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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
has_limit_cospan_forget_of_right : has_limit (cospan g f ⋙ forget) | begin
apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,
change has_limit (cospan (forget .map g) (forget .map f)),
apply_instance
end | instance | algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_cospan_forget_of_right' :
has_limit (cospan ((cospan g f ⋙ forget).map hom.inl)
((cospan g f ⋙ forget).map hom.inr)) | show has_limit (cospan (forget .map g) (forget .map f)), from infer_instance | instance | algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_right' | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_creates_pullback_of_left : creates_limit (cospan f g) forget | creates_limit_of_fully_faithful_of_iso
(PresheafedSpace.is_open_immersion.to_Scheme Y
(@pullback.snd LocallyRingedSpace _ _ _ _ f g _).1)
(eq_to_iso (by simp) ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm) | instance | algebraic_geometry.is_open_immersion.forget_creates_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_creates_pullback_of_right : creates_limit (cospan g f) forget | creates_limit_of_fully_faithful_of_iso
(PresheafedSpace.is_open_immersion.to_Scheme Y
(@pullback.fst LocallyRingedSpace _ _ _ _ g f _).1)
(eq_to_iso (by simp) ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm) | instance | algebraic_geometry.is_open_immersion.forget_creates_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_of_left : preserves_limit (cospan f g) forget | category_theory.preserves_limit_of_creates_limit_and_has_limit _ _ | instance | algebraic_geometry.is_open_immersion.forget_preserves_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"category_theory.preserves_limit_of_creates_limit_and_has_limit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_of_right : preserves_limit (cospan g f) forget | preserves_pullback_symmetry _ _ _ | instance | algebraic_geometry.is_open_immersion.forget_preserves_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullback_of_left : has_pullback f g | has_limit_of_created (cospan f g) forget | instance | algebraic_geometry.is_open_immersion.has_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullback_of_right : has_pullback g f | has_limit_of_created (cospan g f) forget | instance | algebraic_geometry.is_open_immersion.has_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_snd_of_left : is_open_immersion (pullback.snd : pullback f g ⟶ _) | begin
have := preserves_pullback.iso_hom_snd forget f g,
dsimp only [Scheme.forget_to_LocallyRingedSpace, induced_functor_map] at this,
rw ← this,
change LocallyRingedSpace.is_open_immersion _,
apply_instance
end | instance | algebraic_geometry.is_open_immersion.pullback_snd_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_of_right : is_open_immersion (pullback.fst : pullback g f ⟶ _) | begin
rw ← pullback_symmetry_hom_comp_snd,
apply_instance
end | instance | algebraic_geometry.is_open_immersion.pullback_fst_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_to_base [is_open_immersion g] :
is_open_immersion (limit.π (cospan f g) walking_cospan.one) | begin
rw ← limit.w (cospan f g) walking_cospan.hom.inl,
change is_open_immersion (_ ≫ f),
apply_instance
end | instance | algebraic_geometry.is_open_immersion.pullback_to_base | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Top_preserves_of_left :
preserves_limit (cospan f g) Scheme.forget_to_Top | begin
apply_with limits.comp_preserves_limit { instances := ff },
apply_instance,
apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,
dsimp [LocallyRingedSpace.forget_to_Top],
apply_instance
end | instance | algebraic_geometry.is_open_immersion.forget_to_Top_preserves_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Top_preserves_of_right :
preserves_limit (cospan g f) Scheme.forget_to_Top | preserves_pullback_symmetry _ _ _ | instance | algebraic_geometry.is_open_immersion.forget_to_Top_preserves_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_snd_of_left :
set.range (pullback.snd : pullback f g ⟶ Y).1.base =
(opens.map g.1.base).obj ⟨set.range f.1.base, H.base_open.open_range⟩ | begin
rw [← (show _ = (pullback.snd : pullback f g ⟶ _).1.base,
from preserves_pullback.iso_hom_snd Scheme.forget_to_Top f g), coe_comp, set.range_comp,
set.range_iff_surjective.mpr,
← @set.preimage_univ _ _ (pullback.fst : pullback f.1.base g.1.base ⟶ _),
Top.pullback_snd_image_fst_preimage, set.imag... | lemma | algebraic_geometry.is_open_immersion.range_pullback_snd_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.epi_iff_surjective",
"Top.pullback_snd_image_fst_preimage",
"set.image_univ",
"set.preimage_univ",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_fst_of_right :
set.range (pullback.fst : pullback g f ⟶ Y).1.base =
(opens.map g.1.base).obj ⟨set.range f.1.base, H.base_open.open_range⟩ | begin
rw [← (show _ = (pullback.fst : pullback g f ⟶ _).1.base,
from preserves_pullback.iso_hom_fst Scheme.forget_to_Top g f), coe_comp, set.range_comp,
set.range_iff_surjective.mpr,
← @set.preimage_univ _ _ (pullback.snd : pullback g.1.base f.1.base ⟶ _),
Top.pullback_fst_image_snd_preimage, set.imag... | lemma | algebraic_geometry.is_open_immersion.range_pullback_fst_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.epi_iff_surjective",
"Top.pullback_fst_image_snd_preimage",
"set.image_univ",
"set.preimage_univ",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_to_base_of_left :
set.range (pullback.fst ≫ f : pullback f g ⟶ Z).1.base =
set.range f.1.base ∩ set.range g.1.base | begin
rw [pullback.condition, Scheme.comp_val_base, coe_comp, set.range_comp,
range_pullback_snd_of_left, opens.map_obj, opens.coe_mk, set.image_preimage_eq_inter_range,
set.inter_comm],
end | lemma | algebraic_geometry.is_open_immersion.range_pullback_to_base_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.image_preimage_eq_inter_range",
"set.inter_comm",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_to_base_of_right :
set.range (pullback.fst ≫ g : pullback g f ⟶ Z).1.base =
set.range g.1.base ∩ set.range f.1.base | begin
rw [Scheme.comp_val_base, coe_comp, set.range_comp, range_pullback_fst_of_right, opens.map_obj,
opens.coe_mk, set.image_preimage_eq_inter_range, set.inter_comm],
end | lemma | algebraic_geometry.is_open_immersion.range_pullback_to_base_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.image_preimage_eq_inter_range",
