Open immersions of schemes

@[reducible, inline]

abbrev AlgebraicGeometry.IsOpenImmersion {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : Prop

A morphism of Schemes is an open immersion if it is an open immersion as a morphism of LocallyRingedSpaces

Equations

• AlgebraicGeometry.IsOpenImmersion f = AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f

instance AlgebraicGeometry.IsOpenImmersion.comp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] : AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp f g)

Equations

• ⋯ = ⋯

def AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.scheme (X : AlgebraicGeometry.LocallyRingedSpace) (h : ∀ (x : ↑X.toTopCat), ∃ (R : CommRingCat) (f : AlgebraicGeometry.Spec.toLocallyRingedSpace.obj { unop := R } ⟶ X), x ∈ Set.range ⇑f.val.base ∧ AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f) : AlgebraicGeometry.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.

Equations

• AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.scheme X h = { toLocallyRingedSpace := X, local_affine := ⋯ }

theorem AlgebraicGeometry.IsOpenImmersion.isOpen_range {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] : IsOpen (Set.range ⇑f.val.base)

@[deprecated AlgebraicGeometry.IsOpenImmersion.isOpen_range]

theorem AlgebraicGeometry.IsOpenImmersion.open_range {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] : IsOpen (Set.range ⇑f.val.base)

Alias of AlgebraicGeometry.IsOpenImmersion.isOpen_range.

theorem AlgebraicGeometry.Scheme.Hom.openEmbedding {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] : OpenEmbedding ⇑f.val.base

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.opensRange_coe {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] : ↑f.opensRange = Set.range ⇑f.val.base

def AlgebraicGeometry.Scheme.Hom.opensRange {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace

The image of an open immersion as an open set.

Equations

• f.opensRange = { carrier := Set.range ⇑f.val.base, is_open' := ⋯ }

@[reducible, inline]

abbrev AlgebraicGeometry.Scheme.Hom.opensFunctor {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Functor (TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (TopologicalSpace.Opens ↑↑Y.toPresheafedSpace)

The functor opens X ⥤ opens Y associated with an open immersion f : X ⟶ Y.

Equations

• f.opensFunctor = AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.opensFunctor f

def AlgebraicGeometry.«term_''ᵁ_» : Lean.TrailingParserDescr

f ''ᵁ U is notation for the image (as an open set) of U under an open immersion f.

Equations

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

def AlgebraicGeometry.«term_''ᵁ_».delab : Lean.PrettyPrinter.Delaborator.Delab

Pretty printer defined by notation3 command.

Equations

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

def AlgebraicGeometry.Scheme.Hom.invApp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : X.presheaf.obj { unop := U } ⟶ Y.presheaf.obj { unop := f.opensFunctor.obj U }

The isomorphism Γ(X, U) ⟶ Γ(Y, f(U)) induced by an open immersion f : X ⟶ Y.

Equations

• f.invApp U = AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp f U

instance AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatInvApp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : CategoryTheory.IsIso (f.invApp U)

Equations

• ⋯ = ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.invApp_naturality_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} {V : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (i : { unop := U } ⟶ { unop := V }) {Z : CommRingCat} (h : Y.presheaf.obj { unop := f.opensFunctor.obj V } ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.presheaf.map i) (CategoryTheory.CategoryStruct.comp (f.invApp V) h) = CategoryTheory.CategoryStruct.comp (f.invApp U) (CategoryTheory.CategoryStruct.comp (Y.presheaf.map (f.opensFunctor.op.map i)) h)

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.invApp_naturality {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} {V : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (i : { unop := U } ⟶ { unop := V }) : CategoryTheory.CategoryStruct.comp (X.presheaf.map i) (f.invApp V) = CategoryTheory.CategoryStruct.comp (f.invApp U) (Y.presheaf.map (f.opensFunctor.op.map i))

theorem AlgebraicGeometry.Scheme.Hom.inv_invApp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : CategoryTheory.inv (f.invApp U) = CategoryTheory.CategoryStruct.comp (f.app (f.opensFunctor.obj U)) (X.presheaf.map (CategoryTheory.eqToHom ⋯).op)

