statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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coe_base_change_Δ' : ↑(E.base_change A).Δ' = algebra_map R A E.Δ' | rfl | lemma | elliptic_curve.coe_base_change_Δ' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_base_change_Δ' : ↑(E.base_change A).Δ'⁻¹ = algebra_map R A ↑E.Δ'⁻¹ | rfl | lemma | elliptic_curve.coe_inv_base_change_Δ' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_j : (E.base_change A).j = algebra_map R A E.j | by { simp only [j, coe_inv_base_change_Δ', base_change_to_weierstrass_curve, E.base_change_c₄],
map_simp } | lemma | elliptic_curve.base_change_j | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_exists_rep {c : cocone F} (hc : is_colimit c) (x : c.X) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x | concrete.is_colimit_exists_rep (F ⋙ SheafedSpace.forget _)
(is_colimit_of_preserves (SheafedSpace.forget _) hc) x | lemma | algebraic_geometry.SheafedSpace.is_colimit_exists_rep | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_exists_rep (x : colimit F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x | concrete.is_colimit_exists_rep (F ⋙ SheafedSpace.forget _)
(is_colimit_of_preserves (SheafedSpace.forget _) (colimit.is_colimit F)) x | lemma | algebraic_geometry.SheafedSpace.colimit_exists_rep | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coproduct : LocallyRingedSpace | { to_SheafedSpace := colimit (F ⋙ forget_to_SheafedSpace : _),
local_ring := λ x, begin
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forget_to_SheafedSpace) x,
haveI : _root_.local_ring (((F ⋙ forget_to_SheafedSpace).obj i).to_PresheafedSpace.stalk y) :=
(F.obj i).local_ring _,
exact (a... | def | algebraic_geometry.LocallyRingedSpace.coproduct | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"local_ring"
] | The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproduct_cofan : cocone F | { X := coproduct F,
ι :=
{ app := λ j, ⟨colimit.ι (F ⋙ forget_to_SheafedSpace) j, infer_instance⟩,
naturality' := λ j j' f, by { cases j, cases j', tidy, }, } } | def | algebraic_geometry.LocallyRingedSpace.coproduct_cofan | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproduct_cofan_is_colimit : is_colimit (coproduct_cofan F) | { desc := λ s, ⟨colimit.desc (F ⋙ forget_to_SheafedSpace) (forget_to_SheafedSpace.map_cocone s),
begin
intro x,
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forget_to_SheafedSpace) x,
have := PresheafedSpace.stalk_map.comp (colimit.ι (F ⋙ forget_to_SheafedSpace) i : _)
(colimit.desc (F ... | def | algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"is_local_ring_hom"
] | The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_π_app_is_local_ring_hom
(U : topological_space.opens ((coequalizer f.val g.val).carrier)) :
is_local_ring_hom ((coequalizer.π f.val g.val : _).c.app (op U)) | begin
have := ι_comp_coequalizer_comparison f.1 g.1 SheafedSpace.forget_to_PresheafedSpace,
rw ← preserves_coequalizer.iso_hom at this,
erw SheafedSpace.congr_app this.symm (op U),
rw [PresheafedSpace.comp_c_app,
← PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_π],
apply_instance
end | instance | algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"is_local_ring_hom",
"topological_space.opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_basic_open : opens Y | (Y.to_RingedSpace.basic_open
(show Y.presheaf.obj (op (unop _)), from ((coequalizer.π f.1 g.1).c.app (op U)) s)) | def | algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | (Implementation). The basic open set of the section `π꙳ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_basic_open_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base ''
(image_basic_open f g U s).1) = (image_basic_open f g U s).1 | begin
fapply types.coequalizer_preimage_image_eq_of_preimage_eq f.1.base g.1.base,
{ ext,
simp_rw [types_comp_apply, ← Top.comp_app, ← PresheafedSpace.comp_base],
congr' 2,
exact coequalizer.condition f.1 g.1 },
{ apply is_colimit_cofork_map_of_is_colimit (forget Top),
apply is_colimit_cofork_map_... | lemma | algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"Top",
"Top.comp_app",
"topological_space.opens.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_basic_open_image_open :
is_open ((coequalizer.π f.1 g.1).base '' (image_basic_open f g U s).1) | begin
rw [← (Top.homeo_of_iso (preserves_coequalizer.iso (SheafedSpace.forget _) f.1 g.1))
.is_open_preimage, Top.coequalizer_is_open_iff, ← set.preimage_comp],
erw ← coe_comp,
rw [preserves_coequalizer.iso_hom, ι_comp_coequalizer_comparison],
dsimp only [SheafedSpace.forget],
rw image_basic_open_image_... | lemma | algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"Top.coequalizer_is_open_iff",
"Top.homeo_of_iso",
"is_open",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_π_stalk_is_local_ring_hom (x : Y) :
is_local_ring_hom (PresheafedSpace.stalk_map (coequalizer.π f.val g.val : _) x) | begin
constructor,
rintros a ha,
rcases Top.presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩,
erw PresheafedSpace.stalk_map_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩ at ha,
let V := image_basic_open f g U s,
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
... | instance | algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"Top.presheaf.germ_exist",
"Top.presheaf.pushforward_obj_map",
"is_local_ring_hom",
"is_open",
"is_unit_map_iff",
"ring_hom.is_unit_map",
"set.mem_image_of_mem",
"set_like.ext'",
"topological_space.opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer : LocallyRingedSpace | { to_SheafedSpace := coequalizer f.1 g.1,
local_ring := λ x,
begin
obtain ⟨y, rfl⟩ :=
(Top.epi_iff_surjective (coequalizer.π f.val g.val).base).mp infer_instance x,
exact (PresheafedSpace.stalk_map (coequalizer.π f.val g.val : _) y).domain_local_ring
end } | def | algebraic_geometry.LocallyRingedSpace.coequalizer | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"Top.epi_iff_surjective",
"local_ring"
] | The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_cofork : cofork f g | @cofork.of_π _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1, infer_instance⟩
(LocallyRingedSpace.hom.ext _ _ (coequalizer.condition f.1 g.1)) | def | algebraic_geometry.LocallyRingedSpace.coequalizer_cofork | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | The explicit coequalizer cofork of locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_ring_hom_stalk_map_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g)
(x) (h : is_local_ring_hom (PresheafedSpace.stalk_map f x)) :