"set.inter_comm",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (H' : set.range g.1.base ⊆ set.range f.1.base) : Y ⟶ X | LocallyRingedSpace.is_open_immersion.lift f g H' | def | algebraic_geometry.is_open_immersion.lift | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"lift",
"set.range"
] | The universal property of open immersions:
For an open immersion `f : X ⟶ Z`, given any morphism of schemes `g : Y ⟶ Z` whose topological
image is contained in the image of `f`, we can lift this morphism to a unique `Y ⟶ X` that
commutes with these maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_fac (H' : set.range g.1.base ⊆ set.range f.1.base) :
lift f g H' ≫ f = g | LocallyRingedSpace.is_open_immersion.lift_fac f g H' | lemma | algebraic_geometry.is_open_immersion.lift_fac | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"lift",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_uniq (H' : set.range g.1.base ⊆ set.range f.1.base) (l : Y ⟶ X)
(hl : l ≫ f = g) : l = lift f g H' | LocallyRingedSpace.is_open_immersion.lift_uniq f g H' l hl | lemma | algebraic_geometry.is_open_immersion.lift_uniq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"lift",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_range_eq [is_open_immersion g] (e : set.range f.1.base = set.range g.1.base) :
X ≅ Y | { hom := lift g f (le_of_eq e),
inv := lift f g (le_of_eq e.symm),
hom_inv_id' := by { rw ← cancel_mono f, simp },
inv_hom_id' := by { rw ← cancel_mono g, simp } } | def | algebraic_geometry.is_open_immersion.iso_of_range_eq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"lift",
"set.range"
] | Two open immersions with equal range are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.algebraic_geometry.Scheme.hom.opens_functor {X Y : Scheme} (f : X ⟶ Y)
[H : is_open_immersion f] :
opens X.carrier ⥤ opens Y.carrier | H.open_functor | abbreviation | algebraic_geometry.Scheme.hom.opens_functor | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | The functor `opens X ⥤ opens Y` associated with an open immersion `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.algebraic_geometry.Scheme.hom.inv_app {X Y : Scheme} (f : X ⟶ Y)
[H : is_open_immersion f] (U) :
X.presheaf.obj (op U) ⟶ Y.presheaf.obj (op (f.opens_functor.obj U)) | H.inv_app U | def | algebraic_geometry.Scheme.hom.inv_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | The isomorphism `Γ(X, U) ⟶ Γ(Y, f(U))` induced by an open immersion `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_eq_inv_app_app_of_comp_eq_aux {X Y U : Scheme} (f : Y ⟶ U) (g : U ⟶ X)
(fg : Y ⟶ X) (H : fg = f ≫ g) [h : is_open_immersion g] (V : opens U.carrier) :
(opens.map f.1.base).obj V = (opens.map fg.1.base).obj (g.opens_functor.obj V) | begin
subst H,
rw [Scheme.comp_val_base, opens.map_comp_obj],
congr' 1,
ext1,
exact (set.preimage_image_eq _ h.base_open.inj).symm
end | lemma | algebraic_geometry.is_open_immersion.app_eq_inv_app_app_of_comp_eq_aux | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_eq_inv_app_app_of_comp_eq {X Y U : Scheme} (f : Y ⟶ U) (g : U ⟶ X)
(fg : Y ⟶ X) (H : fg = f ≫ g) [h : is_open_immersion g] (V : opens U.carrier) :
f.1.c.app (op V) = g.inv_app _ ≫ fg.1.c.app _ ≫ Y.presheaf.map (eq_to_hom $
is_open_immersion.app_eq_inv_app_app_of_comp_eq_aux f g fg H V).op | begin
subst H,
rw [Scheme.comp_val_c_app, category.assoc, Scheme.hom.inv_app,
PresheafedSpace.is_open_immersion.inv_app_app_assoc,
f.val.c.naturality_assoc, Top.presheaf.pushforward_obj_map, ← functor.map_comp],
convert (category.comp_id _).symm,
convert Y.presheaf.map_id _,
end | lemma | algebraic_geometry.is_open_immersion.app_eq_inv_app_app_of_comp_eq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.presheaf.pushforward_obj_map"
] | The `fg` argument is to avoid nasty stuff about dependent types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_app {X Y U : Scheme} (f : U ⟶ Y) (g : X ⟶ Y)
[h : is_open_immersion f] (H) (V : opens U.carrier) :
(is_open_immersion.lift f g H).1.c.app (op V) = f.inv_app _ ≫ g.1.c.app _ ≫
X.presheaf.map (eq_to_hom $ is_open_immersion.app_eq_inv_app_app_of_comp_eq_aux _ _ _
(is_open_immersion.lift_fac f g H).symm ... | is_open_immersion.app_eq_inv_app_app_of_comp_eq _ _ _ _ _ | lemma | algebraic_geometry.is_open_immersion.lift_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_basic_open {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]
{U : opens X.carrier} (r : X.presheaf.obj (op U)) :
f.opens_functor.obj (X.basic_open r) = Y.basic_open (f.inv_app U r) | begin
have e := Scheme.preimage_basic_open f (f.inv_app U r),
rw [Scheme.hom.inv_app, PresheafedSpace.is_open_immersion.inv_app_app_apply,
Scheme.basic_open_res, inf_eq_right.mpr _] at e,
rw ← e,
ext1,
refine set.image_preimage_eq_inter_range.trans _,
erw set.inter_eq_left_iff_subset,
refine set.subse... | lemma | algebraic_geometry.Scheme.image_basic_open | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.image_subset_range",
"set.inter_eq_left_iff_subset",
"set.preimage_image_eq",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.opens_range {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f] : opens Y.carrier | ⟨_, H.base_open.open_range⟩ | def | algebraic_geometry.Scheme.hom.opens_range | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | The image of an open immersion as an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.restrict_functor : opens X.carrier ⥤ over X | { obj := λ U, over.mk (X.of_restrict U.open_embedding),
map := λ U V i, over.hom_mk (is_open_immersion.lift (X.of_restrict _) (X.of_restrict _)
(by { change set.range coe ⊆ set.range coe, simp_rw [subtype.range_coe], exact i.le }))
(is_open_immersion.lift_fac _ _ _),
map_id' := λ U, by begin
ext1,
d... | def | algebraic_geometry.Scheme.restrict_functor | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.range",
"subtype.range_coe"
] | The functor taking open subsets of `X` to open subschemes of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.restrict_functor_map_of_restrict {U V : opens X.carrier} (i : U ⟶ V) :
(X.restrict_functor.map i).1 ≫ X.of_restrict _ = X.of_restrict _ | is_open_immersion.lift_fac _ _ _ | lemma | algebraic_geometry.Scheme.restrict_functor_map_of_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.restrict_functor_map_base {U V : opens X.carrier} (i : U ⟶ V) :
(X.restrict_functor.map i).1.1.base = (opens.to_Top _).map i | begin
ext a,
exact (congr_arg (λ f : X.restrict U.open_embedding ⟶ X, by exact f.1.base a)
(X.restrict_functor_map_of_restrict i) : _),
