Restriction of Schemes and Morphisms

Main definition

• AlgebraicGeometry.Scheme.restrict: The restriction of a scheme along an open embedding. The map X.restrict f ⟶ X is AlgebraicGeometry.Scheme.ofRestrict. U : X.Opens has a coercion to Scheme and U.ι is a shorthand for X.restrict U.open_embedding : U ⟶ X.
• AlgebraicGeometry.morphism_restrict: The restriction of X ⟶ Y to X ∣_ᵤ f ⁻¹ᵁ U ⟶ Y ∣_ᵤ U.

def AlgebraicGeometry.Scheme.Opens.toScheme {X : Scheme} (U : X.Opens) : Scheme

Open subset of a scheme as a scheme.

Equations

• ↑U = X.restrict ⋯

instance AlgebraicGeometry.Scheme.Opens.instCoeOut {X : Scheme} : CoeOut X.Opens Scheme

Equations

• AlgebraicGeometry.Scheme.Opens.instCoeOut = { coe := AlgebraicGeometry.Scheme.Opens.toScheme }

def AlgebraicGeometry.Scheme.Opens.ι {X : Scheme} (U : X.Opens) : ↑U ⟶ X

The restriction of a scheme to an open subset.

Equations

• U.ι = X.ofRestrict ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_base_apply {X : Scheme} (U : X.Opens) (x : ↥U) : (CategoryTheory.ConcreteCategory.hom U.ι.base) x = ↑x

instance AlgebraicGeometry.Scheme.Opens.instIsOpenImmersionι {X : Scheme} (U : X.Opens) : IsOpenImmersion U.ι

instance AlgebraicGeometry.Scheme.Opens.instCanonicallyOver {X : Scheme} (U : X.Opens) : CanonicallyOver X

Equations

• U.instCanonicallyOver = CategoryTheory.CanonicallyOverClass.mk

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.instCanonicallyOver_over {X : Scheme} (U : X.Opens) : ↑U ↘ X = U.ι

instance AlgebraicGeometry.Scheme.Opens.instIsOverι {X : Scheme} (U : X.Opens) : Hom.IsOver U.ι X

theorem AlgebraicGeometry.Scheme.Opens.toScheme_carrier {X : Scheme} (U : X.Opens) : ↑↑(↑U).toPresheafedSpace = ↑↑U

theorem AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj {X : Scheme} (U : X.Opens) (V : (↑U).Opens) : (↑U).presheaf.obj (Opposite.op V) = X.presheaf.obj (Opposite.op ((Hom.opensFunctor U.ι).obj V))

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map {X : Scheme} (U : X.Opens) {V W : (TopologicalSpace.Opens ↑↑(↑U).toPresheafedSpace)ᵒᵖ} (i : V ⟶ W) : (↑U).presheaf.map i = X.presheaf.map ((Hom.opensFunctor U.ι).map i.unop).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_app {X : Scheme} (U V : X.Opens) : Hom.app U.ι V = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_appTop {X : Scheme} (U : X.Opens) : Hom.appTop U.ι = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_appLE {X : Scheme} (U V : X.Opens) (W : (↑U).Opens) (e : W ≤ (TopologicalSpace.Opens.map U.ι.base).obj V) : Hom.appLE U.ι V W e = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_appIso {X : Scheme} (U : X.Opens) (V : (↑U).Opens) : Hom.appIso U.ι V = CategoryTheory.Iso.refl (X.presheaf.obj (Opposite.op ((Hom.opensFunctor U.ι).obj V)))

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.opensRange_ι {X : Scheme} (U : X.Opens) : Hom.opensRange U.ι = U

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.range_ι {X : Scheme} (U : X.Opens) : Set.range ⇑(CategoryTheory.ConcreteCategory.hom U.ι.base) = ↑U

theorem AlgebraicGeometry.Scheme.Opens.ι_image_top {X : Scheme} (U : X.Opens) : (Hom.opensFunctor U.ι).obj ⊤ = U

theorem AlgebraicGeometry.Scheme.Opens.ι_image_le {X : Scheme} (U : X.Opens) (W : (↑U).Opens) : (Hom.opensFunctor U.ι).obj W ≤ U

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.ι_preimage_self {X : Scheme} (U : X.Opens) : (TopologicalSpace.Opens.map U.ι.base).obj U = ⊤

instance AlgebraicGeometry.Scheme.Opens.ι_appLE_isIso {X : Scheme} (U : X.Opens) : CategoryTheory.IsIso (Hom.appLE U.ι U ⊤ ⋯)

theorem AlgebraicGeometry.Scheme.Opens.ι_app_self {X : Scheme} (U : X.Opens) : Hom.app U.ι U = X.presheaf.map (CategoryTheory.eqToHom ⋯).op

theorem AlgebraicGeometry.Scheme.Opens.eq_presheaf_map_eqToHom {X : Scheme} (U : X.Opens) {V W : (↑U).Opens} (e : (Hom.opensFunctor U.ι).obj V = (Hom.opensFunctor U.ι).obj W) : X.presheaf.map (CategoryTheory.eqToHom e).op = (↑U).presheaf.map (CategoryTheory.eqToHom ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.nonempty_iff {X : Scheme} (U : X.Opens) : Nonempty ↑↑(↑U).toPresheafedSpace ↔ (↑U).Nonempty

def AlgebraicGeometry.Scheme.Opens.topIso {X : Scheme} (U : X.Opens) : (↑U).presheaf.obj (Opposite.op ⊤) ≅ X.presheaf.obj (Opposite.op U)

The global sections of the restriction is isomorphic to the sections on the open set.

Equations

• U.topIso = CategoryTheory.Functor.mapIso X.presheaf (CategoryTheory.eqToIso ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.topIso_inv {X : Scheme} (U : X.Opens) : U.topIso.inv = X.presheaf.map (CategoryTheory.eqToHom ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.topIso_hom {X : Scheme} (U : X.Opens) : U.topIso.hom = X.presheaf.map (CategoryTheory.eqToHom ⋯).op

def AlgebraicGeometry.Scheme.Opens.stalkIso {X : Scheme} (U : X.Opens) (x : ↥U) : (↑U).presheaf.stalk x ≅ X.presheaf.stalk ↑x

The stalks of an open subscheme are isomorphic to the stalks of the original scheme.