is_local_ring_hom (PresheafedSpace.stalk_map g x) | by { rw PresheafedSpace.stalk_map.congr_hom _ _ H.symm x, apply_instance } | lemma | algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"is_local_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_cofork_is_colimit : is_colimit (coequalizer_cofork f g) | begin
apply cofork.is_colimit.mk',
intro s,
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition,
use coequalizer.desc s.π.1 e,
{ intro x,
rcases (Top.epi_iff_surjective (coequalizer.π f.val g.val).base).mp
infer_instance x with ⟨y, rfl⟩,
apply is_local_ring_hom_of_comp _ (Pres... | def | algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [
"Top.epi_iff_surjective",
"is_local_ring_hom",
"is_local_ring_hom_of_comp"
] | The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_coequalizer :
preserves_colimits_of_shape walking_parallel_pair forget_to_SheafedSpace.{v} | ⟨λ F, begin
apply preserves_colimit_of_iso_diagram _ (diagram_iso_parallel_pair F).symm,
apply preserves_colimit_of_preserves_colimit_cocone (coequalizer_cofork_is_colimit _ _),
apply (is_colimit_map_cocone_cofork_equiv _ _).symm _,
dsimp only [forget_to_SheafedSpace],
exact coequalizer_is_coequalizer _ _
end... | instance | algebraic_geometry.LocallyRingedSpace.preserves_coequalizer | algebraic_geometry.locally_ringed_space | src/algebraic_geometry/locally_ringed_space/has_colimits.lean | [
"algebraic_geometry.locally_ringed_space",
"algebra.category.Ring.constructions",
"algebraic_geometry.open_immersion.basic",
"category_theory.limits.constructions.limits_of_products_and_equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property | ∀ ⦃X Y : Scheme⦄ (f : X ⟶ Y) [is_affine Y], Prop | def | algebraic_geometry.affine_target_morphism_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | An `affine_target_morphism_property` is a class of morphisms from an arbitrary scheme into an
affine scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.is_iso : morphism_property Scheme | @is_iso Scheme _ | def | algebraic_geometry.Scheme.is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | `is_iso` as a `morphism_property`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.affine_target_is_iso : affine_target_morphism_property | λ X Y f H, is_iso f | def | algebraic_geometry.Scheme.affine_target_is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | `is_iso` as an `affine_morphism_property`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_target_morphism_property.to_property (P : affine_target_morphism_property) :
morphism_property Scheme | λ X Y f, ∃ h, @@P f h | def | algebraic_geometry.affine_target_morphism_property.to_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | A `affine_target_morphism_property` can be extended to a `morphism_property` such that it
*never* holds when the target is not affine | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_target_morphism_property.to_property_apply (P : affine_target_morphism_property)
{X Y : Scheme} (f : X ⟶ Y) [is_affine Y] :
P.to_property f ↔ P f | by { delta affine_target_morphism_property.to_property, simp [*] } | lemma | algebraic_geometry.affine_target_morphism_property.to_property_apply | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_cancel_left_is_iso {P : affine_target_morphism_property}
(hP : P.to_property.respects_iso) {X Y Z : Scheme} (f : X ⟶ Y)
(g : Y ⟶ Z) [is_iso f] [is_affine Z] : P (f ≫ g) ↔ P g | by rw [← P.to_property_apply, ← P.to_property_apply, hP.cancel_left_is_iso] | lemma | algebraic_geometry.affine_cancel_left_is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_cancel_right_is_iso
{P : affine_target_morphism_property} (hP : P.to_property.respects_iso) {X Y Z : Scheme}
(f : X ⟶ Y) (g : Y ⟶ Z) [is_iso g] [is_affine Z] [is_affine Y] : P (f ≫ g) ↔ P f | by rw [← P.to_property_apply, ← P.to_property_apply, hP.cancel_right_is_iso] | lemma | algebraic_geometry.affine_cancel_right_is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.respects_iso_mk {P : affine_target_morphism_property}
(h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [is_affine Z], by exactI P f → P (e.hom ≫ f))
(h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [h : is_affine Y],
by exactI P f → @@P (f ≫ e.hom) (is_affine_of_iso e.inv)) : P.to_property.respec... | begin
split,
{ rintros X Y Z e f ⟨a, h⟩, exactI ⟨a, h₁ e f h⟩ },
{ rintros X Y Z e f ⟨a, h⟩, exactI ⟨is_affine_of_iso e.inv, h₂ e f h⟩ },
end | lemma | algebraic_geometry.affine_target_morphism_property.respects_iso_mk | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
target_affine_locally (P : affine_target_morphism_property) : morphism_property Scheme | λ {X Y : Scheme} (f : X ⟶ Y), ∀ (U : Y.affine_opens), @@P (f ∣_ U) U.prop | def | algebraic_geometry.target_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | For a `P : affine_target_morphism_property`, `target_affine_locally P` holds for
`f : X ⟶ Y` whenever `P` holds for the restriction of `f` on every affine open subset of `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine_open.map_is_iso {X Y : Scheme} {U : opens Y.carrier} (hU : is_affine_open U)
(f : X ⟶ Y) [is_iso f] : is_affine_open ((opens.map f.1.base).obj U) | begin
haveI : is_affine _ := hU,
exact is_affine_of_iso (f ∣_ U),
end | lemma | algebraic_geometry.is_affine_open.map_is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
target_affine_locally_respects_iso {P : affine_target_morphism_property}
(hP : P.to_property.respects_iso) : (target_affine_locally P).respects_iso | begin
split,
{ introv H U,
rw [morphism_restrict_comp, affine_cancel_left_is_iso hP],
exact H U },
{ introv H,
rintro ⟨U, hU : is_affine_open U⟩, dsimp,
haveI : is_affine _ := hU,
haveI : is_affine _ := hU.map_is_iso e.hom,
rw [morphism_restrict_comp, affine_cancel_right_is_iso hP],
ex... | lemma | algebraic_geometry.target_affine_locally_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local (P : affine_target_morphism_property) : Prop | (respects_iso : P.to_property.respects_iso)
(to_basic_open : ∀ {X Y : Scheme} [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj $ op ⊤),
by exactI P f →
@@P (f ∣_ (Y.basic_open r)) ((top_is_affine_open Y).basic_open_is_affine _))
(of_basic_open_cover : ∀ {X Y : Scheme} [is_affine Y] (f : X ⟶ Y)
(s : finset (Y.presh... | structure | algebraic_geometry.affine_target_morphism_property.is_local | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"finset",
"ideal.span"
] | We say that `P : affine_target_morphism_property` is a local property if
1. `P` respects isomorphisms.