end | lemma | algebraic_geometry.Scheme.restrict_functor_map_base | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.restrict_functor_map_app_aux {U V : opens X.carrier} (i : U ⟶ V) (W : opens V) :
U.open_embedding.is_open_map.functor.obj
((opens.map (X.restrict_functor.map i).1.val.base).obj W) ≤
V.open_embedding.is_open_map.functor.obj W | begin
simp only [← set_like.coe_subset_coe, is_open_map.functor_obj_coe, set.image_subset_iff,
Scheme.restrict_functor_map_base, opens.map_coe, opens.inclusion_apply],
rintros _ h,
exact ⟨_, h, rfl⟩,
end | lemma | algebraic_geometry.Scheme.restrict_functor_map_app_aux | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.image_subset_iff",
"set_like.coe_subset_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.restrict_functor_map_app {U V : opens X.carrier} (i : U ⟶ V) (W : opens V) :
(X.restrict_functor.map i).1.1.c.app (op W) = X.presheaf.map
(hom_of_le $ X.restrict_functor_map_app_aux i W).op | begin
have e₁ := Scheme.congr_app (X.restrict_functor_map_of_restrict i)
(op $ V.open_embedding.is_open_map.functor.obj W),
rw Scheme.comp_val_c_app at e₁,
have e₂ := (X.restrict_functor.map i).1.val.c.naturality (eq_to_hom W.map_functor_eq).op,
rw ← is_iso.eq_inv_comp at e₂,
dsimp at e₁ e₂ ⊢,
rw [e₂, W... | lemma | algebraic_geometry.Scheme.restrict_functor_map_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"is_open_map.functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.restrict_functor_Γ :
X.restrict_functor.op ⋙ (over.forget X).op ⋙ Scheme.Γ ≅ X.presheaf | nat_iso.of_components
(λ U, X.presheaf.map_iso ((eq_to_iso (unop U).open_embedding_obj_top).symm.op : _))
begin
intros U V i,
dsimp [-subtype.val_eq_coe, -Scheme.restrict_functor_map_left],
rw [X.restrict_functor_map_app, ← functor.map_comp, ← functor.map_comp],
congr' 1
end | def | algebraic_geometry.Scheme.restrict_functor_Γ | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"subtype.val_eq_coe"
] | The functor that restricts to open subschemes and then takes global section is
isomorphic to the structure sheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.restrict_map_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] (U : opens Y.carrier) :
X.restrict ((opens.map f.1.base).obj U).open_embedding ≅ Y.restrict U.open_embedding | begin
refine is_open_immersion.iso_of_range_eq (X.of_restrict _ ≫ f) (Y.of_restrict _) _,
dsimp [opens.inclusion],
rw [coe_comp, set.range_comp],
dsimp,
rw [subtype.range_coe, subtype.range_coe],
refine @set.image_preimage_eq _ _ f.1.base U.1 _,
rw ← Top.epi_iff_surjective,
apply_instance
end | abbreviation | algebraic_geometry.Scheme.restrict_map_iso | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.epi_iff_surjective",
"open_embedding",
"set.image_preimage_eq",
"set.range_comp",
"subtype.range_coe"
] | The restriction of an isomorphism onto an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.open_cover.pullback_cover {X : Scheme} (𝒰 : X.open_cover) {W : Scheme} (f : W ⟶ X) :
W.open_cover | { J := 𝒰.J,
obj := λ x, pullback f (𝒰.map x),
map := λ x, pullback.fst,
f := λ x, 𝒰.f (f.1.base x),
covers := λ x, begin
rw ← (show _ = (pullback.fst : pullback f (𝒰.map (𝒰.f (f.1.base x))) ⟶ _).1.base,
from preserves_pullback.iso_hom_fst Scheme.forget_to_Top f
(𝒰.map (𝒰.f (f.1.base x))))... | def | algebraic_geometry.Scheme.open_cover.pullback_cover | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.epi_iff_surjective",
"Top.pullback_fst_range",
"set.image_univ",
"set.range_comp"
] | Given an open cover on `X`, we may pull them back along a morphism `W ⟶ X` to obtain
an open cover of `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.open_cover.Union_range {X : Scheme} (𝒰 : X.open_cover) :
(⋃ i, set.range (𝒰.map i).1.base) = set.univ | begin
rw set.eq_univ_iff_forall,
intros x,
rw set.mem_Union,
exact ⟨𝒰.f x, 𝒰.covers x⟩
end | lemma | algebraic_geometry.Scheme.open_cover.Union_range | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.eq_univ_iff_forall",
"set.mem_Union",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.open_cover.supr_opens_range {X : Scheme} (𝒰 : X.open_cover) :
(⨆ i, (𝒰.map i).opens_range) = ⊤ | opens.ext $ by { rw opens.coe_supr, exact 𝒰.Union_range } | lemma | algebraic_geometry.Scheme.open_cover.supr_opens_range | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.open_cover.compact_space {X : Scheme} (𝒰 : X.open_cover) [finite 𝒰.J]
[H : ∀ i, compact_space (𝒰.obj i).carrier] : compact_space X.carrier | begin
casesI nonempty_fintype 𝒰.J,
rw [← is_compact_univ_iff, ← 𝒰.Union_range],
apply is_compact_Union,
intro i,
rw is_compact_iff_compact_space,
exact @@homeomorph.compact_space _ _ (H i)
(Top.homeo_of_iso (as_iso (is_open_immersion.iso_of_range_eq (𝒰.map i)
(X.of_restrict (opens.open_embedding ... | lemma | algebraic_geometry.Scheme.open_cover.compact_space | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.homeo_of_iso",
"compact_space",
"finite",
"homeomorph.compact_space",
"is_compact_Union",
"is_compact_iff_compact_space",
"is_compact_univ_iff",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.open_cover.inter {X : Scheme.{u}} (𝒰₁ : Scheme.open_cover.{v₁} X)
(𝒰₂ : Scheme.open_cover.{v₂} X) : X.open_cover | { J := 𝒰₁.J × 𝒰₂.J,
obj := λ ij, pullback (𝒰₁.map ij.1) (𝒰₂.map ij.2),
map := λ ij, pullback.fst ≫ 𝒰₁.map ij.1,
f := λ x, ⟨𝒰₁.f x, 𝒰₂.f x⟩,
covers := λ x, by { rw is_open_immersion.range_pullback_to_base_of_left,
exact ⟨𝒰₁.covers x, 𝒰₂.covers x⟩ } } | def | algebraic_geometry.Scheme.open_cover.inter | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | Given open covers `{ Uᵢ }` and `{ Uⱼ }`, we may form the open cover `{ Uᵢ ∩ Uⱼ }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.open_cover_of_supr_eq_top {s : Type*} (X : Scheme) (U : s → opens X.carrier)
(hU : (⨆ i, U i) = ⊤) : X.open_cover | { J := s,
obj := λ i, X.restrict (U i).open_embedding,
map := λ i, X.of_restrict (U i).open_embedding,
f := λ x, begin
have : x ∈ ⨆ i, U i := hU.symm ▸ (show x ∈ (⊤ : opens X.carrier), by triv),
exact (opens.mem_supr.mp this).some,
end,
covers := λ x, begin
erw subtype.range_coe,
have : x ∈ ⨆ ... | def | algebraic_geometry.Scheme.open_cover_of_supr_eq_top | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"open_embedding",
"subtype.range_coe"
] | If `U` is a family of open sets that covers `X`, then `X.restrict U` forms an `X.open_cover`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_restrict_iso_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
pullback f (Y.of_restrict U.open_embedding) ≅