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.app_invApp_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace) {Z : CommRingCat} (h : Y.presheaf.obj { unop := f.opensFunctor.obj ((TopologicalSpace.Opens.map f.val.base).obj U) } ⟶ Z) : CategoryTheory.CategoryStruct.comp (f.app U) (CategoryTheory.CategoryStruct.comp (f.invApp ((TopologicalSpace.Opens.map f.val.base).obj U)) h) = CategoryTheory.CategoryStruct.comp (Y.presheaf.map (CategoryTheory.homOfLE ⋯).op) h

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.app_invApp {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace) : CategoryTheory.CategoryStruct.comp (f.app U) (f.invApp ((TopologicalSpace.Opens.map f.val.base).obj U)) = Y.presheaf.map (CategoryTheory.homOfLE ⋯).op

theorem AlgebraicGeometry.Scheme.Hom.app_invApp'_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace) (hU : U ≤ f.opensRange) {Z : CommRingCat} (h : Y.presheaf.obj { unop := f.opensFunctor.obj ((TopologicalSpace.Opens.map f.val.base).obj U) } ⟶ Z) : CategoryTheory.CategoryStruct.comp (f.app U) (CategoryTheory.CategoryStruct.comp (f.invApp ((TopologicalSpace.Opens.map f.val.base).obj U)) h) = CategoryTheory.CategoryStruct.comp (Y.presheaf.map (CategoryTheory.eqToHom ⋯).op) h

theorem AlgebraicGeometry.Scheme.Hom.app_invApp' {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace) (hU : U ≤ f.opensRange) : CategoryTheory.CategoryStruct.comp (f.app U) (f.invApp ((TopologicalSpace.Opens.map f.val.base).obj U)) = Y.presheaf.map (CategoryTheory.eqToHom ⋯).op

A variant of app_invApp that gives an eqToHom instead of homOfLE.

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.invApp_app_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) {Z : CommRingCat} (h : X.presheaf.obj { unop := (TopologicalSpace.Opens.map f.val.base).obj (f.opensFunctor.obj U) } ⟶ Z) : CategoryTheory.CategoryStruct.comp (f.invApp U) (CategoryTheory.CategoryStruct.comp (f.app (f.opensFunctor.obj U)) h) = CategoryTheory.CategoryStruct.comp (X.presheaf.map (CategoryTheory.eqToHom ⋯).op) h

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.invApp_app {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : CategoryTheory.CategoryStruct.comp (f.invApp U) (f.app (f.opensFunctor.obj U)) = X.presheaf.map (CategoryTheory.eqToHom ⋯).op

@[simp]

theorem AlgebraicGeometry.IsOpenImmersion.opensEquiv_apply_coe {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : ↑((AlgebraicGeometry.IsOpenImmersion.opensEquiv f) U) = (AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj U

@[simp]

theorem AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (U : { U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace // U ≤ AlgebraicGeometry.Scheme.Hom.opensRange f }) : (AlgebraicGeometry.IsOpenImmersion.opensEquiv f).symm U = (TopologicalSpace.Opens.map f.val.base).obj ↑U

def AlgebraicGeometry.IsOpenImmersion.opensEquiv {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] : TopologicalSpace.Opens ↑↑X.toPresheafedSpace ≃ { U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace // U ≤ AlgebraicGeometry.Scheme.Hom.opensRange f }

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

Equations

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

instance AlgebraicGeometry.Scheme.basic_open_isOpenImmersion {R : CommRingCat} (f : ↑R) : AlgebraicGeometry.IsOpenImmersion (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away f))))

Equations

• ⋯ = ⋯

theorem AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset {X : AlgebraicGeometry.Scheme} {x : ↑↑X.toPresheafedSpace} {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (hxU : x ∈ U) : ∃ (R : CommRingCat) (f : AlgebraicGeometry.Spec R ⟶ X), AlgebraicGeometry.IsOpenImmersion f ∧ x ∈ Set.range ⇑f.val.base ∧ Set.range ⇑f.val.base ⊆ ↑U

def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme {X : AlgebraicGeometry.PresheafedSpace CommRingCat} (Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y.toPresheafedSpace) [H : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f] : AlgebraicGeometry.Scheme

If X ⟶ Y is an open immersion, and Y is a scheme, then so is X.