Equations

• U.stalkIso x = X.restrictStalkIso ⋯ x

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom {X : Scheme} (U : X.Opens) {V : (↑U).Opens} (x : ↥U) (hx : x ∈ V) : CategoryTheory.CategoryStruct.comp ((↑U).presheaf.germ V x hx) (U.stalkIso x).hom = X.presheaf.germ ((Hom.opensFunctor U.ι).obj V) ↑x ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom_assoc {X : Scheme} (U : X.Opens) {V : (↑U).Opens} (x : ↥U) (hx : x ∈ V) {Z : CommRingCat} (h : X.presheaf.stalk ↑x ⟶ Z) : CategoryTheory.CategoryStruct.comp ((↑U).presheaf.germ V x hx) (CategoryTheory.CategoryStruct.comp (U.stalkIso x).hom h) = CategoryTheory.CategoryStruct.comp (X.presheaf.germ ((Hom.opensFunctor U.ι).obj V) ↑x ⋯) h

theorem AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv {X : Scheme} (U : X.Opens) (V : (↑U).Opens) (x : ↥U) (hx : x ∈ V) : CategoryTheory.CategoryStruct.comp (X.presheaf.germ ((Hom.opensFunctor U.ι).obj V) ↑x ⋯) (U.stalkIso x).inv = (↑U).presheaf.germ V x hx

theorem AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc {X : Scheme} (U : X.Opens) (V : (↑U).Opens) (x : ↥U) (hx : x ∈ V) {Z : CommRingCat} (h : (↑U).presheaf.stalk x ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.presheaf.germ ((Hom.opensFunctor U.ι).obj V) ↑x ⋯) (CategoryTheory.CategoryStruct.comp (U.stalkIso x).inv h) = CategoryTheory.CategoryStruct.comp ((↑U).presheaf.germ V x hx) h

theorem AlgebraicGeometry.Scheme.Opens.stalkIso_inv {X : Scheme} (U : X.Opens) (x : ↥U) : (U.stalkIso x).inv = Hom.stalkMap U.ι x

def AlgebraicGeometry.Scheme.openCoverOfISupEqTop {s : Type u_1} (X : Scheme) (U : s → X.Opens) (hU : ⨆ (i : s), U i = ⊤) : X.OpenCover

If U is a family of open sets that covers X, then X.restrict U forms an X.open_cover.

Equations

• X.openCoverOfISupEqTop U hU = { J := s, obj := fun (i : s) => ↑(U i), map := fun (i : s) => (U i).ι, f := fun (x : ↑↑X.toPresheafedSpace) => ⋯.choose, covers := ⋯, map_prop := ⋯ }

@[simp]

theorem AlgebraicGeometry.Scheme.openCoverOfISupEqTop_J {s : Type u_1} (X : Scheme) (U : s → X.Opens) (hU : ⨆ (i : s), U i = ⊤) : (X.openCoverOfISupEqTop U hU).J = s

@[simp]

theorem AlgebraicGeometry.Scheme.openCoverOfISupEqTop_obj {s : Type u_1} (X : Scheme) (U : s → X.Opens) (hU : ⨆ (i : s), U i = ⊤) (i : s) : (X.openCoverOfISupEqTop U hU).obj i = ↑(U i)

@[simp]

theorem AlgebraicGeometry.Scheme.openCoverOfISupEqTop_map {s : Type u_1} (X : Scheme) (U : s → X.Opens) (hU : ⨆ (i : s), U i = ⊤) (i : s) : (X.openCoverOfISupEqTop U hU).map i = (U i).ι

def AlgebraicGeometry.opensRestrict {X : Scheme} (U : X.Opens) : (↑U).Opens ≃ { V : X.Opens // V ≤ U }

The open sets of an open subscheme corresponds to the open sets containing in the subset.

Equations

• AlgebraicGeometry.opensRestrict U = (AlgebraicGeometry.IsOpenImmersion.opensEquiv U.ι).trans (Equiv.subtypeEquivProp ⋯)

@[simp]

theorem AlgebraicGeometry.coe_opensRestrict_symm_apply {X : Scheme} (U : X.Opens) (a✝ : { V : X.Opens // V ≤ U }) : ↑((opensRestrict U).symm a✝) = ⇑(CategoryTheory.ConcreteCategory.hom U.ι.base) ⁻¹' ↑↑((Equiv.subtypeEquivProp ⋯).symm a✝)

@[simp]

theorem AlgebraicGeometry.coe_opensRestrict_apply_coe {X : Scheme} (U : X.Opens) (a✝ : (↑U).Opens) : ↑↑((opensRestrict U) a✝) = (fun (a : ↑↑(↑U).toPresheafedSpace) => ↑a) '' ↑a✝

instance AlgebraicGeometry.ΓRestrictAlgebra {X : Scheme} (U : X.Opens) : Algebra ↑(X.presheaf.obj (Opposite.op ⊤)) ↑((↑U).presheaf.obj (Opposite.op ⊤))

Equations

• AlgebraicGeometry.ΓRestrictAlgebra U = (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appTop U.ι)).toAlgebra

theorem AlgebraicGeometry.Scheme.map_basicOpen {X : Scheme} (U : X.Opens) (r : ↑((↑U).presheaf.obj (Opposite.op ⊤))) : (Hom.opensFunctor U.ι).obj ((↑U).basicOpen r) = X.basicOpen ((CategoryTheory.ConcreteCategory.hom (X.presheaf.map (CategoryTheory.eqToHom ⋯).op)) r)

@[deprecated AlgebraicGeometry.Scheme.map_basicOpen (since := "2024-10-23")]

theorem AlgebraicGeometry.Scheme.map_basicOpen' {X : Scheme} (U : X.Opens) (r : ↑((↑U).presheaf.obj (Opposite.op ⊤))) : (Hom.opensFunctor U.ι).obj ((↑U).basicOpen r) = X.basicOpen ((CategoryTheory.ConcreteCategory.hom (X.presheaf.map (CategoryTheory.eqToHom ⋯).op)) r)