X.restrict ((opens.map f.1.base).obj U).open_embedding | begin
refine is_open_immersion.iso_of_range_eq pullback.fst (X.of_restrict _) _,
rw is_open_immersion.range_pullback_fst_of_right,
dsimp [opens.inclusion],
rw [subtype.range_coe, subtype.range_coe],
refl,
end | def | algebraic_geometry.pullback_restrict_iso_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"open_embedding",
"subtype.range_coe"
] | Given a morphism `f : X ⟶ Y` and an open set `U ⊆ Y`, we have `X ×[Y] U ≅ X |_{f ⁻¹ U}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_restrict_iso_restrict_inv_fst {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
(pullback_restrict_iso_restrict f U).inv ≫ pullback.fst = X.of_restrict _ | by { delta pullback_restrict_iso_restrict, simp } | lemma | algebraic_geometry.pullback_restrict_iso_restrict_inv_fst | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_restrict_iso_restrict_hom_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
(pullback_restrict_iso_restrict f U).hom ≫ X.of_restrict _ = pullback.fst | by { delta pullback_restrict_iso_restrict, simp } | lemma | algebraic_geometry.pullback_restrict_iso_restrict_hom_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
X.restrict ((opens.map f.1.base).obj U).open_embedding ⟶ Y.restrict U.open_embedding | (pullback_restrict_iso_restrict f U).inv ≫ pullback.snd | def | algebraic_geometry.morphism_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"open_embedding"
] | The restriction of a morphism `X ⟶ Y` onto `X |_{f ⁻¹ U} ⟶ Y |_ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_restrict_iso_restrict_hom_morphism_restrict {X Y : Scheme} (f : X ⟶ Y)
(U : opens Y.carrier) :
(pullback_restrict_iso_restrict f U).hom ≫ f ∣_ U = pullback.snd | iso.hom_inv_id_assoc _ _ | lemma | algebraic_geometry.pullback_restrict_iso_restrict_hom_morphism_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_ι {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
f ∣_ U ≫ Y.of_restrict U.open_embedding = X.of_restrict _ ≫ f | by { delta morphism_restrict,
rw [category.assoc, pullback.condition.symm, pullback_restrict_iso_restrict_inv_fst_assoc] } | lemma | algebraic_geometry.morphism_restrict_ι | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pullback_morphism_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
is_pullback (f ∣_ U) (X.of_restrict _) (Y.of_restrict _) f | begin
delta morphism_restrict,
nth_rewrite 0 ← category.id_comp f,
refine (is_pullback.of_horiz_is_iso ⟨_⟩).paste_horiz
(is_pullback.of_has_pullback f (Y.of_restrict U.open_embedding)).flip,
rw [pullback_restrict_iso_restrict_inv_fst, category.comp_id],
end | lemma | algebraic_geometry.is_pullback_morphism_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U : opens Z.carrier) :
(f ≫ g) ∣_ U = (f ∣_ ((opens.map g.val.base).obj U) ≫ g ∣_ U : _) | begin
delta morphism_restrict,
rw ← pullback_right_pullback_fst_iso_inv_snd_snd,
simp_rw ← category.assoc,
congr' 1,
rw ← cancel_mono pullback.fst,
simp_rw category.assoc,
rw [pullback_restrict_iso_restrict_inv_fst, pullback_right_pullback_fst_iso_inv_snd_fst,
← pullback.condition, pullback_restrict_i... | lemma | algebraic_geometry.morphism_restrict_comp | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_base_coe {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) (x) :
@coe U Y.carrier _ ((f ∣_ U).1.base x) = f.1.base x.1 | congr_arg (λ f, PresheafedSpace.hom.base (LocallyRingedSpace.hom.val f) x) (morphism_restrict_ι f U) | lemma | algebraic_geometry.morphism_restrict_base_coe | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_val_base {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
⇑(f ∣_ U).1.base = U.1.restrict_preimage f.1.base | funext (λ x, subtype.ext (morphism_restrict_base_coe f U x)) | lemma | algebraic_geometry.morphism_restrict_val_base | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_morphism_restrict_preimage {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier)
(V : opens U) :
((opens.map f.val.base).obj U).open_embedding.is_open_map.functor.obj
((opens.map (f ∣_ U).val.base).obj V) =
(opens.map f.val.base).obj (U.open_embedding.is_open_map.functor.obj V) | begin
ext1,
ext x,
split,
{ rintro ⟨⟨x, hx⟩, (hx' : (f ∣_ U).1.base _ ∈ _), rfl⟩,
refine ⟨⟨_, hx⟩, _, rfl⟩,
convert hx',
ext1,
exact (morphism_restrict_base_coe f U ⟨x, hx⟩).symm },
{ rintro ⟨⟨x, hx⟩, hx', (rfl : x = _)⟩,
refine ⟨⟨_, hx⟩, (_: ((f ∣_ U).1.base ⟨x, hx⟩) ∈ V.1), rfl⟩,
con... | lemma | algebraic_geometry.image_morphism_restrict_preimage | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_c_app {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) (V : opens U) :
(f ∣_ U).1.c.app (op V) = f.1.c.app (op (U.open_embedding.is_open_map.functor.obj V)) ≫
X.presheaf.map (eq_to_hom (image_morphism_restrict_preimage f U V)).op | begin
have := Scheme.congr_app (morphism_restrict_ι f U)
(op (U.open_embedding.is_open_map.functor.obj V)),
rw [Scheme.comp_val_c_app, Scheme.comp_val_c_app_assoc] at this,
have e : (opens.map U.inclusion).obj (U.open_embedding.is_open_map.functor.obj V) = V,
{ ext1, exact set.preimage_image_eq _ subtype.co... | lemma | algebraic_geometry.morphism_restrict_c_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.preimage_image_eq",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map_morphism_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
Scheme.Γ.map (f ∣_ U).op = Y.presheaf.map (eq_to_hom $ U.open_embedding_obj_top.symm).op ≫
f.1.c.app (op U) ≫
X.presheaf.map (eq_to_hom $ ((opens.map f.val.base).obj U).open_embedding_obj_top).op | begin
rw [Scheme.Γ_map_op, morphism_restrict_c_app f U ⊤, f.val.c.naturality_assoc],
erw ← X.presheaf.map_comp,
congr,
end | lemma | algebraic_geometry.Γ_map_morphism_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_opens_range
{X Y U : Scheme} (f : X ⟶ Y) (g : U ⟶ Y) [hg : is_open_immersion g] :
arrow.mk (f ∣_ g.opens_range) ≅ arrow.mk (pullback.snd : pullback f g ⟶ _) | begin
let V : opens Y.carrier := g.opens_range,
let e := is_open_immersion.iso_of_range_eq g (Y.of_restrict V.open_embedding)
(by exact subtype.range_coe.symm),
let t : pullback f g ⟶ pullback f (Y.of_restrict V.open_embedding) :=
pullback.map _ _ _ _ (𝟙 _) e.hom (𝟙 _) (by rw [category.comp_id, category... | def | algebraic_geometry.morphism_restrict_opens_range | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | Restricting a morphism onto the the image of an open immersion is isomorphic to the base change