2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ Y.basic_open r` for any
global section `r`.
3. If `P` holds for `f ∣_ Y.basic_open r` for all `r` in a spanning set of the global sections,
then `P` holds ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
target_affine_locally_of_open_cover {P : affine_target_morphism_property}
(hP : P.is_local)
{X Y : Scheme} (f : X ⟶ Y) (𝒰 : Y.open_cover) [∀ i, is_affine (𝒰.obj i)]
(h𝒰 : ∀ i, P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) :
target_affine_locally P f | begin
classical,
let S := λ i, (⟨⟨set.range (𝒰.map i).1.base, (𝒰.is_open i).base_open.open_range⟩,
range_is_affine_open_of_open_immersion (𝒰.map i)⟩ : Y.affine_opens),
intro U,
apply of_affine_open_cover U (set.range S),
{ intros U r h,
haveI : is_affine _ := U.2,
have := hP.2 (f ∣_ U.1),
r... | lemma | algebraic_geometry.target_affine_locally_of_open_cover | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"CommRing",
"finset.coe_image",
"ideal.comap",
"ideal.comap_top",
"ideal.span",
"set.eq_univ_iff_forall",
"set.mem_Union",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local.affine_open_cover_tfae
{P : affine_target_morphism_property}
(hP : P.is_local) {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [target_affine_locally P f,
∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], ∀ (i : 𝒰.J),
by exactI P (pullback.snd : (𝒰.pullback_c... | begin
tfae_have : 1 → 4,
{ intros H U g h₁ h₂,
resetI,
replace H := H ⟨⟨_, h₂.base_open.open_range⟩,
range_is_affine_open_of_open_immersion g⟩,
rw ← P.to_property_apply at H ⊢,
rwa ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _) },
tfae_have : 4 → 3,
{ intros H 𝒰 h𝒰 i,
re... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.affine_open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"eq_top_iff",
"subtype.range_coe",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local_of_open_cover_imply
(P : affine_target_morphism_property) (hP : P.to_property.respects_iso)
(H : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y),
(∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], ∀ (i : 𝒰.J),
by exactI P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶... | begin
refine ⟨hP, _, _⟩,
{ introv h,
resetI,
haveI : is_affine _ := (top_is_affine_open Y).basic_open_is_affine r,
delta morphism_restrict,
rw affine_cancel_left_is_iso hP,
refine @@H f ⟨Scheme.open_cover_of_is_iso (𝟙 Y), _, _⟩ (Y.of_restrict _) _inst _,
{ intro i, dsimp, apply_instance },
... | lemma | algebraic_geometry.affine_target_morphism_property.is_local_of_open_cover_imply | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local.affine_open_cover_iff
{P : affine_target_morphism_property} (hP : P.is_local)
{X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y) [h𝒰 : ∀ i, is_affine (𝒰.obj i)] :
target_affine_locally P f ↔ ∀ i, @@P (pullback.snd : pullback f (𝒰.map i) ⟶ _) (h𝒰 i) | ⟨λ H, let h := ((hP.affine_open_cover_tfae f).out 0 2).mp H in h 𝒰,
λ H, let h := ((hP.affine_open_cover_tfae f).out 1 0).mp in h ⟨𝒰, infer_instance, H⟩⟩ | lemma | algebraic_geometry.affine_target_morphism_property.is_local.affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local.affine_target_iff
{P : affine_target_morphism_property} (hP : P.is_local)
{X Y : Scheme.{u}} (f : X ⟶ Y) [is_affine Y] :
target_affine_locally P f ↔ P f | begin
rw hP.affine_open_cover_iff f _,
swap, { exact Scheme.open_cover_of_is_iso (𝟙 Y) },
swap, { intro _, dsimp, apply_instance },
transitivity (P (pullback.snd : pullback f (𝟙 _) ⟶ _)),
{ exact ⟨λ H, H punit.star, λ H _, H⟩ },
rw [← category.comp_id pullback.snd, ← pullback.condition, affine_cancel_left... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.affine_target_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
property_is_local_at_target (P : morphism_property Scheme) : Prop | (respects_iso : P.respects_iso)
(restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier), P f → P (f ∣_ U))
(of_open_cover : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y),
(∀ (i : 𝒰.J), P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) → P f) | structure | algebraic_geometry.property_is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | We say that `P : morphism_property Scheme` is local at the target if
1. `P` respects isomorphisms.
2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`.