Alias of AlgebraicGeometry.Scheme.map_basicOpen.

theorem AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen {X : Scheme} (U : X.Opens) (r : ↑((↑U).presheaf.obj (Opposite.op ⊤))) : (Hom.opensFunctor U.ι).obj ((↑U).basicOpen r) = X.basicOpen r

theorem AlgebraicGeometry.Scheme.map_basicOpen_map {X : Scheme} (U : X.Opens) (r : ↑(X.presheaf.obj (Opposite.op U))) : (Hom.opensFunctor U.ι).obj ((↑U).basicOpen ((CategoryTheory.ConcreteCategory.hom U.topIso.inv) r)) = X.basicOpen r

noncomputable def AlgebraicGeometry.Scheme.homOfLE (X : Scheme) {U V : X.Opens} (e : U ≤ V) : ↑U ⟶ ↑V

If U ≤ V, then U is also a subscheme of V.

Equations

• X.homOfLE e = AlgebraicGeometry.IsOpenImmersion.lift V.ι U.ι ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_ι (X : Scheme) {U V : X.Opens} (e : U ≤ V) : CategoryTheory.CategoryStruct.comp (X.homOfLE e) V.ι = U.ι

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_ι_assoc (X : Scheme) {U V : X.Opens} (e : U ≤ V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.homOfLE e) (CategoryTheory.CategoryStruct.comp V.ι h) = CategoryTheory.CategoryStruct.comp U.ι h

instance AlgebraicGeometry.instIsOverHomOfLE {X : Scheme} {U V : X.Opens} (h : U ≤ V) : Scheme.Hom.IsOver (X.homOfLE h) X

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_rfl (X : Scheme) (U : X.Opens) : X.homOfLE ⋯ = CategoryTheory.CategoryStruct.id ↑U

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_homOfLE (X : Scheme) {U V W : X.Opens} (e₁ : U ≤ V) (e₂ : V ≤ W) : CategoryTheory.CategoryStruct.comp (X.homOfLE e₁) (X.homOfLE e₂) = X.homOfLE ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_homOfLE_assoc (X : Scheme) {U V W : X.Opens} (e₁ : U ≤ V) (e₂ : V ≤ W) {Z : Scheme} (h : ↑W ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.homOfLE e₁) (CategoryTheory.CategoryStruct.comp (X.homOfLE e₂) h) = CategoryTheory.CategoryStruct.comp (X.homOfLE ⋯) h

theorem AlgebraicGeometry.Scheme.homOfLE_base {X : Scheme} {U V : X.Opens} (e : U ≤ V) : (X.homOfLE e).base = (TopologicalSpace.Opens.toTopCat ↑X.toPresheafedSpace).map (CategoryTheory.homOfLE e)

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_apply {X : Scheme} {U V : X.Opens} (e : U ≤ V) (x : ↥U) : ↑((CategoryTheory.ConcreteCategory.hom (X.homOfLE e).base) x) = ↑x

theorem AlgebraicGeometry.Scheme.ι_image_homOfLE_le_ι_image {X : Scheme} {U V : X.Opens} (e : U ≤ V) (W : (↑V).Opens) : (Hom.opensFunctor U.ι).obj ((TopologicalSpace.Opens.map (X.homOfLE e).base).obj W) ≤ (Hom.opensFunctor V.ι).obj W

@[simp]

theorem AlgebraicGeometry.Scheme.homOfLE_app {X : Scheme} {U V : X.Opens} (e : U ≤ V) (W : (↑V).Opens) : Hom.app (X.homOfLE e) W = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

theorem AlgebraicGeometry.Scheme.homOfLE_appTop {X : Scheme} {U V : X.Opens} (e : U ≤ V) : Hom.appTop (X.homOfLE e) = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

instance AlgebraicGeometry.instIsOpenImmersionHomOfLE (X : Scheme) {U V : X.Opens} (e : U ≤ V) : IsOpenImmersion (X.homOfLE e)

def AlgebraicGeometry.Scheme.restrictFunctor (X : Scheme) : CategoryTheory.Functor X.Opens (CategoryTheory.Over X)

The functor taking open subsets of X to open subschemes of X.

Equations

• X.restrictFunctor = { obj := fun (U : X.Opens) => CategoryTheory.Over.mk U.ι, map := fun {U V : X.Opens} (i : U ⟶ V) => CategoryTheory.Over.homMk (X.homOfLE ⋯) ⋯, map_id := ⋯, map_comp := ⋯ }

@[simp]

theorem AlgebraicGeometry.Scheme.restrictFunctor_obj_left (X : Scheme) (U : X.Opens) : (X.restrictFunctor.obj U).left = ↑U

@[simp]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_left (X : Scheme) {U V : X.Opens} (i : U ⟶ V) : (X.restrictFunctor.map i).left = X.homOfLE ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.restrictFunctor_obj_hom (X : Scheme) (U : X.Opens) : (X.restrictFunctor.obj U).hom = U.ι

@[deprecated AlgebraicGeometry.Scheme.homOfLE_ι (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_ofRestrict (X : Scheme) {U V : X.Opens} (e : U ≤ V) : CategoryTheory.CategoryStruct.comp (X.homOfLE e) V.ι = U.ι

Alias of AlgebraicGeometry.Scheme.homOfLE_ι.

@[deprecated AlgebraicGeometry.Scheme.homOfLE_ι_assoc (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_ofRestrict_assoc (X : Scheme) {U V : X.Opens} (e : U ≤ V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.homOfLE e) (CategoryTheory.CategoryStruct.comp V.ι h) = CategoryTheory.CategoryStruct.comp U.ι h

Alias of AlgebraicGeometry.Scheme.homOfLE_ι_assoc.

@[deprecated AlgebraicGeometry.Scheme.homOfLE_base (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_base {X : Scheme} {U V : X.Opens} (e : U ≤ V) : (X.homOfLE e).base = (TopologicalSpace.Opens.toTopCat ↑X.toPresheafedSpace).map (CategoryTheory.homOfLE e)

Alias of AlgebraicGeometry.Scheme.homOfLE_base.