along the immersion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_restrict_eq {X Y : Scheme} (f : X ⟶ Y) {U V : opens Y.carrier} (e : U = V) :
arrow.mk (f ∣_ U) ≅ arrow.mk (f ∣_ V) | eq_to_iso (by subst e) | def | algebraic_geometry.morphism_restrict_eq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_restrict_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) (V : opens U) :
arrow.mk (f ∣_ U ∣_ V) ≅ arrow.mk (f ∣_ (U.open_embedding.is_open_map.functor.obj V)) | begin
have : (f ∣_ U ∣_ V) ≫ (iso.refl _).hom =
(as_iso $ (pullback_restrict_iso_restrict (f ∣_ U) V).inv ≫ (pullback_symmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _)
((pullback_restrict_iso_restrict f U).inv ≫ (pullback_symmetry _ _).hom) (𝟙 _)
((category.comp_id _).trans (category.id_comp _).symm) (... | def | algebraic_geometry.morphism_restrict_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.range_comp",
"subtype.range_coe"
] | Restricting a morphism twice is isomorpic to one restriction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_restrict_restrict_basic_open {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier)
(r : Y.presheaf.obj (op U)) :
arrow.mk (f ∣_ U ∣_ (Y.restrict _).basic_open
(Y.presheaf.map (eq_to_hom U.open_embedding_obj_top).op r)) ≅ arrow.mk (f ∣_ Y.basic_open r) | begin
refine morphism_restrict_restrict _ _ _ ≪≫ morphism_restrict_eq _ _,
have e := Scheme.preimage_basic_open (Y.of_restrict U.open_embedding) r,
erw [Scheme.of_restrict_val_c_app, opens.adjunction_counit_app_self, eq_to_hom_op] at e,
rw [← (Y.restrict U.open_embedding).basic_open_res_eq _
(eq_to_hom U.in... | def | algebraic_geometry.morphism_restrict_restrict_basic_open | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"set.image_preimage_eq_inter_range",
"set.inter_eq_left_iff_subset",
"subtype.range_coe"
] | Restricting a morphism twice onto a basic open set is isomorphic to one restriction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_restrict_stalk_map {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) (x) :
arrow.mk (PresheafedSpace.stalk_map (f ∣_ U).1 x) ≅
arrow.mk (PresheafedSpace.stalk_map f.1 x.1) | begin
fapply arrow.iso_mk',
{ refine Y.restrict_stalk_iso U.open_embedding ((f ∣_ U).1 x) ≪≫ Top.presheaf.stalk_congr _ _,
apply inseparable.of_eq,
exact morphism_restrict_base_coe f U x },
{ exact X.restrict_stalk_iso _ _ },
{ apply Top.presheaf.stalk_hom_ext,
intros V hxV,
simp only [Top.presh... | def | algebraic_geometry.morphism_restrict_stalk_map | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/Scheme.lean | [
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.Scheme",
"category_theory.limits.shapes.comm_sq"
] | [
"Top.presheaf.germ_res",
"Top.presheaf.stalk_congr",
"Top.presheaf.stalk_hom_ext",
"inseparable.of_eq"
] | The stalk map of a restriction of a morphism is isomorphic to the stalk map of the original map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_data extends glue_data (PresheafedSpace.{v} C) | (f_open : ∀ i j, is_open_immersion (f i j)) | structure | algebraic_geometry.PresheafedSpace.glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | A family of gluing data consists of
1. An index type `J`
2. A presheafed space `U i` for each `i : J`.
3. A presheafed space `V i j` for each `i j : J`.
(Note that this is `J × J → PresheafedSpace C` rather than `J → J → PresheafedSpace C` to
connect to the limits library easier.)
4. An open immersion `f i j : V i ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Top_glue_data : Top.glue_data | { f_open := λ i j, (D.f_open i j).base_open,
to_glue_data := 𝖣 .map_glue_data (forget C) } | abbreviation | algebraic_geometry.PresheafedSpace.glue_data.to_Top_glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.glue_data"
] | The glue data of topological spaces associated to a family of glue data of PresheafedSpaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_open_embedding [has_limits C] (i : D.J) : open_embedding (𝖣 .ι i).base | begin
rw ← (show _ = (𝖣 .ι i).base, from 𝖣 .ι_glued_iso_inv (PresheafedSpace.forget _) _),
exact open_embedding.comp (Top.homeo_of_iso
(𝖣 .glued_iso (PresheafedSpace.forget _)).symm).open_embedding
(D.to_Top_glue_data.ι_open_embedding i)
end | lemma | algebraic_geometry.PresheafedSpace.glue_data.ι_open_embedding | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.homeo_of_iso",
"open_embedding",
"open_embedding.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_base (i j k : D.J) (S : set (D.V (i, j)).carrier) :
(π₂ i, j, k) '' ((π₁ i, j, k) ⁻¹' S) = D.f i k ⁻¹' (D.f i j '' S) | begin
have eq₁ : _ = (π₁ i, j, k).base := preserves_pullback.iso_hom_fst (forget C) _ _,
have eq₂ : _ = (π₂ i, j, k).base := preserves_pullback.iso_hom_snd (forget C) _ _,
rw [coe_to_fun_eq, coe_to_fun_eq, ← eq₁, ← eq₂, coe_comp, set.image_comp, coe_comp,
set.preimage_comp, set.image_preimage_eq, Top.pullback... | lemma | algebraic_geometry.PresheafedSpace.glue_data.pullback_base | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.epi_iff_surjective",
"Top.pullback_snd_image_fst_preimage",
"set.image_comp",
"set.image_preimage_eq",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
f_inv_app_f_app (i j k : D.J) (U : (opens (D.V (i, j)).carrier)) :
(D.f_open i j).inv_app U ≫ (D.f i k).c.app _ =
(π₁ i, j, k).c.app (op U) ≫ (π₂⁻¹ i, j, k) (unop _) ≫ (D.V _).presheaf.map (eq_to_hom
(begin
delta is_open_immersion.open_functor,
dsimp only [functor.op, is_open_map.functor, o... | begin
have := PresheafedSpace.congr_app (@pullback.condition _ _ _ _ _ (D.f i j) (D.f i k) _),
dsimp only [comp_c_app] at this,
rw [← cancel_epi (inv ((D.f_open i j).inv_app U)), is_iso.inv_hom_id_assoc,
is_open_immersion.inv_inv_app],
simp_rw category.assoc,
erw [(π₁ i, j, k).c.naturality_assoc,
reas... | lemma | algebraic_geometry.PresheafedSpace.glue_data.f_inv_app_f_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"is_open_map.functor"
] | The red and the blue arrows in  commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_inv_app_t_app' (i j k : D.J) (U : opens (pullback (D.f i j) (D.f i k)).carrier) :
∃ eq, (π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ ≫ (D.V (k, i)).presheaf.map (eq_to_hom eq) =
(D.t' k i j).c.app _ ≫ (π₁⁻¹ k, j, i) (unop _) | begin
split,
rw [← is_iso.eq_inv_comp, is_open_immersion.inv_inv_app, category.assoc,
(D.t' k i j).c.naturality_assoc],
simp_rw ← category.assoc,
erw ← comp_c_app,
rw [congr_app (D.t_fac k i j), comp_c_app],
simp_rw category.assoc,