3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_target_morphism_property.is_local.target_affine_locally_is_local
{P : affine_target_morphism_property} (hP : P.is_local) :
property_is_local_at_target (target_affine_locally P) | begin
constructor,
{ exact target_affine_locally_respects_iso hP.1 },
{ intros X Y f U H V,
rw [← P.to_property_apply, hP.1.arrow_mk_iso_iff (morphism_restrict_restrict f _ _)],
convert H ⟨_, is_affine_open.image_is_open_immersion V.2 (Y.of_restrict _)⟩,
rw ← P.to_property_apply,
refl },
{ rintr... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.target_affine_locally_is_local | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
property_is_local_at_target.open_cover_tfae
{P : morphism_property Scheme}
(hP : property_is_local_at_target P)
{X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [P f,
∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J),
P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i),
∀ (𝒰 : Scheme.open_cover.{u} Y) (i... | begin
tfae_have : 2 → 1,
{ rintro ⟨𝒰, H⟩, exact hP.3 f 𝒰 H },
tfae_have : 1 → 4,
{ intros H U, exact hP.2 f U H },
tfae_have : 4 → 3,
{ intros H 𝒰 i,
rw ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _),
exact H (𝒰.map i).opens_range },
tfae_have : 3 → 2,
{ exact λ H, ⟨Y.affine_cov... | lemma | algebraic_geometry.property_is_local_at_target.open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"csupr_const",
"subtype.range_coe",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
property_is_local_at_target.open_cover_iff
{P : morphism_property Scheme} (hP : property_is_local_at_target P)
{X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y) :
P f ↔ ∀ i, P (pullback.snd : pullback f (𝒰.map i) ⟶ _) | ⟨λ H, let h := ((hP.open_cover_tfae f).out 0 2).mp H in h 𝒰,
λ H, let h := ((hP.open_cover_tfae f).out 1 0).mp in h ⟨𝒰, H⟩⟩ | lemma | algebraic_geometry.property_is_local_at_target.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change
(P : affine_target_morphism_property) : Prop | ∀ ⦃X Y S : Scheme⦄ [is_affine S] [is_affine X] (f : X ⟶ S) (g : Y ⟶ S),
by exactI P g → P (pullback.fst : pullback f g ⟶ X) | def | algebraic_geometry.affine_target_morphism_property.stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | A `P : affine_target_morphism_property` is stable under base change if `P` holds for `Y ⟶ S`
implies that `P` holds for `X ×ₛ Y ⟶ X` with `X` and `S` affine schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local.target_affine_locally_pullback_fst_of_right_of_stable_under_base_change
{P : affine_target_morphism_property} (hP : P.is_local) (hP' : P.stable_under_base_change)
{X Y S : Scheme} (f : X ⟶ S) (g : Y ⟶ S) [is_affine S] (H : P g) :
target_affine_locally P (pullback.fst : pullback f g ⟶ X) | begin
rw (hP.affine_open_cover_tfae (pullback.fst : pullback f g ⟶ X)).out 0 1,
use [X.affine_cover, infer_instance],
intro i,
let e := pullback_symmetry _ _ ≪≫ pullback_right_pullback_fst_iso f g (X.affine_cover.map i),
have : e.hom ≫ pullback.fst = pullback.snd := by simp,
rw [← this, affine_cancel_left_i... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.target_affine_locally_pullback_fst_of_right_of_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local.stable_under_base_change
{P : affine_target_morphism_property} (hP : P.is_local) (hP' : P.stable_under_base_change) :
(target_affine_locally P).stable_under_base_change | morphism_property.stable_under_base_change.mk (target_affine_locally_respects_iso hP.respects_iso)
begin
intros X Y S f g H,
rw (hP.target_affine_locally_is_local.open_cover_tfae (pullback.fst : pullback f g ⟶ X)).out 0 1,
use S.affine_cover.pullback_cover f,
intro i,
rw (hP.affine_open_cover_tfae g).out 0 3 ... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.diagonal (P : affine_target_morphism_property) :
affine_target_morphism_property | λ X Y f hf, ∀ {U₁ U₂ : Scheme} (f₁ : U₁ ⟶ X) (f₂ : U₂ ⟶ X) [is_affine U₁] [is_affine U₂]
[is_open_immersion f₁] [is_open_immersion f₂],
by exactI P (pullback.map_desc f₁ f₂ f) | def | algebraic_geometry.affine_target_morphism_property.diagonal | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | The `affine_target_morphism_property` associated to `(target_affine_locally P).diagonal`.
See `diagonal_target_affine_locally_eq_target_affine_locally`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_target_morphism_property.diagonal_respects_iso (P : affine_target_morphism_property)
(hP : P.to_property.respects_iso) :
P.diagonal.to_property.respects_iso | begin
delta affine_target_morphism_property.diagonal,
apply affine_target_morphism_property.respects_iso_mk,
{ introv H _ _,
resetI,
rw [pullback.map_desc_comp, affine_cancel_left_is_iso hP, affine_cancel_right_is_iso hP],
apply H },
{ introv H _ _,
resetI,
rw [pullback.map_desc_comp, affine... | lemma | algebraic_geometry.affine_target_morphism_property.diagonal_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_target_affine_locally_of_open_cover (P : affine_target_morphism_property)
(hP : P.is_local)
{X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} Y)
[∀ i, is_affine (𝒰.obj i)] (𝒰' : Π i, Scheme.open_cover.{u} (pullback f (𝒰.map i)))
[∀ i j, is_affine ((𝒰' i).obj j)]
(h𝒰' : ∀ i j k, P (pul... | begin
refine (hP.affine_open_cover_iff _ _).mpr _,
{ exact ((Scheme.pullback.open_cover_of_base 𝒰 f f).bind (λ i,
Scheme.pullback.open_cover_of_left_right.{u u} (𝒰' i) (𝒰' i) pullback.snd pullback.snd)) },
{ intro i,
dsimp at *,
apply_instance },
{ rintro ⟨i, j, k⟩,
dsimp,
convert (affi... | lemma | algebraic_geometry.diagonal_target_affine_locally_of_open_cover | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.diagonal_of_target_affine_locally
(P : affine_target_morphism_property)
(hP : P.is_local) {X Y U : Scheme.{u}} (f : X ⟶ Y) (g : U ⟶ Y)
[is_affine U] [is_open_immersion g] (H : (target_affine_locally P).diagonal f) :
P.diagonal (pullback.snd : pullback f g ⟶ _) | begin
rintros U V f₁ f₂ _ _ _ _,
resetI,
replace H := ((hP.affine_open_cover_tfae (pullback.diagonal f)).out 0 3).mp H,
let g₁ := pullback.map (f₁ ≫ pullback.snd)
(f₂ ≫ pullback.snd) f f
(f₁ ≫ pullback.fst)
(f₂ ≫ pullback.fst) g
(by rw [category.assoc, category.assoc, pullback.condition])