@[deprecated AlgebraicGeometry.Scheme.ι_image_homOfLE_le_ι_image (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_app_aux {X : Scheme} {U V : X.Opens} (e : U ≤ V) (W : (↑V).Opens) : (Hom.opensFunctor U.ι).obj ((TopologicalSpace.Opens.map (X.homOfLE e).base).obj W) ≤ (Hom.opensFunctor V.ι).obj W

Alias of AlgebraicGeometry.Scheme.ι_image_homOfLE_le_ι_image.

@[deprecated AlgebraicGeometry.Scheme.homOfLE_app (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictFunctor_map_app {X : Scheme} {U V : X.Opens} (e : U ≤ V) (W : (↑V).Opens) : Hom.app (X.homOfLE e) W = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

Alias of AlgebraicGeometry.Scheme.homOfLE_app.

def AlgebraicGeometry.Scheme.restrictFunctorΓ {X : Scheme} : X.restrictFunctor.op.comp ((CategoryTheory.Over.forget X).op.comp Γ) ≅ X.presheaf

The functor that restricts to open subschemes and then takes global section is isomorphic to the structure sheaf.

Equations

• AlgebraicGeometry.Scheme.restrictFunctorΓ = CategoryTheory.NatIso.ofComponents (fun (U : X.Opensᵒᵖ) => CategoryTheory.Functor.mapIso X.presheaf (CategoryTheory.eqToIso ⋯).symm.op) ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app {X : Scheme} (X✝ : X.Opensᵒᵖ) : restrictFunctorΓ.inv.app X✝ = X.presheaf.map (CategoryTheory.eqToHom ⋯)

@[simp]

theorem AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app {X : Scheme} (X✝ : X.Opensᵒᵖ) : restrictFunctorΓ.hom.app X✝ = X.presheaf.map (CategoryTheory.eqToHom ⋯)

noncomputable def AlgebraicGeometry.Scheme.restrictRestrictComm (X : Scheme) (U V : X.Opens) : ↑((TopologicalSpace.Opens.map U.ι.base).obj V) ≅ ↑((TopologicalSpace.Opens.map V.ι.base).obj U)

X ∣_ U ∣_ V is isomorphic to X ∣_ V ∣_ U

Equations

• One or more equations did not get rendered due to their size.

noncomputable def AlgebraicGeometry.Scheme.Hom.isoImage {X Y : Scheme} (f : X.Hom Y) [IsOpenImmersion f] (U : X.Opens) : ↑U ≅ ↑(f.opensFunctor.obj U)

If f : X ⟶ Y is an open immersion, then for any U : X.Opens, we have the isomorphism U ≅ f ''ᵁ U.

Equations

• f.isoImage U = AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq (CategoryTheory.CategoryStruct.comp U.ι f) (f.opensFunctor.obj U).ι ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) : CategoryTheory.CategoryStruct.comp (isoImage f U).hom ((opensFunctor f).obj U).ι = CategoryTheory.CategoryStruct.comp U.ι f

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι_assoc {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoImage f U).hom (CategoryTheory.CategoryStruct.comp ((opensFunctor f).obj U).ι h) = CategoryTheory.CategoryStruct.comp U.ι (CategoryTheory.CategoryStruct.comp f h)

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) : CategoryTheory.CategoryStruct.comp (isoImage f U).inv (CategoryTheory.CategoryStruct.comp U.ι f) = ((opensFunctor f).obj U).ι

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι_assoc {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoImage f U).inv (CategoryTheory.CategoryStruct.comp U.ι (CategoryTheory.CategoryStruct.comp f h)) = CategoryTheory.CategoryStruct.comp ((opensFunctor f).obj U).ι h

@[deprecated AlgebraicGeometry.Scheme.Hom.isoImage (since := "2024-10-20")]

def AlgebraicGeometry.Scheme.restrictRestrict {X Y : Scheme} (f : X.Hom Y) [IsOpenImmersion f] (U : X.Opens) : ↑U ≅ ↑(f.opensFunctor.obj U)

Alias of AlgebraicGeometry.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.

Equations

• @AlgebraicGeometry.Scheme.restrictRestrict = @AlgebraicGeometry.Scheme.Hom.isoImage

@[deprecated AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictRestrict_hom_restrict {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) : CategoryTheory.CategoryStruct.comp (Hom.isoImage f U).hom ((Hom.opensFunctor f).obj U).ι = CategoryTheory.CategoryStruct.comp U.ι f

Alias of AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι.

@[deprecated AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictRestrict_inv_restrict_restrict {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) : CategoryTheory.CategoryStruct.comp (Hom.isoImage f U).inv (CategoryTheory.CategoryStruct.comp U.ι f) = ((Hom.opensFunctor f).obj U).ι

Alias of AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι.

@[deprecated AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι_assoc (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictRestrict_hom_restrict_assoc {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (Hom.isoImage f U).hom (CategoryTheory.CategoryStruct.comp ((Hom.opensFunctor f).obj U).ι h) = CategoryTheory.CategoryStruct.comp U.ι (CategoryTheory.CategoryStruct.comp f h)

Alias of AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι_assoc.

@[deprecated AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι_assoc (since := "2024-10-20")]

theorem AlgebraicGeometry.Scheme.restrictRestrict_inv_restrict_restrict_assoc {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (U : X.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (Hom.isoImage f U).inv (CategoryTheory.CategoryStruct.comp U.ι (CategoryTheory.CategoryStruct.comp f h)) = CategoryTheory.CategoryStruct.comp ((Hom.opensFunctor f).obj U).ι h

Alias of AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι_assoc.

def AlgebraicGeometry.Scheme.topIso (X : Scheme) : ↑⊤ ≅ X

(⊤ : X.Opens) as a scheme is isomorphic to X.