erw [is_open_immersion.inv_naturality, is_open_immersion.inv_naturalit... | lemma | algebraic_geometry.PresheafedSpace.glue_data.snd_inv_app_t_app' | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.id_app"
] | We can prove the `eq` along with the lemma. Thus this is bundled together here, and the
lemma itself is separated below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_inv_app_t_app (i j k : D.J) (U : opens (pullback (D.f i j) (D.f i k)).carrier) :
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ = (D.t' k i j).c.app _ ≫ (π₁⁻¹ k, j, i) (unop _) ≫
(D.V (k, i)).presheaf.map (eq_to_hom (D.snd_inv_app_t_app' i j k U).some.symm) | begin
have e := (D.snd_inv_app_t_app' i j k U).some_spec,
reassoc! e,
rw ← e,
simp [eq_to_hom_map],
end | lemma | algebraic_geometry.PresheafedSpace.glue_data.snd_inv_app_t_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The red and the blue arrows in  commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_image_preimage_eq (i j : D.J) (U : opens (D.U i).carrier) :
(opens.map (𝖣 .ι j).base).obj ((D.ι_open_embedding i).is_open_map.functor.obj U) =
(D.f_open j i).open_functor.obj ((opens.map (𝖣 .t j i).base).obj
((opens.map (𝖣 .f i j).base).obj U)) | begin
ext1,
dsimp only [opens.map_coe, is_open_map.functor_obj_coe],
rw [← (show _ = (𝖣 .ι i).base, from 𝖣 .ι_glued_iso_inv (PresheafedSpace.forget _) i),
← (show _ = (𝖣 .ι j).base, from 𝖣 .ι_glued_iso_inv (PresheafedSpace.forget _) j),
coe_comp, coe_comp, set.image_comp, set.preimage_comp, set.preima... | lemma | algebraic_geometry.PresheafedSpace.glue_data.ι_image_preimage_eq | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.homeo_of_iso",
"Top.mono_iff_injective",
"homeomorph.bijective",
"set.eq_preimage_iff_image_eq",
"set.image_comp",
"set.preimage_comp",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_image_preimage_map (i j : D.J) (U : opens (D.U i).carrier) :
(D.U i).presheaf.obj (op U) ⟶ (D.U j).presheaf.obj _ | (D.f i j).c.app (op U) ≫ (D.t j i).c.app _ ≫ (D.f_open j i).inv_app (unop _) ≫
(𝖣 .U j).presheaf.map (eq_to_hom (D.ι_image_preimage_eq i j U)).op | def | algebraic_geometry.PresheafedSpace.glue_data.opens_image_preimage_map | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | (Implementation). The map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{U_j}, 𝖣.ι j ⁻¹' (𝖣.ι i '' U))` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens_image_preimage_map_app' (i j k : D.J) (U : opens (D.U i).carrier) :
∃ eq, D.opens_image_preimage_map i j U ≫ (D.f j k).c.app _ =
((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫ (π₂⁻¹ j, i, k) (unop _) ≫
(D.V (j, k)).presheaf.map (eq_to_hom eq) | begin
split,
delta opens_image_preimage_map,
simp_rw category.assoc,
rw [(D.f j k).c.naturality, f_inv_app_f_app_assoc],
erw ← (D.V (j, k)).presheaf.map_comp,
simp_rw ← category.assoc,
erw [← comp_c_app, ← comp_c_app],
simp_rw category.assoc,
dsimp only [functor.op, unop_op, quiver.hom.unop_op],
rw ... | lemma | algebraic_geometry.PresheafedSpace.glue_data.opens_image_preimage_map_app' | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_image_preimage_map_app (i j k : D.J) (U : opens (D.U i).carrier) :
D.opens_image_preimage_map i j U ≫ (D.f j k).c.app _ =
((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫ (π₂⁻¹ j, i, k) (unop _) ≫
(D.V (j, k)).presheaf.map (eq_to_hom ((opens_image_preimage_map_app' D i j k U).some)) | (opens_image_preimage_map_app' D i j k U).some_spec | lemma | algebraic_geometry.PresheafedSpace.glue_data.opens_image_preimage_map_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The red and the blue arrows in  commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens_image_preimage_map_app_assoc (i j k : D.J) (U : opens (D.U i).carrier)
{X' : C} (f' : _ ⟶ X') :
D.opens_image_preimage_map i j U ≫ (D.f j k).c.app _ ≫ f' =
((π₁ j, i, k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫ (π₂⁻¹ j, i, k) (unop _) ≫
(D.V (j, k)).presheaf.map (eq_to_hom ((opens_image_preimage_map_app' D... | by simpa only [category.assoc]
using congr_arg (λ g, g ≫ f') (opens_image_preimage_map_app D i j k U) | lemma | algebraic_geometry.PresheafedSpace.glue_data.opens_image_preimage_map_app_assoc | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_over_open {i : D.J} (U : opens (D.U i).carrier) :
(walking_multispan _ _)ᵒᵖ ⥤ C | componentwise_diagram 𝖣 .diagram.multispan ((D.ι_open_embedding i).is_open_map.functor.obj U) | abbreviation | algebraic_geometry.PresheafedSpace.glue_data.diagram_over_open | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | (Implementation) Given an open subset of one of the spaces `U ⊆ Uᵢ`, the sheaf component of
the image `ι '' U` in the glued space is the limit of this diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_over_open_π {i : D.J} (U : opens (D.U i).carrier) (j : D.J) | limit.π (D.diagram_over_open U) (op (walking_multispan.right j)) | abbreviation | algebraic_geometry.PresheafedSpace.glue_data.diagram_over_open_π | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | (Implementation)
The projection from the limit of `diagram_over_open` to a component of `D.U j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_inv_app_π_app {i : D.J} (U : opens (D.U i).carrier) (j) :
(𝖣 .U i).presheaf.obj (op U) ⟶ (D.diagram_over_open U).obj (op j) | begin
rcases j with (⟨j, k⟩|j),
{ refine D.opens_image_preimage_map i j U ≫ (D.f j k).c.app _ ≫
(D.V (j, k)).presheaf.map (eq_to_hom _),
rw [functor.op_obj],
congr' 1, ext1,
dsimp only [functor.op_obj, opens.map_coe, unop_op, is_open_map.functor_obj_coe],
rw set.preimage_preimage,
change (... | def | algebraic_geometry.PresheafedSpace.glue_data.ι_inv_app_π_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"set.preimage_preimage"
] | (Implementation) We construct the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing
diagram. We will lift these maps into `ι_inv_app`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_inv_app {i : D.J} (U : opens (D.U i).carrier) :
(D.U i).presheaf.obj (op U) ⟶ limit (D.diagram_over_open U) | limit.lift (D.diagram_over_open U)
{ X := (D.U i).presheaf.obj (op U),
π := { app := λ j, D.ι_inv_app_π_app U (unop j),
naturality' := λ X Y f', begin
induction X using opposite.rec,
induction Y using opposite.rec,
let f : Y ⟶ X := f'.unop, have : f' = f.op := rfl, clear_value f, subst this,
rcases ... | def | algebraic_geometry.PresheafedSpace.glue_data.ι_inv_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"opposite.rec"
] | (Implementation) The natural map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, 𝖣.ι i '' U)`.