(b... | lemma | algebraic_geometry.affine_target_morphism_property.diagonal_of_target_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local.diagonal_affine_open_cover_tfae
{P : affine_target_morphism_property}
(hP : P.is_local) {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [(target_affine_locally P).diagonal f,
∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], by exactI
∀ (i : 𝒰.J), P.diagonal (... | begin
tfae_have : 1 → 4,
{ introv H hU hg _ _, resetI, apply P.diagonal_of_target_affine_locally; assumption },
tfae_have : 4 → 3,
{ introv H h𝒰, resetI, apply H },
tfae_have : 3 → 2,
{ exact λ H, ⟨Y.affine_cover, infer_instance, H Y.affine_cover⟩ },
tfae_have : 2 → 5,
{ rintro ⟨𝒰, h𝒰, H⟩,
resetI... | lemma | algebraic_geometry.affine_target_morphism_property.is_local.diagonal_affine_open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_target_morphism_property.is_local.diagonal {P : affine_target_morphism_property}
(hP : P.is_local) : P.diagonal.is_local | affine_target_morphism_property.is_local_of_open_cover_imply
P.diagonal
(P.diagonal_respects_iso hP.1)
(λ _ _ f, ((hP.diagonal_affine_open_cover_tfae f).out 1 3).mp) | lemma | algebraic_geometry.affine_target_morphism_property.is_local.diagonal | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_target_affine_locally_eq_target_affine_locally (P : affine_target_morphism_property)
(hP : P.is_local) :
(target_affine_locally P).diagonal = target_affine_locally P.diagonal | begin
ext _ _ f,
exact ((hP.diagonal_affine_open_cover_tfae f).out 0 1).trans
((hP.diagonal.affine_open_cover_tfae f).out 1 0),
end | lemma | algebraic_geometry.diagonal_target_affine_locally_eq_target_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_is_local_at_target (P : morphism_property Scheme)
(hP : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y),
(∀ (i : 𝒰.J), P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) → P f) :
property_is_local_at_target P.universally | begin
refine ⟨P.universally_respects_iso, λ X Y f U, P.universally_stable_under_base_change
(is_pullback_morphism_restrict f U).flip, _⟩,
intros X Y f 𝒰 h X' Y' i₁ i₂ f' H,
apply hP _ (𝒰.pullback_cover i₂),
intro i,
dsimp,
apply h i (pullback.lift (pullback.fst ≫ i₁) (pullback.snd ≫ pullback.snd) _) p... | lemma | algebraic_geometry.universally_is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_is_local_at_target_of_morphism_restrict (P : morphism_property Scheme)
(hP₁ : P.respects_iso)
(hP₂ : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → opens Y.carrier) (hU : supr U = ⊤),
(∀ i, P (f ∣_ (U i))) → P f) :
property_is_local_at_target P.universally | universally_is_local_at_target P
begin
intros X Y f 𝒰 h𝒰,
apply hP₂ f (λ (i : 𝒰.J), (𝒰.map i).opens_range) 𝒰.supr_opens_range,
simp_rw hP₁.arrow_mk_iso_iff (morphism_restrict_opens_range f _),
exact h𝒰
end | lemma | algebraic_geometry.universally_is_local_at_target_of_morphism_restrict | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_property.topologically
(P : ∀ {α β : Type u} [topological_space α] [topological_space β] (f : α → β), Prop) :
morphism_property Scheme.{u} | λ X Y f, P f.1.base | def | algebraic_geometry.morphism_property.topologically | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/basic.lean | [
"algebraic_geometry.AffineScheme",
"algebraic_geometry.pullbacks",
"category_theory.morphism_property"
] | [
"topological_space"
] | `topologically P` holds for a morphism if the underlying topological map satisfies `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_of_finite_type (f : X ⟶ Y) : Prop | (finite_type_of_affine_subset :
∀ (U : Y.affine_opens) (V : X.affine_opens) (e : V.1 ≤ (opens.map f.1.base).obj U.1),
(f.app_le e).finite_type) | class | algebraic_geometry.locally_of_finite_type | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | A morphism of schemes `f : X ⟶ Y` is locally of finite type if for each affine `U ⊆ Y` and
`V ⊆ f ⁻¹' U`, The induced map `Γ(Y, U) ⟶ Γ(X, V)` is of finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_of_finite_type_eq :
@locally_of_finite_type = affine_locally @ring_hom.finite_type | begin
ext X Y f,
rw [locally_of_finite_type_iff, affine_locally_iff_affine_opens_le],
exact ring_hom.finite_type_respects_iso
end | lemma | algebraic_geometry.locally_of_finite_type_eq | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [
"ring_hom.finite_type",
"ring_hom.finite_type_respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type_of_is_open_immersion {X Y : Scheme} (f : X ⟶ Y)
[is_open_immersion f] : locally_of_finite_type f | locally_of_finite_type_eq.symm ▸ ring_hom.finite_type_is_local.affine_locally_of_is_open_immersion f | instance | algebraic_geometry.locally_of_finite_type_of_is_open_immersion | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type_stable_under_composition :
morphism_property.stable_under_composition @locally_of_finite_type | locally_of_finite_type_eq.symm ▸
ring_hom.finite_type_is_local.affine_locally_stable_under_composition | lemma | algebraic_geometry.locally_of_finite_type_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : locally_of_finite_type f] [hg : locally_of_finite_type g] :
locally_of_finite_type (f ≫ g) | locally_of_finite_type_stable_under_composition f g hf hg | instance | algebraic_geometry.locally_of_finite_type_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type_of_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : locally_of_finite_type (f ≫ g)] :
locally_of_finite_type f | begin
unfreezingI { revert hf },
rw [locally_of_finite_type_eq],
apply ring_hom.finite_type_is_local.affine_locally_of_comp,
introv H,
exactI ring_hom.finite_type.of_comp_finite_type H,
end | lemma | algebraic_geometry.locally_of_finite_type_of_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [
"ring_hom.finite_type.of_comp_finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type.affine_open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)]
(𝒰' : ∀ i, Scheme.open_cover.{u} ((𝒰.pullback_cover f).obj i))
[∀ i j, is_affine ((𝒰' i).obj j)] :
locally_of_finite_type f ↔
(∀ i j, (Scheme.Γ.map ((𝒰' i).map j ≫ pullb... | locally_of_finite_type_eq.symm ▸ ring_hom.finite_type_is_local.affine_open_cover_iff f 𝒰 𝒰' | lemma | algebraic_geometry.locally_of_finite_type.affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type.source_open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} X) :