Equations

• X.topIso = { hom := ⊤.ι, inv := { toHom_1 := X.restrictTopIso.inv }, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem AlgebraicGeometry.Scheme.topIso_hom (X : Scheme) : X.topIso.hom = ⊤.ι

@[simp]

theorem AlgebraicGeometry.Scheme.toIso_inv_ι (X : Scheme) : CategoryTheory.CategoryStruct.comp X.topIso.inv ⊤.ι = CategoryTheory.CategoryStruct.id X

@[simp]

theorem AlgebraicGeometry.Scheme.toIso_inv_ι_assoc (X : Scheme) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp X.topIso.inv (CategoryTheory.CategoryStruct.comp ⊤.ι h) = h

@[simp]

theorem AlgebraicGeometry.Scheme.ι_toIso_inv (X : Scheme) : CategoryTheory.CategoryStruct.comp ⊤.ι X.topIso.inv = CategoryTheory.CategoryStruct.id ↑⊤

@[simp]

theorem AlgebraicGeometry.Scheme.ι_toIso_inv_assoc (X : Scheme) {Z : Scheme} (h : ↑⊤ ⟶ Z) : CategoryTheory.CategoryStruct.comp ⊤.ι (CategoryTheory.CategoryStruct.comp X.topIso.inv h) = h

noncomputable def AlgebraicGeometry.Scheme.isoOfEq (X : Scheme) {U V : X.Opens} (e : U = V) : ↑U ≅ ↑V

If U = V, then X ∣_ U is isomorphic to X ∣_ V.

Equations

• X.isoOfEq e = AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq U.ι V.ι ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.isoOfEq_hom_ι (X : Scheme) {U V : X.Opens} (e : U = V) : CategoryTheory.CategoryStruct.comp (X.isoOfEq e).hom V.ι = U.ι

@[simp]

theorem AlgebraicGeometry.Scheme.isoOfEq_hom_ι_assoc (X : Scheme) {U V : X.Opens} (e : U = V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.isoOfEq e).hom (CategoryTheory.CategoryStruct.comp V.ι h) = CategoryTheory.CategoryStruct.comp U.ι h

@[simp]

theorem AlgebraicGeometry.Scheme.isoOfEq_inv_ι (X : Scheme) {U V : X.Opens} (e : U = V) : CategoryTheory.CategoryStruct.comp (X.isoOfEq e).inv U.ι = V.ι

@[simp]

theorem AlgebraicGeometry.Scheme.isoOfEq_inv_ι_assoc (X : Scheme) {U V : X.Opens} (e : U = V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.isoOfEq e).inv (CategoryTheory.CategoryStruct.comp U.ι h) = CategoryTheory.CategoryStruct.comp V.ι h

@[simp]

theorem AlgebraicGeometry.Scheme.isoOfEq_rfl (X : Scheme) (U : X.Opens) : X.isoOfEq ⋯ = CategoryTheory.Iso.refl ↑U

@[deprecated AlgebraicGeometry.Scheme.isoOfEq (since := "2024-10-20")]

def AlgebraicGeometry.Scheme.restrictIsoOfEq (X : Scheme) {U V : X.Opens} (e : U = V) : ↑U ≅ ↑V

Alias of AlgebraicGeometry.Scheme.isoOfEq.

---

If U = V, then X ∣_ U is isomorphic to X ∣_ V.

Equations

• AlgebraicGeometry.Scheme.restrictIsoOfEq = AlgebraicGeometry.Scheme.isoOfEq

noncomputable def AlgebraicGeometry.Scheme.Hom.preimageIso {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) : ↑((TopologicalSpace.Opens.map f.base).obj U) ≅ ↑U

The restriction of an isomorphism onto an open set.

Equations

• f.preimageIso U = AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι f) U.ι ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (f.preimageIso U).hom U.ι = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι f

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι_assoc {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (f.preimageIso U).hom (CategoryTheory.CategoryStruct.comp U.ι h) = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι (CategoryTheory.CategoryStruct.comp f h)

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (f.preimageIso U).inv (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι f) = U.ι

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι_assoc {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (f.preimageIso U).inv (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι (CategoryTheory.CategoryStruct.comp f h)) = CategoryTheory.CategoryStruct.comp U.ι h

@[deprecated AlgebraicGeometry.Scheme.Hom.preimageIso (since := "2024-10-20")]

def AlgebraicGeometry.Scheme.restrictMapIso {X Y : Scheme} (f : X.Hom Y) [CategoryTheory.IsIso f] (U : Y.Opens) : ↑((TopologicalSpace.Opens.map f.base).obj U) ≅ ↑U

Alias of AlgebraicGeometry.Scheme.Hom.preimageIso.

---

The restriction of an isomorphism onto an open set.

Equations

• @AlgebraicGeometry.Scheme.restrictMapIso = @AlgebraicGeometry.Scheme.Hom.preimageIso

noncomputable def AlgebraicGeometry.Scheme.Opens.isoOfLE {X : Scheme} {U V : X.Opens} (hUV : U ≤ V) : ↑((TopologicalSpace.Opens.map V.ι.base).obj U) ≅ ↑U

If U ≤ V are opens of X, the restriction of U to V is isomorphic to U.

Equations

• AlgebraicGeometry.Scheme.Opens.isoOfLE hUV = AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map V.ι.base).obj U).ι V.ι) U.ι ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.isoOfLE_hom_ι {X : Scheme} {U V : X.Opens} (hUV : U ≤ V) : CategoryTheory.CategoryStruct.comp (isoOfLE hUV).hom U.ι = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map V.ι.base).obj U).ι V.ι

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.isoOfLE_hom_ι_assoc {X : Scheme} {U V : X.Opens} (hUV : U ≤ V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoOfLE hUV).hom (CategoryTheory.CategoryStruct.comp U.ι h) = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map V.ι.base).obj U).ι (CategoryTheory.CategoryStruct.comp V.ι h)

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.isoOfLE_inv_ι {X : Scheme} {U V : X.Opens} (hUV : U ≤ V) : CategoryTheory.CategoryStruct.comp (isoOfLE hUV).inv (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map V.ι.base).obj U).ι V.ι) = U.ι

