cospan g f ⋙ forget).map Hom.inl) ((cospan g f ⋙ forget).map Hom.inr)) :=
show HasLimit (cospan ((forget).map g) ((forget).map f)) from inferInstance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasLimit_cospan_forget_of_right' | null |
forgetCreatesPullbackOfLeft : CreatesLimit (cospan f g) forget :=
createsLimitOfFullyFaithfulOfIso
(PresheafedSpace.IsOpenImmersion.toScheme Y (pullback.snd f.toLRSHom g.toLRSHom).toShHom)
(eqToIso (by simp) ≪≫ HasLimit.isoOfNatIso (diagramIsoCospan _).symm) | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forgetCreatesPullbackOfLeft | null |
forgetCreatesPullbackOfRight : CreatesLimit (cospan g f) forget :=
createsLimitOfFullyFaithfulOfIso
(PresheafedSpace.IsOpenImmersion.toScheme Y (pullback.fst g.toLRSHom f.toLRSHom).1)
(eqToIso (by simp) ≪≫ HasLimit.isoOfNatIso (diagramIsoCospan _).symm) | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forgetCreatesPullbackOfRight | null |
forget_preservesOfLeft : PreservesLimit (cospan f g) forget :=
CategoryTheory.preservesLimit_of_createsLimit_and_hasLimit _ _ | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forget_preservesOfLeft | null |
forget_preservesOfRight : PreservesLimit (cospan g f) forget :=
preservesPullback_symmetry _ _ _ | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forget_preservesOfRight | null |
hasPullback_of_left : HasPullback f g :=
hasLimit_of_created (cospan f g) forget | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasPullback_of_left | null |
hasPullback_of_right : HasPullback g f :=
hasLimit_of_created (cospan g f) forget | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | hasPullback_of_right | null |
pullback_snd_of_left : IsOpenImmersion (pullback.snd f g) := by
have := PreservesPullback.iso_hom_snd forget f g
dsimp only [Scheme.forgetToLocallyRingedSpace, inducedFunctor_map] at this
change LocallyRingedSpace.IsOpenImmersion _
rw [← this]
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | pullback_snd_of_left | null |
pullback_fst_of_right : IsOpenImmersion (pullback.fst g f) := by
rw [← pullbackSymmetry_hom_comp_snd]
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | pullback_fst_of_right | null |
pullback_to_base [IsOpenImmersion g] :
IsOpenImmersion (limit.π (cospan f g) WalkingCospan.one) := by
rw [← limit.w (cospan f g) WalkingCospan.Hom.inl]
change IsOpenImmersion (_ ≫ f)
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | pullback_to_base | null |
forgetToTop_preserves_of_left : PreservesLimit (cospan f g) Scheme.forgetToTop := by
delta Scheme.forgetToTop
refine @Limits.comp_preservesLimit _ _ _ _ _ _ (K := cospan f g) _ _ (F := forget)
(G := LocallyRingedSpace.forgetToTop) ?_ ?_
· infer_instance
refine @preservesLimit_of_iso_diagram _ _ _ _ _ _ _ _ ... | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forgetToTop_preserves_of_left | null |
forgetToTop_preserves_of_right : PreservesLimit (cospan g f) Scheme.forgetToTop :=
preservesPullback_symmetry _ _ _ | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | forgetToTop_preserves_of_right | null |
range_pullback_snd_of_left :
Set.range (pullback.snd f g).base = (g ⁻¹ᵁ f.opensRange).1 := by
rw [← show _ = (pullback.snd f g).base from
PreservesPullback.iso_hom_snd Scheme.forgetToTop f g, TopCat.coe_comp, Set.range_comp,
Set.range_eq_univ.mpr, ← @Set.preimage_univ _ _ (pullback.fst f.base g.base)]
·... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | range_pullback_snd_of_left | null |
opensRange_pullback_snd_of_left :
(pullback.snd f g).opensRange = g ⁻¹ᵁ f.opensRange :=
Opens.ext (range_pullback_snd_of_left f g) | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | opensRange_pullback_snd_of_left | null |
range_pullback_fst_of_right :
Set.range (pullback.fst g f).base =
((Opens.map g.base).obj ⟨Set.range f.base, H.base_open.isOpen_range⟩).1 := by
rw [← show _ = (pullback.fst g f).base from
PreservesPullback.iso_hom_fst Scheme.forgetToTop g f, TopCat.coe_comp, Set.range_comp,
Set.range_eq_univ.mpr, ← ... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | range_pullback_fst_of_right | null |
opensRange_pullback_fst_of_right :
(pullback.fst g f).opensRange = g ⁻¹ᵁ f.opensRange :=