locally_of_finite_type f ↔ (∀ i, locally_of_finite_type (𝒰.map i ≫ f)) | locally_of_finite_type_eq.symm ▸ ring_hom.finite_type_is_local.source_open_cover_iff f 𝒰 | lemma | algebraic_geometry.locally_of_finite_type.source_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type.open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} Y) :
locally_of_finite_type f ↔
(∀ i, locally_of_finite_type (pullback.snd : pullback f (𝒰.map i) ⟶ _)) | locally_of_finite_type_eq.symm ▸
ring_hom.finite_type_is_local.is_local_affine_locally.open_cover_iff f 𝒰 | lemma | algebraic_geometry.locally_of_finite_type.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_of_finite_type_respects_iso :
morphism_property.respects_iso @locally_of_finite_type | locally_of_finite_type_eq.symm ▸ target_affine_locally_respects_iso
(source_affine_locally_respects_iso ring_hom.finite_type_respects_iso) | lemma | algebraic_geometry.locally_of_finite_type_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/finite_type.lean | [
"algebraic_geometry.morphisms.ring_hom_properties",
"ring_theory.ring_hom.finite_type"
] | [
"ring_hom.finite_type_respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_iff_stalk {f : X ⟶ Y} :
is_open_immersion f ↔
open_embedding f.1.base ∧ ∀ x, is_iso (PresheafedSpace.stalk_map f.1 x) | begin
split,
{ intro h, exactI ⟨h.1, infer_instance⟩ },
{ rintro ⟨h₁, h₂⟩, exactI is_open_immersion.of_stalk_iso f h₁ }
end | lemma | algebraic_geometry.is_open_immersion_iff_stalk | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_stable_under_composition :
morphism_property.stable_under_composition @is_open_immersion | begin
introsI X Y Z f g h₁ h₂, apply_instance
end | lemma | algebraic_geometry.is_open_immersion_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_respects_iso :
morphism_property.respects_iso @is_open_immersion | begin
apply is_open_immersion_stable_under_composition.respects_iso,
intros _ _ _, apply_instance
end | lemma | algebraic_geometry.is_open_immersion_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_is_local_at_target : property_is_local_at_target @is_open_immersion | begin
constructor,
{ exact is_open_immersion_respects_iso },
{ introsI, apply_instance },
{ intros X Y f 𝒰 H,
rw is_open_immersion_iff_stalk,
split,
{ apply (open_embedding_iff_open_embedding_of_supr_eq_top
𝒰.supr_opens_range f.1.base.2).mpr,
intro i,
have := ((is_open_immersio... | lemma | algebraic_geometry.is_open_immersion_is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [
"open_embedding_iff_open_embedding_of_supr_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion.open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [is_open_immersion f,
∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J),
is_open_immersion (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i),
∀ (𝒰 : Scheme.open_cover.{u} Y) (i : 𝒰.J),
is_open_immersion (pullback.snd : (�... | is_open_immersion_is_local_at_target.open_cover_tfae f | lemma | algebraic_geometry.is_open_immersion.open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion.open_cover_iff {X Y : Scheme.{u}}
(𝒰 : Scheme.open_cover.{u} Y) (f : X ⟶ Y) :
is_open_immersion f ↔ ∀ i, is_open_immersion (pullback.snd : pullback f (𝒰.map i) ⟶ _) | is_open_immersion_is_local_at_target.open_cover_iff f 𝒰 | lemma | algebraic_geometry.is_open_immersion.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_stable_under_base_change :
morphism_property.stable_under_base_change @is_open_immersion | morphism_property.stable_under_base_change.mk is_open_immersion_respects_iso $
by { introsI X Y Z f g H, apply_instance } | lemma | algebraic_geometry.is_open_immersion_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/open_immersion.lean | [
"topology.local_at_target",
"algebraic_geometry.morphisms.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact (f : X ⟶ Y) : Prop | (is_compact_preimage : ∀ U : set Y.carrier, is_open U → is_compact U → is_compact (f.1.base ⁻¹' U)) | class | algebraic_geometry.quasi_compact | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"is_compact",
"is_open"
] | A morphism is `quasi-compact` if the underlying map of topological spaces is, i.e. if the preimages
of quasi-compact open sets are quasi-compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_compact_iff_spectral : quasi_compact f ↔ is_spectral_map f.1.base | ⟨λ ⟨h⟩, ⟨by continuity, h⟩, λ h, ⟨h.2⟩⟩ | lemma | algebraic_geometry.quasi_compact_iff_spectral | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"continuity",
"is_spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_property : affine_target_morphism_property | λ X Y f hf, compact_space X.carrier | def | algebraic_geometry.quasi_compact.affine_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space"
] | The `affine_target_morphism_property` corresponding to `quasi_compact`, asserting that the
domain is a quasi-compact scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_compact_of_is_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] : quasi_compact f | begin
constructor,
intros U hU hU',
convert hU'.image (inv f.1.base).continuous_to_fun using 1,
rw set.image_eq_preimage_of_inverse,
delta function.left_inverse,
exacts [is_iso.inv_hom_id_apply f.1.base, is_iso.hom_inv_id_apply f.1.base]
end | instance | algebraic_geometry.quasi_compact_of_is_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"set.image_eq_preimage_of_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[quasi_compact f] [quasi_compact g] : quasi_compact (f ≫ g) | begin
constructor,
intros U hU hU',
rw [Scheme.comp_val_base, coe_comp, set.preimage_comp],
apply quasi_compact.is_compact_preimage,
{ exact continuous.is_open_preimage (by continuity) _ hU },
apply quasi_compact.is_compact_preimage; assumption
end | instance | algebraic_geometry.quasi_compact_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"continuity",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_open_iff_eq_finset_affine_union {X : Scheme} (U : set X.carrier) :
is_compact U ∧ is_open U ↔
∃ (s : set X.affine_opens), s.finite ∧ U = ⋃ (i : X.affine_opens) (h : i ∈ s), i | begin