@[simp]

theorem AlgebraicGeometry.Scheme.Opens.isoOfLE_inv_ι_assoc {X : Scheme} {U V : X.Opens} (hUV : U ≤ V) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (isoOfLE hUV).inv (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map V.ι.base).obj U).ι (CategoryTheory.CategoryStruct.comp V.ι h)) = CategoryTheory.CategoryStruct.comp U.ι h

def AlgebraicGeometry.pullbackRestrictIsoRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.Limits.pullback f U.ι ≅ ↑((TopologicalSpace.Opens.map f.base).obj U)

Given a morphism f : X ⟶ Y and an open set U ⊆ Y, we have X ×[Y] U ≅ X |_{f ⁻¹ U}

Equations

• AlgebraicGeometry.pullbackRestrictIsoRestrict f U = AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq (CategoryTheory.Limits.pullback.fst f U.ι) ((TopologicalSpace.Opens.map f.base).obj U).ι ⋯

@[simp]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).inv (CategoryTheory.Limits.pullback.fst f U.ι) = ((TopologicalSpace.Opens.map f.base).obj U).ι

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst f U.ι) h) = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι h

@[simp]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom ((TopologicalSpace.Opens.map f.base).obj U).ι = CategoryTheory.Limits.pullback.fst f U.ι

@[simp]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst f U.ι) h

@[deprecated AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι (since := "2024-10-20")]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_restrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom ((TopologicalSpace.Opens.map f.base).obj U).ι = CategoryTheory.Limits.pullback.fst f U.ι

Alias of AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι.

@[deprecated AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc (since := "2024-10-20")]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_restrict_assoc {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) {Z : Scheme} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom (CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst f U.ι) h

Alias of AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc.

def AlgebraicGeometry.morphismRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : ↑((TopologicalSpace.Opens.map f.base).obj U) ⟶ ↑U

The restriction of a morphism X ⟶ Y onto X |_{f ⁻¹ U} ⟶ Y |_ U.

Equations

• f ∣_ U = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.pullbackRestrictIsoRestrict f U).inv (CategoryTheory.Limits.pullback.snd f U.ι)

def AlgebraicGeometry.«term_∣__» : Lean.TrailingParserDescr

the notation for restricting a morphism of scheme to an open subset of the target scheme

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom (f ∣_ U) = CategoryTheory.Limits.pullback.snd f U.ι

@[simp]

theorem AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) {Z : Scheme} (h : ↑U ⟶ Z) : CategoryTheory.CategoryStruct.comp (pullbackRestrictIsoRestrict f U).hom (CategoryTheory.CategoryStruct.comp (f ∣_ U) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.snd f U.ι) h

@[simp]

theorem AlgebraicGeometry.morphismRestrict_ι {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.CategoryStruct.comp (f ∣_ U) U.ι = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι f

@[simp]

theorem AlgebraicGeometry.morphismRestrict_ι_assoc {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (f ∣_ U) (CategoryTheory.CategoryStruct.comp U.ι h) = CategoryTheory.CategoryStruct.comp ((TopologicalSpace.Opens.map f.base).obj U).ι (CategoryTheory.CategoryStruct.comp f h)

theorem AlgebraicGeometry.isPullback_morphismRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : CategoryTheory.IsPullback (f ∣_ U) ((TopologicalSpace.Opens.map f.base).obj U).ι U.ι f

theorem AlgebraicGeometry.isPullback_opens_inf_le {X : Scheme} {U V W : X.Opens} (hU : U ≤ W) (hV : V ≤ W) : CategoryTheory.IsPullback (X.homOfLE ⋯) (X.homOfLE ⋯) (X.homOfLE hU) (X.homOfLE hV)

theorem AlgebraicGeometry.isPullback_opens_inf {X : Scheme} (U V : X.Opens) : CategoryTheory.IsPullback (X.homOfLE ⋯) (X.homOfLE ⋯) U.ι V.ι

@[simp]

theorem AlgebraicGeometry.morphismRestrict_id {X : Scheme} (U : X.Opens) : CategoryTheory.CategoryStruct.id X ∣_ U = CategoryTheory.CategoryStruct.id ↑((TopologicalSpace.Opens.map (CategoryTheory.CategoryStruct.id X).base).obj U)

theorem AlgebraicGeometry.morphismRestrict_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U : TopologicalSpace.Opens ↑↑Z.toPresheafedSpace) : CategoryTheory.CategoryStruct.comp f g ∣_ U = CategoryTheory.CategoryStruct.comp (f ∣_ (TopologicalSpace.Opens.map g.base).obj U) (g ∣_ U)

instance AlgebraicGeometry.instIsIsoSchemeMorphismRestrict {X Y : Scheme} (f : X ⟶ Y) [CategoryTheory.IsIso f] (U : Y.Opens) : CategoryTheory.IsIso (f ∣_ U)

theorem AlgebraicGeometry.morphismRestrict_base_coe {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (x : ↑↑(↑((TopologicalSpace.Opens.map f.base).obj U)).toPresheafedSpace) : Coe.coe ((CategoryTheory.ConcreteCategory.hom (f ∣_ U).base) x) = (CategoryTheory.ConcreteCategory.hom f.base) ↑x

theorem AlgebraicGeometry.morphismRestrict_base {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : ⇑(CategoryTheory.ConcreteCategory.hom (f ∣_ U).base) = U.carrier.restrictPreimage ⇑(CategoryTheory.ConcreteCategory.hom f.base)

theorem AlgebraicGeometry.image_morphismRestrict_preimage {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : TopologicalSpace.Opens ↑↑(↑U).toPresheafedSpace) : (Scheme.Hom.opensFunctor ((TopologicalSpace.Opens.map f.base).obj U).ι).obj ((TopologicalSpace.Opens.map (f ∣_ U).base).obj V) = (TopologicalSpace.Opens.map f.base).obj ((Scheme.Hom.opensFunctor U.ι).obj V)

theorem AlgebraicGeometry.eqToHom_eq_homOfLE {C : Type u_1} [Preorder C] {X Y : C} (e : X = Y) : CategoryTheory.eqToHom e = CategoryTheory.homOfLE ⋯

theorem AlgebraicGeometry.morphismRestrict_app {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : (↑U).Opens) : Scheme.Hom.app (f ∣_ U) V = CategoryTheory.CategoryStruct.comp (Scheme.Hom.app f ((Scheme.Hom.opensFunctor U.ι).obj V)) (X.presheaf.map (CategoryTheory.eqToHom ⋯).op)

theorem AlgebraicGeometry.morphismRestrict_appTop {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : Scheme.Hom.appTop (f ∣_ U) = CategoryTheory.CategoryStruct.comp (Scheme.Hom.app f ((Scheme.Hom.opensFunctor U.ι).obj ⊤)) (X.presheaf.map (CategoryTheory.eqToHom ⋯).op)

