Documentation

Mathlib.AlgebraicGeometry.Restrict

Restriction of Schemes and Morphisms #

Main definition #

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    X ∣_ᵤ U is notation for X.restrict U.openEmbedding, the restriction of X to an open set U of X.

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      @[reducible, inline]
      abbrev AlgebraicGeometry.Scheme.ιOpens {X : AlgebraicGeometry.Scheme} (U : TopologicalSpace.Opens X.toPresheafedSpace) :
      X ∣_ᵤ U X

      The restriction of a scheme to an open subset.

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        @[simp]
        theorem AlgebraicGeometry.opensRestrict_apply_coe_coe {X : AlgebraicGeometry.Scheme} (U : TopologicalSpace.Opens X.toPresheafedSpace) :
        ∀ (a : TopologicalSpace.Opens (X.restrict ).toPresheafedSpace), ((AlgebraicGeometry.opensRestrict U) a) = U.inclusion '' a
        @[simp]
        theorem AlgebraicGeometry.opensRestrict_symm_apply_coe {X : AlgebraicGeometry.Scheme} (U : TopologicalSpace.Opens X.toPresheafedSpace) :
        ∀ (a : { V : TopologicalSpace.Opens X.toPresheafedSpace // V U }), ((AlgebraicGeometry.opensRestrict U).symm a) = U.inclusion ⁻¹' ((Equiv.subtypeEquivProp ).symm a)
        def AlgebraicGeometry.opensRestrict {X : AlgebraicGeometry.Scheme} (U : TopologicalSpace.Opens X.toPresheafedSpace) :
        TopologicalSpace.Opens (X ∣_ᵤ U).toPresheafedSpace { V : TopologicalSpace.Opens X.toPresheafedSpace // V U }

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

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          instance AlgebraicGeometry.ΓRestrictAlgebra {X : AlgebraicGeometry.Scheme} {Y : TopCat} {f : Y X.toPresheafedSpace} (hf : OpenEmbedding f) :
          Algebra (AlgebraicGeometry.Scheme.Γ.obj { unop := X }) (AlgebraicGeometry.Scheme.Γ.obj { unop := X.restrict hf })
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          theorem AlgebraicGeometry.Scheme.map_basicOpen' (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) (r : ((X ∣_ᵤ U).presheaf.obj { unop := })) :
          (AlgebraicGeometry.Scheme.Hom.opensFunctor (AlgebraicGeometry.Scheme.ιOpens U)).obj ((X ∣_ᵤ U).basicOpen r) = X.basicOpen ((X.presheaf.map (CategoryTheory.eqToHom ).op) r)
          theorem AlgebraicGeometry.Scheme.map_basicOpen (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) (r : ((X ∣_ᵤ U).presheaf.obj { unop := })) :
          (AlgebraicGeometry.Scheme.Hom.opensFunctor (AlgebraicGeometry.Scheme.ιOpens U)).obj ((X ∣_ᵤ U).basicOpen r) = X.basicOpen r
          theorem AlgebraicGeometry.Scheme.map_basicOpen_map (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) (r : (X.presheaf.obj { unop := U })) :
          (AlgebraicGeometry.Scheme.Hom.opensFunctor (AlgebraicGeometry.Scheme.ιOpens U)).obj ((X ∣_ᵤ U).basicOpen ((X.presheaf.map (CategoryTheory.eqToHom ).op) r)) = X.basicOpen r