Opens.ext (range_pullback_fst_of_right f g) | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | opensRange_pullback_fst_of_right | null |
range_pullback_to_base_of_left :
Set.range (pullback.fst f g ≫ f).base =
Set.range f.base ∩ Set.range g.base := by
rw [pullback.condition, Scheme.comp_base, TopCat.coe_comp, Set.range_comp,
range_pullback_snd_of_left, Opens.carrier_eq_coe, Opens.map_obj, Opens.coe_mk,
Set.image_preimage_eq_inter_ran... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | range_pullback_to_base_of_left | null |
range_pullback_to_base_of_right :
Set.range (pullback.fst g f ≫ g).base =
Set.range g.base ∩ Set.range f.base := by
rw [Scheme.comp_base, TopCat.coe_comp, Set.range_comp, range_pullback_fst_of_right,
Opens.map_obj, Opens.carrier_eq_coe, Opens.coe_mk, Set.image_preimage_eq_inter_range,
Set.inter_comm... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | range_pullback_to_base_of_right | null |
image_preimage_eq_preimage_image_of_isPullback {X Y U V : Scheme.{u}}
{f : X ⟶ Y} {f' : U ⟶ V} {iU : U ⟶ X} {iV : V ⟶ Y} [IsOpenImmersion iV] [IsOpenImmersion iU]
(H : IsPullback f' iU iV f) (W : V.Opens) : iU ''ᵁ f' ⁻¹ᵁ W = f ⁻¹ᵁ iV ''ᵁ W := by
ext x
by_cases hx : x ∈ Set.range iU.base
· obtain ⟨x, rfl⟩ ... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | image_preimage_eq_preimage_image_of_isPullback | null |
lift (H' : Set.range g.base ⊆ Set.range f.base) : Y ⟶ X :=
⟨LocallyRingedSpace.IsOpenImmersion.lift f.toLRSHom g.toLRSHom H'⟩
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lift | The universal property of open immersions:
For an open immersion `f : X ⟶ Z`, given any morphism of schemes `g : Y ⟶ Z` whose topological
image is contained in the image of `f`, we can lift this morphism to a unique `Y ⟶ X` that
commutes with these maps. |
lift_fac (H' : Set.range g.base ⊆ Set.range f.base) : lift f g H' ≫ f = g :=
Scheme.Hom.ext' <| LocallyRingedSpace.IsOpenImmersion.lift_fac f.toLRSHom g.toLRSHom H' | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lift_fac | null |
lift_uniq (H' : Set.range g.base ⊆ Set.range f.base) (l : Y ⟶ X) (hl : l ≫ f = g) :
l = lift f g H' :=
Scheme.Hom.ext' <| LocallyRingedSpace.IsOpenImmersion.lift_uniq
f.toLRSHom g.toLRSHom H' l.toLRSHom congr(($hl).toLRSHom)
@[reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lift_uniq | null |
comp_lift {Y' : Scheme} (g' : Y' ⟶ Y) (H : Set.range g.base ⊆ Set.range f.base) :
g' ≫ lift f g H = lift f (g' ≫ g) (.trans (by simp [Set.range_comp_subset_range]) H) := by
simp [← cancel_mono f] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | comp_lift | null |
isPullback_lift_id
{X U Y : Scheme.{u}} (f : X ⟶ Y) (g : U ⟶ Y) [IsOpenImmersion g]
(H : Set.range f.base ⊆ Set.range g.base) :
IsPullback (IsOpenImmersion.lift g f H) (𝟙 _) g f := by
convert IsPullback.of_id_snd.paste_horiz (IsKernelPair.id_of_mono g)
· exact (Category.comp_id _).symm
· simp | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isPullback_lift_id | null |
isoOfRangeEq [IsOpenImmersion g] (e : Set.range f.base = Set.range g.base) : X ≅ Y where
hom := lift g f (le_of_eq e)
inv := lift f g (le_of_eq e.symm)
hom_inv_id := by rw [← cancel_mono f]; simp
inv_hom_id := by rw [← cancel_mono g]; simp
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isoOfRangeEq | Two open immersions with equal range are isomorphic. |
isoOfRangeEq_hom_fac {X Y Z : Scheme.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)
[IsOpenImmersion f] [IsOpenImmersion g] (e : Set.range f.base = Set.range g.base) :
(isoOfRangeEq f g e).hom ≫ g = f :=
lift_fac _ _ (le_of_eq e)
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isoOfRangeEq_hom_fac | null |
isoOfRangeEq_inv_fac {X Y Z : Scheme.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)
[IsOpenImmersion f] [IsOpenImmersion g] (e : Set.range f.base = Set.range g.base) :
(isoOfRangeEq f g e).inv ≫ f = g :=
lift_fac _ _ (le_of_eq e.symm) | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isoOfRangeEq_inv_fac | null |
app_eq_invApp_app_of_comp_eq_aux {X Y U : Scheme.{u}} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X)
(H : fg = f ≫ g) [h : IsOpenImmersion g] (V : U.Opens) :