apply opens.is_basis.is_compact_open_iff_eq_finite_Union
(coe : X.affine_opens → opens X.carrier),
{ rw subtype.range_coe, exact is_basis_affine_open X },
{ exact λ i, i.2.is_compact }
end | lemma | algebraic_geometry.is_compact_open_iff_eq_finset_affine_union | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"is_compact",
"is_open",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_open_iff_eq_basic_open_union {X : Scheme} [is_affine X] (U : set X.carrier) :
is_compact U ∧ is_open U ↔
∃ (s : set (X.presheaf.obj (op ⊤))), s.finite ∧
U = ⋃ (i : X.presheaf.obj (op ⊤)) (h : i ∈ s), X.basic_open i | (is_basis_basic_open X).is_compact_open_iff_eq_finite_Union _
(λ i, ((top_is_affine_open _).basic_open_is_affine _).is_compact) _ | lemma | algebraic_geometry.is_compact_open_iff_eq_basic_open_union | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_iff_forall_affine : quasi_compact f ↔
∀ U : opens Y.carrier, is_affine_open U → is_compact (f.1.base ⁻¹' (U : set Y.carrier)) | begin
rw quasi_compact_iff,
refine ⟨λ H U hU, H U U.is_open hU.is_compact, _⟩,
intros H U hU hU',
obtain ⟨S, hS, rfl⟩ := (is_compact_open_iff_eq_finset_affine_union U).mp ⟨hU', hU⟩,
simp only [set.preimage_Union, subtype.val_eq_coe],
exact hS.is_compact_bUnion (λ i _, H i i.prop)
end | lemma | algebraic_geometry.quasi_compact_iff_forall_affine | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"is_compact",
"set.preimage_Union",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_property_to_property {X Y : Scheme} (f : X ⟶ Y) :
(quasi_compact.affine_property : _).to_property f ↔
is_affine Y ∧ compact_space X.carrier | by { delta affine_target_morphism_property.to_property quasi_compact.affine_property, simp } | lemma | algebraic_geometry.quasi_compact.affine_property_to_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_iff_affine_property :
quasi_compact f ↔ target_affine_locally quasi_compact.affine_property f | begin
rw quasi_compact_iff_forall_affine,
transitivity (∀ U : Y.affine_opens, is_compact (f.1.base ⁻¹' (U : set Y.carrier))),
{ exact ⟨λ h U, h U U.prop, λ h U hU, h ⟨U, hU⟩⟩ },
apply forall_congr,
exact λ _, is_compact_iff_compact_space,
end | lemma | algebraic_geometry.quasi_compact_iff_affine_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"is_compact",
"is_compact_iff_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_eq_affine_property :
@quasi_compact = target_affine_locally quasi_compact.affine_property | by { ext, exact quasi_compact_iff_affine_property _ } | lemma | algebraic_geometry.quasi_compact_eq_affine_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_basic_open (X : Scheme) {U : opens X.carrier} (hU : is_compact (U : set X.carrier))
(f : X.presheaf.obj (op U)) : is_compact (X.basic_open f : set X.carrier) | begin
classical,
refine ((is_compact_open_iff_eq_finset_affine_union _).mpr _).1,
obtain ⟨s, hs, e⟩ := (is_compact_open_iff_eq_finset_affine_union _).mp ⟨hU, U.is_open⟩,
let g : s → X.affine_opens,
{ intro V,
use V.1 ⊓ X.basic_open f,
have : V.1.1 ⟶ U,
{ apply hom_of_le, change _ ⊆ (U : set X.carr... | lemma | algebraic_geometry.is_compact_basic_open | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"finite",
"is_compact",
"set.Union₂_inter",
"set.Union₂_subset",
"set.finite_range",
"set.mem_range_self",
"set.subset.rfl",
"set.subset.trans",
"set.subset_Union₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_property_is_local :
(quasi_compact.affine_property : _).is_local | begin
split,
{ apply affine_target_morphism_property.respects_iso_mk; rintros X Y Z _ _ _ H,
exacts [@@homeomorph.compact_space _ _ H (Top.homeo_of_iso (as_iso e.inv.1.base)), H] },
{ introv H,
delta quasi_compact.affine_property at H ⊢,
change compact_space ((opens.map f.val.base).obj (Y.basic_open r... | lemma | algebraic_geometry.quasi_compact.affine_property_is_local | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"Top.homeo_of_iso",
"compact_space",
"homeomorph.compact_space",
"is_compact",
"is_compact_Union",
"is_compact_iff_compact_space",
"is_compact_univ_iff",
"set.preimage_Union",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [quasi_compact f,
∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)],
∀ (i : 𝒰.J), compact_space (pullback f (𝒰.map i)).carrier,
∀ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)] (i : 𝒰.J),
compact_s... | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.affine_open_cover_tfae f | lemma | algebraic_geometry.quasi_compact.affine_open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.is_local_at_target :
property_is_local_at_target @quasi_compact | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.target_affine_locally_is_local | lemma | algebraic_geometry.quasi_compact.is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [quasi_compact f,
∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J),
quasi_compact (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i),
∀ (𝒰 : Scheme.open_cover.{u} Y) (i : 𝒰.J),
quasi_compact (pullback.snd : (𝒰.pullback_cover... | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.target_affine_locally_is_local.open_cover_tfae f | lemma | algebraic_geometry.quasi_compact.open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_over_affine_iff {X Y : Scheme} (f : X ⟶ Y) [is_affine Y] :
quasi_compact f ↔ compact_space X.carrier | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.affine_target_iff f | lemma | algebraic_geometry.quasi_compact_over_affine_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space_iff_quasi_compact (X : Scheme) :
compact_space X.carrier ↔ quasi_compact (terminal.from X) | (quasi_compact_over_affine_iff _).symm | lemma | algebraic_geometry.compact_space_iff_quasi_compact | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_open_cover_iff {X Y : Scheme.{u}} (𝒰 : Scheme.open_cover.{u} Y)
[∀ i, is_affine (𝒰.obj i)] (f : X ⟶ Y) :
quasi_compact f ↔ ∀ i, compact_space (pullback f (𝒰.map i)).carrier | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.affine_open_cover_iff f 𝒰 | lemma | algebraic_geometry.quasi_compact.affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.open_cover_iff {X Y : Scheme.{u}} (𝒰 : Scheme.open_cover.{u} Y) (f : X ⟶ Y) :