Equations

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

@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme_toLocallyRingedSpace {X : AlgebraicGeometry.PresheafedSpace CommRingCat} (Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y.toPresheafedSpace) [H : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f] : (AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme Y f).toLocallyRingedSpace = AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace Y.toLocallyRingedSpace f

def AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom {X : AlgebraicGeometry.PresheafedSpace CommRingCat} (Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y.toPresheafedSpace) [H : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f] : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme Y f ⟶ Y

If X ⟶ Y is an open immersion of PresheafedSpaces, and Y is a Scheme, we can upgrade it into a morphism of Schemes.

Equations

• AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom Y f = AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom Y.toLocallyRingedSpace f

@[simp]

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_val {X : AlgebraicGeometry.PresheafedSpace CommRingCat} (Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y.toPresheafedSpace) [H : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f] : (AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom Y f).val = f

instance AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_isOpenImmersion {X : AlgebraicGeometry.PresheafedSpace CommRingCat} (Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y.toPresheafedSpace) [H : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion f] : AlgebraicGeometry.IsOpenImmersion (AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom Y f)

Equations

• ⋯ = H

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_eq_of_locallyRingedSpace_eq {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (H : X.toLocallyRingedSpace = Y.toLocallyRingedSpace) : X = Y

theorem AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme Y f.val = X

theorem AlgebraicGeometry.Scheme.restrict_carrier {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : ↑(X.restrict h).toPresheafedSpace = U

@[simp]

theorem AlgebraicGeometry.Scheme.restrict_presheaf_obj {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X✝.toPresheafedSpace} (h : OpenEmbedding ⇑f) (X : (TopologicalSpace.Opens ↑U)ᵒᵖ) : (X✝.restrict h).presheaf.obj X = X✝.presheaf.obj { unop := ⋯.functor.obj X.unop }

def AlgebraicGeometry.Scheme.restrict {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : AlgebraicGeometry.Scheme

The restriction of a Scheme along an open embedding.

Equations

• X.restrict h = let __src := AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme X (X.ofRestrict h); { toPresheafedSpace := X.restrict h, IsSheaf := ⋯, localRing := ⋯, local_affine := ⋯ }

theorem AlgebraicGeometry.Scheme.restrict_toPresheafedSpace {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : (X.restrict h).toPresheafedSpace = X.restrict h

@[simp]

theorem AlgebraicGeometry.Scheme.ofRestrict_val_base {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : (X.ofRestrict h).val.base = f

theorem AlgebraicGeometry.Scheme.ofRestrict_val_c_app {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : (TopologicalSpace.Opens ↑↑X.toPresheafedSpace)ᵒᵖ) : (X.ofRestrict h).val.c.app V = X.presheaf.map (⋯.adjunction.counit.app V.unop).op

def AlgebraicGeometry.Scheme.ofRestrict {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : X.restrict h ⟶ X

The canonical map from the restriction to the subspace.

Equations

• X.ofRestrict h = X.ofRestrict h

@[simp]

theorem AlgebraicGeometry.Scheme.ofRestrict_app {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) : AlgebraicGeometry.Scheme.Hom.app (X.ofRestrict h) V = X.presheaf.map (⋯.adjunction.counit.app V).op

instance AlgebraicGeometry.IsOpenImmersion.ofRestrict {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) : AlgebraicGeometry.IsOpenImmersion (X.ofRestrict h)

Equations

• ⋯ = ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.ofRestrict_appLE {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (W : TopologicalSpace.Opens ↑↑(X.restrict h).toPresheafedSpace) (e : W ≤ (TopologicalSpace.Opens.map (X.ofRestrict h).val.base).obj V) : AlgebraicGeometry.Scheme.Hom.appLE (X.ofRestrict h) V W e = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

@[simp]

theorem AlgebraicGeometry.Scheme.restrict_presheaf_map {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : (TopologicalSpace.Opens ↑↑(X.restrict h).toPresheafedSpace)ᵒᵖ) (W : (TopologicalSpace.Opens ↑↑(X.restrict h).toPresheafedSpace)ᵒᵖ) (i : V ⟶ W) : (X.restrict h).presheaf.map i = X.presheaf.map (CategoryTheory.homOfLE ⋯).op