@[simp]

theorem AlgebraicGeometry.morphismRestrict_app' {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : TopologicalSpace.Opens ↑↑(↑U).toPresheafedSpace) : Scheme.Hom.app (f ∣_ U) V = Scheme.Hom.appLE f ((Scheme.Hom.opensFunctor U.ι).obj V) ((Scheme.Hom.opensFunctor ((TopologicalSpace.Opens.map f.base).obj U).ι).obj ((TopologicalSpace.Opens.map (f ∣_ U).base).obj V)) ⋯

@[simp]

theorem AlgebraicGeometry.morphismRestrict_appLE {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : (↑U).Opens) (W : (↑((TopologicalSpace.Opens.map f.base).obj U)).Opens) (e : W ≤ (TopologicalSpace.Opens.map (f ∣_ U).base).obj V) : Scheme.Hom.appLE (f ∣_ U) V W e = Scheme.Hom.appLE f ((Scheme.Hom.opensFunctor U.ι).obj V) ((Scheme.Hom.opensFunctor ((TopologicalSpace.Opens.map f.base).obj U).ι).obj W) ⋯

theorem AlgebraicGeometry.Γ_map_morphismRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) : Scheme.Γ.map (f ∣_ U).op = CategoryTheory.CategoryStruct.comp (Y.presheaf.map (CategoryTheory.eqToHom ⋯).op) (CategoryTheory.CategoryStruct.comp (Scheme.Hom.app f U) (X.presheaf.map (CategoryTheory.eqToHom ⋯).op))

def AlgebraicGeometry.morphismRestrictOpensRange {X Y U : Scheme} (f : X ⟶ Y) (g : U ⟶ Y) [hg : IsOpenImmersion g] : CategoryTheory.Arrow.mk (f ∣_ Scheme.Hom.opensRange g) ≅ CategoryTheory.Arrow.mk (CategoryTheory.Limits.pullback.snd f g)

Restricting a morphism onto the image of an open immersion is isomorphic to the base change along the immersion.

Equations

• One or more equations did not get rendered due to their size.

def AlgebraicGeometry.morphismRestrictEq {X Y : Scheme} (f : X ⟶ Y) {U V : Y.Opens} (e : U = V) : CategoryTheory.Arrow.mk (f ∣_ U) ≅ CategoryTheory.Arrow.mk (f ∣_ V)

The restrictions onto two equal open sets are isomorphic. This currently has bad defeqs when unfolded, but it should not matter for now. Replace this definition if better defeqs are needed.

Equations

• AlgebraicGeometry.morphismRestrictEq f e = CategoryTheory.eqToIso ⋯

def AlgebraicGeometry.morphismRestrictRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : (↑U).Opens) : CategoryTheory.Arrow.mk (f ∣_ U ∣_ V) ≅ CategoryTheory.Arrow.mk (f ∣_ (Scheme.Hom.opensFunctor U.ι).obj V)

Restricting a morphism twice is isomorphic to one restriction.

Equations

• One or more equations did not get rendered due to their size.

def AlgebraicGeometry.morphismRestrictRestrictBasicOpen {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (r : ↑(Y.presheaf.obj (Opposite.op U))) : CategoryTheory.Arrow.mk (f ∣_ U ∣_ (↑U).basicOpen ((CategoryTheory.ConcreteCategory.hom (Y.presheaf.map (CategoryTheory.eqToHom ⋯).op)) r)) ≅ CategoryTheory.Arrow.mk (f ∣_ Y.basicOpen r)

Restricting a morphism twice onto a basic open set is isomorphic to one restriction.

Equations

• One or more equations did not get rendered due to their size.

def AlgebraicGeometry.morphismRestrictStalkMap {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (x : ↑↑(↑((TopologicalSpace.Opens.map f.base).obj U)).toPresheafedSpace) : CategoryTheory.Arrow.mk (Scheme.Hom.stalkMap (f ∣_ U) x) ≅ CategoryTheory.Arrow.mk (Scheme.Hom.stalkMap f ↑x)

The stalk map of a restriction of a morphism is isomorphic to the stalk map of the original map.

Equations

• One or more equations did not get rendered due to their size.

instance AlgebraicGeometry.instIsOpenImmersionMorphismRestrict {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) [IsOpenImmersion f] : IsOpenImmersion (f ∣_ U)

def AlgebraicGeometry.Scheme.Hom.resLE {X Y : Scheme} (f : X.Hom Y) (U : Y.Opens) (V : X.Opens) (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) : ↑V ⟶ ↑U

The restriction of a morphism f : X ⟶ Y to open sets on the source and target.