This forms the inverse of `(𝖣.ι i).c.app (op U)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_inv_app_π {i : D.J} (U : opens (D.U i).carrier) :
∃ eq, D.ι_inv_app U ≫ D.diagram_over_open_π U i = (D.U i).presheaf.map (eq_to_hom eq) | begin
split,
delta ι_inv_app,
rw limit.lift_π,
change D.opens_image_preimage_map i i U = _,
dsimp [opens_image_preimage_map],
rw [congr_app (D.t_id _), id_c_app, ← functor.map_comp],
erw [is_open_immersion.inv_naturality_assoc, is_open_immersion.app_inv_app'_assoc],
simp only [eq_to_hom_op, eq_to_hom_tr... | lemma | algebraic_geometry.PresheafedSpace.glue_data.ι_inv_app_π | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top.epi_iff_surjective"
] | `ι_inv_app` is the left inverse of `D.ι i` on `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_inv_app_π_eq_map {i : D.J} (U : opens (D.U i).carrier) | (D.U i).presheaf.map (eq_to_iso (D.ι_inv_app_π U).some).inv | abbreviation | algebraic_geometry.PresheafedSpace.glue_data.ι_inv_app_π_eq_map | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The `eq_to_hom` given by `ι_inv_app_π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_ι_inv_app_π (i j : D.J) (U : opens (D.U i).carrier) :
D.diagram_over_open_π U i ≫ D.ι_inv_app_π_eq_map U ≫ D.ι_inv_app U ≫
D.diagram_over_open_π U j = D.diagram_over_open_π U j | begin
rw ← cancel_mono ((componentwise_diagram 𝖣 .diagram.multispan _).map
(quiver.hom.op (walking_multispan.hom.snd (i, j))) ≫ (𝟙 _)),
simp_rw category.assoc,
rw limit.w_assoc,
erw limit.lift_π_assoc,
rw [category.comp_id, category.comp_id],
change _ ≫ _ ≫ (_ ≫ _) ≫ _ = _,
rw [congr_app (D.t_id _),... | lemma | algebraic_geometry.PresheafedSpace.glue_data.π_ι_inv_app_π | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"quiver.hom.op"
] | `ι_inv_app` is the right inverse of `D.ι i` on `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_ι_inv_app_eq_id (i : D.J) (U : opens (D.U i).carrier) :
D.diagram_over_open_π U i ≫ D.ι_inv_app_π_eq_map U ≫ D.ι_inv_app U = 𝟙 _ | begin
ext j,
induction j using opposite.rec,
rcases j with (⟨j, k⟩|⟨j⟩),
{ rw [← limit.w (componentwise_diagram 𝖣 .diagram.multispan _)
(quiver.hom.op (walking_multispan.hom.fst (j, k))), ← category.assoc, category.id_comp],
congr' 1,
simp_rw category.assoc,
apply π_ι_inv_app_π },
{ simp_rw... | lemma | algebraic_geometry.PresheafedSpace.glue_data.π_ι_inv_app_eq_id | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"opposite.rec",
"quiver.hom.op"
] | `ι_inv_app` is the inverse of `D.ι i` on `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
componentwise_diagram_π_is_iso (i : D.J) (U : opens (D.U i).carrier) :
is_iso (D.diagram_over_open_π U i) | begin
use D.ι_inv_app_π_eq_map U ≫ D.ι_inv_app U,
split,
{ apply π_ι_inv_app_eq_id },
{ rw [category.assoc, (D.ι_inv_app_π _).some_spec],
exact iso.inv_hom_id ((D.to_glue_data.U i).presheaf.map_iso (eq_to_iso _)) }
end | instance | algebraic_geometry.PresheafedSpace.glue_data.componentwise_diagram_π_is_iso | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_is_open_immersion (i : D.J) :
is_open_immersion (𝖣 .ι i) | { base_open := D.ι_open_embedding i,
c_iso := λ U, by { erw ← colimit_presheaf_obj_iso_componentwise_limit_hom_π, apply_instance } } | instance | algebraic_geometry.PresheafedSpace.glue_data.ι_is_open_immersion | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone_is_limit (i j : D.J) : is_limit (𝖣 .V_pullback_cone i j) | pullback_cone.is_limit_aux' _ $ λ s,
begin
refine ⟨_, _, _, _⟩,
{ refine PresheafedSpace.is_open_immersion.lift (D.f i j) s.fst _,
erw ← D.to_Top_glue_data.preimage_range j i,
have : s.fst.base ≫ D.to_Top_glue_data.to_glue_data.ι i =
s.snd.base ≫ D.to_Top_glue_data.to_glue_data.ι j,
{ rw [← 𝖣 .ι_... | def | algebraic_geometry.PresheafedSpace.glue_data.V_pullback_cone_is_limit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"set.image_comp",
"set.image_subset_iff",
"set.image_subset_range",
"set.image_univ"
] | The following diagram is a pullback, i.e. `Vᵢⱼ` is the intersection of `Uᵢ` and `Uⱼ` in `X`.
Vᵢⱼ ⟶ Uᵢ
| |
↓ ↓
Uⱼ ⟶ X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_jointly_surjective (x : 𝖣 .glued) :
∃ (i : D.J) (y : D.U i), (𝖣 .ι i).base y = x | 𝖣 .ι_jointly_surjective (PresheafedSpace.forget _ ⋙ category_theory.forget Top) x | lemma | algebraic_geometry.PresheafedSpace.glue_data.ι_jointly_surjective | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top",
"category_theory.forget"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_data extends glue_data (SheafedSpace.{v} C) | (f_open : ∀ i j, SheafedSpace.is_open_immersion (f i j)) | structure | algebraic_geometry.SheafedSpace.glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | A family of gluing data consists of
1. An index type `J`
2. A sheafed space `U i` for each `i : J`.
3. A sheafed space `V i j` for each `i j : J`.
(Note that this is `J × J → SheafedSpace C` rather than `J → J → SheafedSpace C` to
connect to the limits library easier.)
4. An open immersion `f i j : V i j ⟶ U i` for... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_PresheafedSpace_glue_data : PresheafedSpace.glue_data C | { f_open := D.f_open,
to_glue_data := 𝖣 .map_glue_data forget_to_PresheafedSpace } | abbreviation | algebraic_geometry.SheafedSpace.glue_data.to_PresheafedSpace_glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The glue data of presheafed spaces associated to a family of glue data of sheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_PresheafedSpace : 𝖣 .glued.to_PresheafedSpace ≅
D.to_PresheafedSpace_glue_data.to_glue_data.glued | 𝖣 .glued_iso forget_to_PresheafedSpace | abbreviation | algebraic_geometry.SheafedSpace.glue_data.iso_PresheafedSpace | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The gluing as sheafed spaces is isomorphic to the gluing as presheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_iso_PresheafedSpace_inv (i : D.J) :
D.to_PresheafedSpace_glue_data.to_glue_data.ι i ≫ D.iso_PresheafedSpace.inv = 𝖣 .ι i | 𝖣 .ι_glued_iso_inv _ _ | lemma | algebraic_geometry.SheafedSpace.glue_data.ι_iso_PresheafedSpace_inv | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_is_open_immersion (i : D.J) :
is_open_immersion (𝖣 .ι i) | by { rw ← D.ι_iso_PresheafedSpace_inv, apply_instance } | instance | algebraic_geometry.SheafedSpace.glue_data.ι_is_open_immersion | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_jointly_surjective (x : 𝖣 .glued) :
∃ (i : D.J) (y : D.U i), (𝖣 .ι i).base y = x | 𝖣 .ι_jointly_surjective (SheafedSpace.forget _ ⋙ category_theory.forget Top) x | lemma | algebraic_geometry.SheafedSpace.glue_data.ι_jointly_surjective | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top",
"category_theory.forget"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone_is_limit (i j : D.J) : is_limit (𝖣 .V_pullback_cone i j) | 𝖣 .V_pullback_cone_is_limit_of_map forget_to_PresheafedSpace i j
(D.to_PresheafedSpace_glue_data.V_pullback_cone_is_limit _ _) | def | algebraic_geometry.SheafedSpace.glue_data.V_pullback_cone_is_limit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The following diagram is a pullback, i.e. `Vᵢⱼ` is the intersection of `Uᵢ` and `Uⱼ` in `X`.