quasi_compact f ↔ ∀ i, quasi_compact (pullback.snd : pullback f (𝒰.map i) ⟶ _) | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.target_affine_locally_is_local.open_cover_iff f 𝒰 | lemma | algebraic_geometry.quasi_compact.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_respects_iso : morphism_property.respects_iso @quasi_compact | quasi_compact_eq_affine_property.symm ▸
target_affine_locally_respects_iso quasi_compact.affine_property_is_local.1 | lemma | algebraic_geometry.quasi_compact_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_stable_under_composition :
morphism_property.stable_under_composition @quasi_compact | λ _ _ _ _ _ _ _, by exactI infer_instance | lemma | algebraic_geometry.quasi_compact_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact.affine_property_stable_under_base_change :
quasi_compact.affine_property.stable_under_base_change | begin
intros X Y S _ _ f g h,
rw quasi_compact.affine_property at h ⊢,
resetI,
let 𝒰 := Scheme.pullback.open_cover_of_right Y.affine_cover.finite_subcover f g,
haveI : finite 𝒰.J,
{ dsimp [𝒰], apply_instance },
haveI : ∀ i, compact_space (𝒰.obj i).carrier,
{ intro i, dsimp, apply_instance },
exact... | lemma | algebraic_geometry.quasi_compact.affine_property_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"compact_space",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_stable_under_base_change :
morphism_property.stable_under_base_change @quasi_compact | quasi_compact_eq_affine_property.symm ▸
quasi_compact.affine_property_is_local.stable_under_base_change
quasi_compact.affine_property_stable_under_base_change | lemma | algebraic_geometry.quasi_compact_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_open_induction_on {P : opens X.carrier → Prop} (S : opens X.carrier)
(hS : is_compact S.1)
(h₁ : P ⊥)
(h₂ : ∀ (S : opens X.carrier) (hS : is_compact S.1) (U : X.affine_opens), P S → P (S ⊔ U)) :
P S | begin
classical,
obtain ⟨s, hs, hs'⟩ := (is_compact_open_iff_eq_finset_affine_union S.1).mp ⟨hS, S.2⟩,
replace hs' : S = supr (λ i : s, (i : opens X.carrier)) := by { ext1, simpa using hs' },
subst hs',
apply hs.induction_on,
{ convert h₁, rw supr_eq_bot, rintro ⟨_, h⟩, exact h.elim },
{ intros x s h₃ hs ... | lemma | algebraic_geometry.compact_open_induction_on | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"coe_coe",
"is_compact",
"sup_comm",
"supr",
"supr_eq_bot",
"supr_insert",
"supr_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open (X : Scheme)
{U : opens X.carrier} (hU : is_affine_open U) (x f : X.presheaf.obj (op U))
(H : x |_ X.basic_open f = 0) :
∃ n : ℕ, f ^ n * x = 0 | begin
rw ← map_zero (X.presheaf.map (hom_of_le $ X.basic_open_le f : X.basic_open f ⟶ U).op) at H,
have := (is_localization_basic_open hU f).3,
obtain ⟨⟨_, n, rfl⟩, e⟩ := this.mp H,
exact ⟨n, by simpa [mul_comm x] using e⟩,
end | lemma | algebraic_geometry.exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_compact (X : Scheme)
{U : opens X.carrier} (hU : is_compact U.1) (x f : X.presheaf.obj (op U))
(H : x |_ X.basic_open f = 0) :
∃ n : ℕ, f ^ n * x = 0 | begin
obtain ⟨s, hs, e⟩ := (is_compact_open_iff_eq_finset_affine_union U.1).mp ⟨hU, U.2⟩,
replace e : U = supr (λ i : s, (i : opens X.carrier)),
{ ext1, simpa using e },
have h₁ : ∀ i : s, i.1.1 ≤ U,
{ intro i, change (i : opens X.carrier) ≤ U, rw e, exact le_supr _ _ },
have H' := λ (i : s), exists_pow_mul... | lemma | algebraic_geometry.exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_compact | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_compact.lean | [
"algebraic_geometry.morphisms.basic",
"topology.spectral.hom",
"algebraic_geometry.limits"
] | [
"Top.presheaf.restrict",
"Top.presheaf.restrict_open",
"finset.le_sup",
"finset.mem_univ",
"is_compact",
"le_supr",
"map_mul",
"map_pow",
"mul_assoc",
"mul_zero",
"nonempty_fintype",
"pow_add",
"set.inter_subset_right",
"supr",
"tsub_add_cancel_of_le"
] | If `x : Γ(X, U)` is zero on `D(f)` for some `f : Γ(X, U)`, and `U` is quasi-compact, then
`f ^ n * x = 0` for some `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_separated (f : X ⟶ Y) : Prop | (diagonal_quasi_compact : quasi_compact (pullback.diagonal f)) | class | algebraic_geometry.quasi_separated | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | A morphism is `quasi_separated` if diagonal map is quasi-compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_separated.affine_property : affine_target_morphism_property | (λ X Y f _, quasi_separated_space X.carrier) | def | algebraic_geometry.quasi_separated.affine_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"quasi_separated_space"
] | The `affine_target_morphism_property` corresponding to `quasi_separated`, asserting that the
domain is a quasi-separated scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_separated_space_iff_affine (X : Scheme) :
quasi_separated_space X.carrier ↔ ∀ (U V : X.affine_opens), is_compact (U ∩ V : set X.carrier) | begin
rw quasi_separated_space_iff,
split,
{ intros H U V, exact H U V U.1.2 U.2.is_compact V.1.2 V.2.is_compact },
{ intros H,
suffices : ∀ (U : opens X.carrier) (hU : is_compact U.1) (V : opens X.carrier)
(hV : is_compact V.1), is_compact (U ⊓ V).1,
{ intros U V hU hU' hV hV', exact this ⟨U, hU⟩... | lemma | algebraic_geometry.quasi_separated_space_iff_affine | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"is_compact",
"quasi_separated_space",
"set.inter_union_distrib_left",
"set.union_inter_distrib_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_affine_property_iff_quasi_separated_space {X Y : Scheme} [is_affine Y]
(f : X ⟶ Y) :
quasi_compact.affine_property.diagonal f ↔ quasi_separated_space X.carrier | begin
delta affine_target_morphism_property.diagonal,
rw quasi_separated_space_iff_affine,
split,
{ intros H U V,
haveI : is_affine _ := U.2,
haveI : is_affine _ := V.2,
let g : pullback (X.of_restrict U.1.open_embedding) (X.of_restrict V.1.open_embedding) ⟶ X :=
pullback.fst ≫ X.of_restrict _... | lemma | algebraic_geometry.quasi_compact_affine_property_iff_quasi_separated_space | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"homeomorph.compact_space",
"homeomorph.of_embedding",
"is_compact_iff_compact_space",
"quasi_separated_space",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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