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

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            @[simp]
            theorem AlgebraicGeometry.Scheme.restrictFunctor_obj_left (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) :
            (X.restrictFunctor.obj U).left = X ∣_ᵤ U
            @[simp]
            theorem AlgebraicGeometry.Scheme.restrictFunctor_map_base (X : AlgebraicGeometry.Scheme) {U : TopologicalSpace.Opens X.toPresheafedSpace} {V : TopologicalSpace.Opens X.toPresheafedSpace} (i : U V) :
            (X.restrictFunctor.map i).left.val.base = (TopologicalSpace.Opens.toTopCat X.toPresheafedSpace).map i
            theorem AlgebraicGeometry.Scheme.restrictFunctor_map_app (X : AlgebraicGeometry.Scheme) {U : TopologicalSpace.Opens X.toPresheafedSpace} {V : TopologicalSpace.Opens X.toPresheafedSpace} (i : U V) (W : TopologicalSpace.Opens V) :
            AlgebraicGeometry.Scheme.Hom.app (X.restrictFunctor.map i).left W = X.presheaf.map (CategoryTheory.homOfLE ).op
            @[simp]
            theorem AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app (X : AlgebraicGeometry.Scheme) (X : (TopologicalSpace.Opens X✝.toPresheafedSpace)ᵒᵖ) :
            X✝.restrictFunctorΓ.inv.app X = X✝.presheaf.map (CategoryTheory.eqToHom )
            @[simp]
            theorem AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app (X : AlgebraicGeometry.Scheme) (X : (TopologicalSpace.Opens X✝.toPresheafedSpace)ᵒᵖ) :
            X✝.restrictFunctorΓ.hom.app X = X✝.presheaf.map (CategoryTheory.eqToHom )

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

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              noncomputable def AlgebraicGeometry.Scheme.restrictRestrictComm (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) (V : TopologicalSpace.Opens X.toPresheafedSpace) :
              X ∣_ᵤ U ∣_ᵤ (TopologicalSpace.Opens.map (AlgebraicGeometry.Scheme.ιOpens U).val.base).obj V X ∣_ᵤ V ∣_ᵤ (TopologicalSpace.Opens.map (AlgebraicGeometry.Scheme.ιOpens V).val.base).obj U

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

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                noncomputable def AlgebraicGeometry.Scheme.restrictRestrict (X : AlgebraicGeometry.Scheme) (U : TopologicalSpace.Opens X.toPresheafedSpace) (V : TopologicalSpace.Opens (X ∣_ᵤ U).toPresheafedSpace) :

                If V is an open subset of U, then X ∣_ U ∣_ V is isomorphic to X ∣_ V.

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                  noncomputable def AlgebraicGeometry.Scheme.restrictIsoOfEq (X : AlgebraicGeometry.Scheme) {U : TopologicalSpace.Opens X.toPresheafedSpace} {V : TopologicalSpace.Opens X.toPresheafedSpace} (e : U = V) :
                  X ∣_ᵤ U X ∣_ᵤ V

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

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                    @[reducible, inline]
                    noncomputable abbrev AlgebraicGeometry.Scheme.restrictMapIso {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X Y) [CategoryTheory.IsIso f] (U : TopologicalSpace.Opens Y.toPresheafedSpace) :
                    X ∣_ᵤ (TopologicalSpace.Opens.map f.val.base).obj U Y ∣_ᵤ U

                    The restriction of an isomorphism onto an open set.

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

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                        def AlgebraicGeometry.morphismRestrict {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X Y) (U : TopologicalSpace.Opens Y.toPresheafedSpace) :
                        X ∣_ᵤ (TopologicalSpace.Opens.map f.val.base).obj U Y ∣_ᵤ U

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

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

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                            theorem AlgebraicGeometry.morphismRestrict_base_coe {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X Y) (U : TopologicalSpace.Opens Y.toPresheafedSpace) (x : (X ∣_ᵤ (TopologicalSpace.Opens.map f.val.base).obj U).toPresheafedSpace) :
                            Coe.coe ((f ∣_ U).val.base x) = f.val.base x
                            theorem AlgebraicGeometry.morphismRestrict_val_base {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X Y) (U : TopologicalSpace.Opens Y.toPresheafedSpace) :
                            (f ∣_ U).val.base = U.carrier.restrictPreimage f.val.base

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

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                              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.

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                                Restricting a morphism twice is isomorphic to one restriction.

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                                  def AlgebraicGeometry.morphismRestrictRestrictBasicOpen {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X Y) (U : TopologicalSpace.Opens Y.toPresheafedSpace) (r : (Y.presheaf.obj { unop := U })) :
                                  CategoryTheory.Arrow.mk (f ∣_ U ∣_ (Y ∣_ᵤ U).basicOpen ((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.

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

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