Vᵢⱼ ⟶ Uᵢ
| |
↓ ↓
Uⱼ ⟶ X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glue_data extends glue_data LocallyRingedSpace | (f_open : ∀ i j, LocallyRingedSpace.is_open_immersion (f i j)) | structure | algebraic_geometry.LocallyRingedSpace.glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | A family of gluing data consists of
1. An index type `J`
2. A locally ringed space `U i` for each `i : J`.
3. A locally ringed space `V i j` for each `i j : J`.
(Note that this is `J × J → LocallyRingedSpace` rather than `J → J → LocallyRingedSpace` to
connect to the limits library easier.)
4. An open immersion `f ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_SheafedSpace_glue_data : SheafedSpace.glue_data CommRing | { f_open := D.f_open,
to_glue_data := 𝖣 .map_glue_data forget_to_SheafedSpace } | abbreviation | algebraic_geometry.LocallyRingedSpace.glue_data.to_SheafedSpace_glue_data | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"CommRing"
] | The glue data of ringed spaces associated to a family of glue data of locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_SheafedSpace : 𝖣 .glued.to_SheafedSpace ≅
D.to_SheafedSpace_glue_data.to_glue_data.glued | 𝖣 .glued_iso forget_to_SheafedSpace | abbreviation | algebraic_geometry.LocallyRingedSpace.glue_data.iso_SheafedSpace | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The gluing as locally ringed spaces is isomorphic to the gluing as ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_iso_SheafedSpace_inv (i : D.J) :
D.to_SheafedSpace_glue_data.to_glue_data.ι i ≫ D.iso_SheafedSpace.inv = (𝖣 .ι i).1 | 𝖣 .ι_glued_iso_inv forget_to_SheafedSpace i | lemma | algebraic_geometry.LocallyRingedSpace.glue_data.ι_iso_SheafedSpace_inv | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_is_open_immersion (i : D.J) :
is_open_immersion (𝖣 .ι i) | by { delta is_open_immersion, rw ← D.ι_iso_SheafedSpace_inv,
apply PresheafedSpace.is_open_immersion.comp } | instance | algebraic_geometry.LocallyRingedSpace.glue_data.ι_is_open_immersion | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_jointly_surjective (x : 𝖣 .glued) :
∃ (i : D.J) (y : D.U i), (𝖣 .ι i).1.base y = x | 𝖣 .ι_jointly_surjective ((LocallyRingedSpace.forget_to_SheafedSpace ⋙
SheafedSpace.forget _) ⋙ forget Top) x | lemma | algebraic_geometry.LocallyRingedSpace.glue_data.ι_jointly_surjective | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone_is_limit (i j : D.J) : is_limit (𝖣 .V_pullback_cone i j) | 𝖣 .V_pullback_cone_is_limit_of_map forget_to_SheafedSpace i j
(D.to_SheafedSpace_glue_data.V_pullback_cone_is_limit _ _) | def | algebraic_geometry.LocallyRingedSpace.glue_data.V_pullback_cone_is_limit | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/gluing.lean | [
"topology.gluing",
"algebraic_geometry.open_immersion.basic",
"algebraic_geometry.locally_ringed_space.has_colimits"
] | [] | The following diagram is a pullback, i.e. `Vᵢⱼ` is the intersection of `Uᵢ` and `Uⱼ` in `X`.
Vᵢⱼ ⟶ Uᵢ
| |
↓ ↓
Uⱼ ⟶ X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_id_c_app (F : J ⥤ PresheafedSpace.{v} C) (j) (U) :
(F.map (𝟙 j)).c.app (op U) =
(pushforward.id (F.obj j).presheaf).inv.app (op U) ≫
(pushforward_eq (by { simp, refl }) (F.obj j).presheaf).hom.app (op U) | begin
cases U,
dsimp,
simp [PresheafedSpace.congr_app (F.map_id j)],
refl,
end | lemma | algebraic_geometry.PresheafedSpace.map_id_c_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_c_app (F : J ⥤ PresheafedSpace.{v} C) {j₁ j₂ j₃} (f : j₁ ⟶ j₂) (g : j₂ ⟶ j₃) (U) :
(F.map (f ≫ g)).c.app (op U) =
(F.map g).c.app (op U) ≫
(pushforward_map (F.map g).base (F.map f).c).app (op U) ≫
(pushforward.comp (F.obj j₁).presheaf (F.map f).base (F.map g).base).inv.app (op U) ≫
(pushforwa... | begin
cases U,
dsimp,
simp only [PresheafedSpace.congr_app (F.map_comp f g)],
dsimp, simp, dsimp, simp, -- See note [dsimp, simp]
end | lemma | algebraic_geometry.PresheafedSpace.map_comp_c_app | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
componentwise_diagram (F : J ⥤ PresheafedSpace.{v} C)
[has_colimit F] (U : opens (limits.colimit F).carrier) : Jᵒᵖ ⥤ C | { obj := λ j, (F.obj (unop j)).presheaf.obj (op ((opens.map (colimit.ι F (unop j)).base).obj U)),
map := λ j k f, (F.map f.unop).c.app _ ≫ (F.obj (unop k)).presheaf.map
(eq_to_hom (by { rw [← colimit.w F f.unop, comp_base], refl })),
map_comp' := λ i j k f g,
begin
cases U,
dsimp,
simp_rw [map_com... | def | algebraic_geometry.PresheafedSpace.componentwise_diagram | algebraic_geometry.presheafed_space | src/algebraic_geometry/presheafed_space/has_colimits.lean | [
"algebraic_geometry.presheafed_space",
"topology.category.Top.limits.basic",
"topology.sheaves.limits"
] | [
"Top.presheaf.pushforward.comp_inv_app",
"Top.presheaf.pushforward_eq_hom_app"
] | Given a diagram of `PresheafedSpace C`s, its colimit is computed by pushing the sheaves onto
the colimit of the underlying spaces, and taking componentwise limit.
This is the componentwise diagram for an open set `U` of the colimit of the underlying spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.