f ⁻¹ᵁ V = fg ⁻¹ᵁ (g ''ᵁ V) := by
simp_all | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | app_eq_invApp_app_of_comp_eq_aux | null |
app_eq_appIso_inv_app_of_comp_eq {X Y U : Scheme.{u}} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X)
(H : fg = f ≫ g) [h : IsOpenImmersion g] (V : U.Opens) :
f.app V = (g.appIso V).inv ≫ fg.app (g ''ᵁ V) ≫ Y.presheaf.map
(eqToHom <| IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux f g fg H V).op := by
subst H
r... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | app_eq_appIso_inv_app_of_comp_eq | The `fg` argument is to avoid nasty stuff about dependent types. |
lift_app {X Y U : Scheme.{u}} (f : U ⟶ Y) (g : X ⟶ Y) [IsOpenImmersion f] (H)
(V : U.Opens) :
(IsOpenImmersion.lift f g H).app V = (f.appIso V).inv ≫ g.app (f ''ᵁ V) ≫
X.presheaf.map (eqToHom <| IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux _ _ _
(IsOpenImmersion.lift_fac f g H).symm V).op :=
... | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | lift_app | null |
noncomputable
ΓIso {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
Γ(X, f⁻¹ᵁ U) ≅ Γ(Y, f.opensRange ⊓ U) :=
(f.appIso (f⁻¹ᵁ U)).symm ≪≫
Y.presheaf.mapIso (eqToIso <| (f.image_preimage_eq_opensRange_inter U).symm).op
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | ΓIso | If `f` is an open immersion `X ⟶ Y`, the global sections of `X`
are naturally isomorphic to the sections of `Y` over the image of `f`. |
ΓIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
(ΓIso f U).inv = f.appLE (f.opensRange ⊓ U) (f⁻¹ᵁ U)
(by rw [← f.image_preimage_eq_opensRange_inter, f.preimage_image_eq]) := by
simp only [ΓIso, Iso.trans_inv, Functor.mapIso_inv, Iso.op_inv, eqToIso.inv, eqToHom_op,
Iso.symm_i... | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | ΓIso_inv | null |
map_ΓIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
Y.presheaf.map (homOfLE inf_le_right).op ≫ (ΓIso f U).inv = f.app U := by
simp [Scheme.Hom.appLE_eq_app]
@[reassoc, elementwise] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | map_ΓIso_inv | null |
ΓIso_hom_map {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
f.app U ≫ (ΓIso f U).hom = Y.presheaf.map (homOfLE inf_le_right).op := by
rw [← map_ΓIso_inv]
simp [-ΓIso_inv] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | ΓIso_hom_map | null |
noncomputable
ΓIsoTop {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] :
Γ(X, ⊤) ≅ Γ(Y, f.opensRange) :=
(f.appIso ⊤).symm ≪≫ Y.presheaf.mapIso (eqToIso f.image_top_eq_opensRange.symm).op | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | ΓIsoTop | Given an open immersion `f : U ⟶ X`, the isomorphism between global sections
of `U` and the sections of `X` at the image of `f`. |
isIso_of_isOpenImmersion_of_opensRange_eq_top {X Y : Scheme.{u}} (f : X ⟶ Y)
[IsOpenImmersion f] (hf : f.opensRange = ⊤) : IsIso f := by
rw [isIso_iff_isOpenImmersion]
refine ⟨inferInstance, ?_⟩
rw [TopCat.epi_iff_surjective, ← Set.range_eq_univ]
exact TopologicalSpace.Opens.ext_iff.mp hf | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isIso_of_isOpenImmersion_of_opensRange_eq_top | null |
isOpenImmersion_isStableUnderComposition :
MorphismProperty.IsStableUnderComposition @IsOpenImmersion where
comp_mem f g _ _ := LocallyRingedSpace.IsOpenImmersion.comp f.toLRSHom g.toLRSHom | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isOpenImmersion_isStableUnderComposition | null |
isOpenImmersion_respectsIso : MorphismProperty.RespectsIso @IsOpenImmersion := by
apply MorphismProperty.respectsIso_of_isStableUnderComposition
intro _ _ f (hf : IsIso f)
have : IsIso f := hf
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isOpenImmersion_respectsIso | null |
isOpenImmersion_isMultiplicative :
MorphismProperty.IsMultiplicative @IsOpenImmersion where
id_mem _ := inferInstance | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.OpenImmersion",
"Mathlib.AlgebraicGeometry.Scheme",
"Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq",
"Mathlib.CategoryTheory.MorphismProperty.Limits"
] | Mathlib/AlgebraicGeometry/OpenImmersion.lean | isOpenImmersion_isMultiplicative | null |
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