@[instance 100]

instance AlgebraicGeometry.IsOpenImmersion.of_isIso {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (g : Y ⟶ Z) [CategoryTheory.IsIso g] : AlgebraicGeometry.IsOpenImmersion g

Equations

• ⋯ = ⋯

theorem AlgebraicGeometry.IsOpenImmersion.to_iso {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [h : AlgebraicGeometry.IsOpenImmersion f] [CategoryTheory.Epi f.val.base] : CategoryTheory.IsIso f

theorem AlgebraicGeometry.IsOpenImmersion.of_stalk_iso {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (hf : OpenEmbedding ⇑f.val.base) [∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.PresheafedSpace.stalkMap f.val x)] : AlgebraicGeometry.IsOpenImmersion f

theorem AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsOpenImmersion f ↔ OpenEmbedding ⇑f.val.base ∧ ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.PresheafedSpace.stalkMap f.val x)

theorem AlgebraicGeometry.isIso_iff_isOpenImmersion {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : CategoryTheory.IsIso f ↔ AlgebraicGeometry.IsOpenImmersion f ∧ CategoryTheory.Epi f.val.base

theorem AlgebraicGeometry.isIso_iff_stalk_iso {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : CategoryTheory.IsIso f ↔ CategoryTheory.IsIso f.val.base ∧ ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.PresheafedSpace.stalkMap f.val x)

def AlgebraicGeometry.IsOpenImmersion.isoRestrict {X : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : X ≅ Z.restrict ⋯

An open immersion induces an isomorphism from the domain onto the image

Equations

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

instance AlgebraicGeometry.IsOpenImmersion.mono {X : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Mono f

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.forget_map_isOpenImmersion {X : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion (AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace.map f)

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasLimit ((CategoryTheory.Limits.cospan f g).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace)

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left' {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.cospan (((CategoryTheory.Limits.cospan f g).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inl) (((CategoryTheory.Limits.cospan f g).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inr))

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasLimit ((CategoryTheory.Limits.cospan g f).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace)

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right' {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.cospan (((CategoryTheory.Limits.cospan g f).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inl) (((CategoryTheory.Limits.cospan g f).comp AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace).map CategoryTheory.Limits.WalkingCospan.Hom.inr))

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.forgetCreatesPullbackOfLeft {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.CreatesLimit (CategoryTheory.Limits.cospan f g) AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace

Equations

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

instance AlgebraicGeometry.IsOpenImmersion.forgetCreatesPullbackOfRight {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.CreatesLimit (CategoryTheory.Limits.cospan g f) AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace

Equations

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

instance AlgebraicGeometry.IsOpenImmersion.forgetPreservesOfLeft {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace

Equations

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

instance AlgebraicGeometry.IsOpenImmersion.forgetPreservesOfRight {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan g f) AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace

Equations

• AlgebraicGeometry.IsOpenImmersion.forgetPreservesOfRight f g = CategoryTheory.Limits.preservesPullbackSymmetry AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace f g

instance AlgebraicGeometry.IsOpenImmersion.hasPullback_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasPullback f g

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.hasPullback_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.HasPullback g f

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.pullback_snd_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.IsOpenImmersion CategoryTheory.Limits.pullback.snd

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.pullback_fst_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.IsOpenImmersion CategoryTheory.Limits.pullback.fst

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.pullback_to_base {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] : AlgebraicGeometry.IsOpenImmersion (CategoryTheory.Limits.limit.π (CategoryTheory.Limits.cospan f g) CategoryTheory.Limits.WalkingCospan.one)

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.IsOpenImmersion.forgetToTopPreservesOfLeft {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f g) AlgebraicGeometry.Scheme.forgetToTop

Equations

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

instance AlgebraicGeometry.IsOpenImmersion.forgetToTopPreservesOfRight {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan g f) AlgebraicGeometry.Scheme.forgetToTop

