Documentation

Mathlib.Geometry.RingedSpace.LocallyRingedSpace

The category of locally ringed spaces #

We define (bundled) locally ringed spaces (as SheafedSpace CommRing along with the fact that the stalks are local rings), and morphisms between these (morphisms in SheafedSpace with IsLocalRingHom on the stalk maps).

A LocallyRingedSpace is a topological space equipped with a sheaf of commutative rings such that all the stalks are local rings.

A morphism of locally ringed spaces is a morphism of ringed spaces such that the morphisms induced on stalks are local ring homomorphisms.

  • carrier : TopCat
  • presheaf : TopCat.Presheaf CommRingCat self.toPresheafedSpace
  • IsSheaf : self.presheaf.IsSheaf
  • localRing : ∀ (x : self.toPresheafedSpace), LocalRing (self.presheaf.stalk x)

    Stalks of a locally ringed space are local rings.

Instances For
theorem AlgebraicGeometry.LocallyRingedSpace.localRing (self : AlgebraicGeometry.LocallyRingedSpace) (x : self.toPresheafedSpace) :
LocalRing (self.presheaf.stalk x)

Stalks of a locally ringed space are local rings.

An alias for toSheafedSpace, where the result type is a RingedSpace. This allows us to use dot-notation for the RingedSpace namespace.

Equations
  • X.toRingedSpace = X.toSheafedSpace

The underlying topological space of a locally ringed space.

Equations
  • X.toTopCat = X.toPresheafedSpace

The structure sheaf of a locally ringed space.

Equations
  • X.𝒪 = X.sheaf

A morphism of locally ringed spaces is a morphism of ringed spaces such that the morphisms induced on stalks are local ring homomorphisms.

Instances For

the underlying morphism induces a local ring homomorphism on stalks

The identity morphism on a locally ringed space.

Equations
Equations
  • X.instInhabitedHom = { default := X.id }

Composition of morphisms of locally ringed spaces.

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The category of locally ringed spaces.

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

Given two locally ringed spaces X and Y, an isomorphism between X and Y as sheafed spaces can be lifted to a morphism X ⟶ Y as locally ringed spaces.

See also isoOfSheafedSpaceIso.

Equations

Given two locally ringed spaces X and Y, an isomorphism between X and Y as sheafed spaces can be lifted to an isomorphism X ⟶ Y as locally ringed spaces.

This is related to the property that the functor forgetToSheafedSpace reflects isomorphisms. In fact, it is slightly stronger as we do not require f to come from a morphism between locally ringed spaces.

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  • One or more equations did not get rendered due to their size.
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrict_carrier {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) :
(X.restrict h).toPresheafedSpace = U
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_obj {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X✝.toTopCat} (h : OpenEmbedding f) (X : (TopologicalSpace.Opens U)ᵒᵖ) :
(X✝.restrict h).presheaf.obj X = X✝.presheaf.obj (Opposite.op (.functor.obj (Opposite.unop X)))
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) :
∀ {X_1 Y : (TopologicalSpace.Opens U)ᵒᵖ} (f_1 : X_1 Y), (X.restrict h).presheaf.map f_1 = X.presheaf.map (.functor.map f_1.unop).op

The restriction of a locally ringed space along an open embedding.

Equations
  • X.restrict h = { toSheafedSpace := X.restrict h, localRing := }

The canonical map from the restriction to the subspace.

Equations
  • X.ofRestrict h = { val := X.ofRestrict h, prop := }
Instances For

The restriction of a locally ringed space X to the top subspace is isomorphic to X itself.

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The empty locally ringed space.

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

The canonical map from the empty locally ringed space.