Equations

• f.resLE U V e = CategoryTheory.CategoryStruct.comp (X.homOfLE e) (f ∣_ U)

theorem AlgebraicGeometry.Scheme.Hom.resLE_eq_morphismRestrict {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} : resLE f U ((TopologicalSpace.Opens.map f.base).obj U) ⋯ = f ∣_ U

theorem AlgebraicGeometry.Scheme.Hom.resLE_id {X : Scheme} {V V' : X.Opens} (i : V ≤ V') : resLE (CategoryTheory.CategoryStruct.id X) V' V i = X.homOfLE i

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.resLE_comp_ι {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) : CategoryTheory.CategoryStruct.comp (resLE f U V e) U.ι = CategoryTheory.CategoryStruct.comp V.ι f

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.resLE_comp_ι_assoc {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (resLE f U V e) (CategoryTheory.CategoryStruct.comp U.ι h) = CategoryTheory.CategoryStruct.comp V.ι (CategoryTheory.CategoryStruct.comp f h)

theorem AlgebraicGeometry.Scheme.Hom.resLE_comp_resLE {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) {Z : Scheme} (g : Y ⟶ Z) {W : Z.Opens} (e' : U ≤ (TopologicalSpace.Opens.map g.base).obj W) : CategoryTheory.CategoryStruct.comp (resLE f U V e) (resLE g W U e') = resLE (CategoryTheory.CategoryStruct.comp f g) W V ⋯

theorem AlgebraicGeometry.Scheme.Hom.resLE_comp_resLE_assoc {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) {Z : Scheme} (g : Y ⟶ Z) {W : Z.Opens} (e' : U ≤ (TopologicalSpace.Opens.map g.base).obj W) {Z✝ : Scheme} (h : ↑W ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (resLE f U V e) (CategoryTheory.CategoryStruct.comp (resLE g W U e') h) = CategoryTheory.CategoryStruct.comp (resLE (CategoryTheory.CategoryStruct.comp f g) W V ⋯) h

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.map_resLE {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (i : V' ≤ V) : CategoryTheory.CategoryStruct.comp (X.homOfLE i) (resLE f U V e) = resLE f U V' ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.map_resLE_assoc {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (i : V' ≤ V) {Z : Scheme} (h : ↑U ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.homOfLE i) (CategoryTheory.CategoryStruct.comp (resLE f U V e) h) = CategoryTheory.CategoryStruct.comp (resLE f U V' ⋯) h

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.resLE_map {X Y : Scheme} (f : X ⟶ Y) {U U' : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (i : U ≤ U') : CategoryTheory.CategoryStruct.comp (resLE f U V e) (Y.homOfLE i) = resLE f U' V ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.resLE_map_assoc {X Y : Scheme} (f : X ⟶ Y) {U U' : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (i : U ≤ U') {Z : Scheme} (h : ↑U' ⟶ Z) : CategoryTheory.CategoryStruct.comp (resLE f U V e) (CategoryTheory.CategoryStruct.comp (Y.homOfLE i) h) = CategoryTheory.CategoryStruct.comp (resLE f U' V ⋯) h

theorem AlgebraicGeometry.Scheme.Hom.resLE_congr {X Y : Scheme} (f : X ⟶ Y) {U U' : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (e₁ : U = U') (e₂ : V = V') (P : CategoryTheory.MorphismProperty Scheme) : P (resLE f U V e) ↔ P (resLE f U' V' ⋯)

theorem AlgebraicGeometry.Scheme.Hom.resLE_preimage {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (O : (↑U).Opens) : (TopologicalSpace.Opens.map (resLE f U V e).base).obj O = (TopologicalSpace.Opens.map V.ι.base).obj ((TopologicalSpace.Opens.map f.base).obj ((opensFunctor U.ι).obj O))

theorem AlgebraicGeometry.Scheme.Hom.le_preimage_resLE_iff {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (O : (↑U).Opens) (W : (↑V).Opens) : W ≤ (TopologicalSpace.Opens.map (resLE f U V e).base).obj O ↔ (opensFunctor V.ι).obj W ≤ (TopologicalSpace.Opens.map f.base).obj ((opensFunctor U.ι).obj O)

theorem AlgebraicGeometry.Scheme.Hom.resLE_appLE {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (O : (↑U).Opens) (W : (↑V).Opens) (e' : W ≤ (TopologicalSpace.Opens.map (resLE f U V e).base).obj O) : appLE (resLE f U V e) O W e' = appLE f ((opensFunctor U.ι).obj O) ((opensFunctor V.ι).obj W) ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.coe_resLE_base {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (x : ↥V) : ↑((CategoryTheory.ConcreteCategory.hom (resLE f U V e).base) x) = (CategoryTheory.ConcreteCategory.hom f.base) ↑x

def AlgebraicGeometry.Scheme.Hom.resLEStalkMap {X Y : Scheme} (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) (x : ↥V) : CategoryTheory.Arrow.mk (stalkMap (resLE f U V e) x) ≅ CategoryTheory.Arrow.mk (stalkMap f ↑x)

The stalk map of f.resLE U V at x : V is is the stalk map of f at x.

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• One or more equations did not get rendered due to their size.

noncomputable def AlgebraicGeometry.arrowResLEAppIso {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens) (V : X.Opens) (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U) : CategoryTheory.Arrow.mk (Scheme.Hom.appTop (Scheme.Hom.resLE f U V e)) ≅ CategoryTheory.Arrow.mk (Scheme.Hom.appLE f U V e)

f.resLE U V induces f.appLE U V on global sections.

Equations

• AlgebraicGeometry.arrowResLEAppIso f U V e = CategoryTheory.Arrow.isoMk U.topIso V.topIso ⋯

noncomputable def AlgebraicGeometry.Scheme.OpenCover.restrict {X : Scheme} (𝒰 : X.OpenCover) (U : X.Opens) : (↑U).OpenCover

The restriction of an open cover to an open subset.

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• One or more equations did not get rendered due to their size.

@[simp]

theorem AlgebraicGeometry.Scheme.OpenCover.restrict_obj {X : Scheme} (𝒰 : X.OpenCover) (U : X.Opens) (x✝ : 𝒰.J) : (𝒰.restrict U).obj x✝ = ↑((TopologicalSpace.Opens.map (𝒰.map x✝).base).obj U)

@[simp]

theorem AlgebraicGeometry.Scheme.OpenCover.restrict_map {X : Scheme} (𝒰 : X.OpenCover) (U : X.Opens) (x✝ : 𝒰.J) : (𝒰.restrict U).map x✝ = 𝒰.map x✝ ∣_ U

@[simp]

theorem AlgebraicGeometry.Scheme.OpenCover.restrict_J {X : Scheme} (𝒰 : X.OpenCover) (U : X.Opens) : (𝒰.restrict U).J = 𝒰.J