Equations

• AlgebraicGeometry.IsOpenImmersion.forgetToTopPreservesOfRight f g = CategoryTheory.Limits.preservesPullbackSymmetry AlgebraicGeometry.Scheme.forgetToTop f g

theorem AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : Set.range ⇑CategoryTheory.Limits.pullback.snd.val.base = ((TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)).carrier

theorem AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.Scheme.Hom.opensRange CategoryTheory.Limits.pullback.snd = (TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)

theorem AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : Set.range ⇑CategoryTheory.Limits.pullback.fst.val.base = ((TopologicalSpace.Opens.map g.val.base).obj { carrier := Set.range ⇑f.val.base, is_open' := ⋯ }).carrier

theorem AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : AlgebraicGeometry.Scheme.Hom.opensRange CategoryTheory.Limits.pullback.fst = (TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)

theorem AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : Set.range ⇑(CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.pullback.fst f).val.base = Set.range ⇑f.val.base ∩ Set.range ⇑g.val.base

theorem AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] : Set.range ⇑(CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.pullback.fst g).val.base = Set.range ⇑g.val.base ∩ Set.range ⇑f.val.base

def AlgebraicGeometry.IsOpenImmersion.lift {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) : Y ⟶ X

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.

Equations

• AlgebraicGeometry.IsOpenImmersion.lift f g H' = AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift f g H'

theorem AlgebraicGeometry.IsOpenImmersion.lift_fac_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z✝) (g : Y ⟶ Z✝) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) {Z : AlgebraicGeometry.Scheme} (h : Z✝ ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.lift f g H') (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp g h

@[simp]

theorem AlgebraicGeometry.IsOpenImmersion.lift_fac {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.lift f g H') f = g

theorem AlgebraicGeometry.IsOpenImmersion.lift_uniq {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) (l : Y ⟶ X) (hl : CategoryTheory.CategoryStruct.comp l f = g) : l = AlgebraicGeometry.IsOpenImmersion.lift f g H'

def AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base) : X ≅ Y

Two open immersions with equal range are isomorphic.

Equations

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

theorem AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z✝) (g : Y ⟶ Z✝) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base) {Z : AlgebraicGeometry.Scheme} (h : Z✝ ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).hom (CategoryTheory.CategoryStruct.comp g h) = CategoryTheory.CategoryStruct.comp f h

@[simp]

theorem AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).hom g = f

theorem AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac_assoc {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z✝) (g : Y ⟶ Z✝) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base) {Z : AlgebraicGeometry.Scheme} (h : Z✝ ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).inv (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp g h

@[simp]

theorem AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).inv f = g

theorem AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {U : AlgebraicGeometry.Scheme} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X) (H : fg = CategoryTheory.CategoryStruct.comp f g) [h : AlgebraicGeometry.IsOpenImmersion g] (V : TopologicalSpace.Opens ↑↑U.toPresheafedSpace) : (TopologicalSpace.Opens.map f.val.base).obj V = (TopologicalSpace.Opens.map fg.val.base).obj ((AlgebraicGeometry.Scheme.Hom.opensFunctor g).obj V)

theorem AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {U : AlgebraicGeometry.Scheme} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X) (H : fg = CategoryTheory.CategoryStruct.comp f g) [h : AlgebraicGeometry.IsOpenImmersion g] (V : TopologicalSpace.Opens ↑↑U.toPresheafedSpace) : AlgebraicGeometry.Scheme.Hom.app f V = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.invApp g V) (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.app fg ((AlgebraicGeometry.Scheme.Hom.opensFunctor g).obj V)) (Y.presheaf.map (CategoryTheory.eqToHom ⋯).op))

The fg argument is to avoid nasty stuff about dependent types.

theorem AlgebraicGeometry.IsOpenImmersion.lift_app {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} {U : AlgebraicGeometry.Scheme} (f : U ⟶ Y) (g : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (H : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) (V : TopologicalSpace.Opens ↑↑U.toPresheafedSpace) : AlgebraicGeometry.Scheme.Hom.app (AlgebraicGeometry.IsOpenImmersion.lift f g H) V = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.invApp f V) (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.app g ((AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj V)) (X.presheaf.map (CategoryTheory.eqToHom ⋯).op))

theorem AlgebraicGeometry.Scheme.image_basicOpen {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (r : ↑(X.presheaf.obj { unop := U })) : (AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj (X.basicOpen r) = Y.basicOpen ((AlgebraicGeometry.Scheme.Hom.invApp f U) r)