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  • One or more equations did not get rendered due to their size.
Equations
  • AlgebraicGeometry.LocallyRingedSpace.instUniqueHomEmptyCollection = { default := X.emptyTo, uniq := }
theorem AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) {U : TopologicalSpace.Opens Y.toTopCat} (s : (Y.presheaf.obj (Opposite.op U))) :
(TopologicalSpace.Opens.map f.val.base).obj (Y.toRingedSpace.basicOpen s) = X.toRingedSpace.basicOpen ((f.val.c.app (Opposite.op U)) s)
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_of_isNilpotent (X : AlgebraicGeometry.LocallyRingedSpace) (U : TopologicalSpace.Opens X.toPresheafedSpace) (f : (X.presheaf.obj (Opposite.op U))) (hf : IsNilpotent f) :
X.toRingedSpace.basicOpen f =
instance AlgebraicGeometry.LocallyRingedSpace.component_nontrivial (X : AlgebraicGeometry.LocallyRingedSpace) (U : TopologicalSpace.Opens X.toPresheafedSpace) [hU : Nonempty { x : X.toPresheafedSpace // x U }] :
Nontrivial (X.presheaf.obj (Opposite.op U))
Equations
  • =
theorem AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (x : X.toTopCat) (x' : X.toTopCat) (h : x x') (y : (Y.presheaf.stalk (f.val.base x'))) :
(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) ((Y.presheaf.stalkSpecializes ) y) = (X.presheaf.stalkSpecializes h) ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x') y)
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (e : X Y) (y : Y.toTopCat) (z : (Y.presheaf.stalk (e.hom.val.base (e.inv.val.base y)))) :
(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv y) ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom (e.inv.val.base y)) z) = (Y.presheaf.stalkSpecializes ) z
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (e : X Y) (x : X.toTopCat) (y : (X.presheaf.stalk (e.inv.val.base (e.hom.val.base x)))) :
(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom x) ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv (e.hom.val.base x)) y) = (X.presheaf.stalkSpecializes ) y
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_assoc {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (U : TopologicalSpace.Opens Y.toTopCat) (x : { x : X.toPresheafedSpace // x (TopologicalSpace.Opens.map f.val.base).obj U }) {Z : CommRingCat} (h : X.presheaf.stalk x Z) :
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (U : TopologicalSpace.Opens Y.toTopCat) (x : { x : X.toPresheafedSpace // x (TopologicalSpace.Opens.map f.val.base).obj U }) :
CategoryTheory.CategoryStruct.comp (Y.presheaf.germ f.val.base x, ) (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) = CategoryTheory.CategoryStruct.comp (f.val.c.app (Opposite.op U)) (X.presheaf.germ x)
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (U : TopologicalSpace.Opens Y.toTopCat) (x : { x : X.toPresheafedSpace // x (TopologicalSpace.Opens.map f.val.base).obj U }) (y : (Y.presheaf.obj (Opposite.op U))) :
(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) ((Y.presheaf.germ f.val.base x, ) y) = (X.presheaf.germ x) ((f.val.c.app (Opposite.op U)) y)
@[simp]
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ' {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (U : TopologicalSpace.Opens Y.toTopCat) (x : X.toTopCat) (hx : f.val.base x U) :
CategoryTheory.CategoryStruct.comp (Y.presheaf.germ f.val.base x, hx) (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) = CategoryTheory.CategoryStruct.comp (f.val.c.app (Opposite.op U)) (X.presheaf.germ x, hx)
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ'_apply {X : AlgebraicGeometry.LocallyRingedSpace} {Y : AlgebraicGeometry.LocallyRingedSpace} (f : X Y) (U : TopologicalSpace.Opens Y.toTopCat) (x : X.toTopCat) (hx : f.val.base x U) (y : (Y.presheaf.obj (Opposite.op U))) :
(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) ((Y.presheaf.germ f.val.base x, hx) y) = (X.presheaf.germ x, hx) ((f.val.c.app (Opposite.op U)) y)
noncomputable def AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (x : U) :
(X.restrict h).presheaf.stalk x X.presheaf.stalk (f x)

For an open embedding f : U ⟶ X and a point x : U, we get an isomorphism between the stalk of X at f x and the stalk of the restriction of X along f at t x.

Equations
  • X.restrictStalkIso h x = X.restrictStalkIso h x
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_assoc {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) {Z : CommRingCat} (h : X.presheaf.stalk (f x) Z) :
CategoryTheory.CategoryStruct.comp ((X.restrict h✝).presheaf.germ x, hx) (CategoryTheory.CategoryStruct.comp (X.restrictStalkIso h✝ x).hom h) = CategoryTheory.CategoryStruct.comp (X.presheaf.germ f x, ) h
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) :
CategoryTheory.CategoryStruct.comp ((X.restrict h).presheaf.germ x, hx) (X.restrictStalkIso h x).hom = X.presheaf.germ f x,
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) (y : ((X.restrict h).presheaf.obj (Opposite.op V))) :
(X.restrictStalkIso h x).hom (((X.restrict h).presheaf.germ x, hx) y) = (X.presheaf.germ f x, ) y
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_assoc {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) {Z : CommRingCat} (h : (X.restrict h✝).presheaf.stalk x Z) :
CategoryTheory.CategoryStruct.comp (X.presheaf.germ f x, ) (CategoryTheory.CategoryStruct.comp (X.restrictStalkIso h✝ x).inv h) = CategoryTheory.CategoryStruct.comp ((X.restrict h✝).presheaf.germ x, hx) h
@[simp]
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) :
CategoryTheory.CategoryStruct.comp (X.presheaf.germ f x, ) (X.restrictStalkIso h x).inv = (X.restrict h).presheaf.germ x, hx
theorem AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U X.toTopCat} (h : OpenEmbedding f) (V : TopologicalSpace.Opens U) (x : U) (hx : x V) (y : (X.presheaf.obj (Opposite.op (.functor.obj V)))) :
(X.restrictStalkIso h x).inv ((X.presheaf.germ f x, ) y) = ((X.restrict h).presheaf.germ x, hx) y