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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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lift_uniq (H : set.range g.base ⊆ set.range f.base) (l : Y ⟶ X)
(hl : l ≫ f = g) : l = lift f g H | by rw [← cancel_mono f, hl, lift_fac] | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.lift_uniq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"lift",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_range_eq [is_open_immersion g] (e : set.range f.base = set.range g.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.PresheafedSpace.is_open_immersion.iso_of_range_eq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"lift",
"set.range"
] | Two open immersions with equal range is isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_SheafedSpace : SheafedSpace C | { is_sheaf :=
begin
apply Top.presheaf.is_sheaf_of_iso (sheaf_iso_of_iso H.iso_restrict.symm).symm,
apply Top.sheaf.pushforward_sheaf_of_sheaf,
exact (Y.restrict H.base_open).is_sheaf
end,
to_PresheafedSpace := X } | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.presheaf.is_sheaf_of_iso",
"Top.sheaf.pushforward_sheaf_of_sheaf"
] | If `X ⟶ Y` is an open immersion, and `Y` is a SheafedSpace, then so is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_SheafedSpace_to_PresheafedSpace : (to_SheafedSpace Y f).to_PresheafedSpace = X | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_to_PresheafedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_SheafedSpace_hom : to_SheafedSpace Y f ⟶ Y | f | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | If `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a SheafedSpace, we can
upgrade it into a morphism of SheafedSpaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_SheafedSpace_hom_base : (to_SheafedSpace_hom Y f).base = f.base | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom_base | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_SheafedSpace_hom_c : (to_SheafedSpace_hom Y f).c = f.c | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom_c | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_SheafedSpace_is_open_immersion :
SheafedSpace.is_open_immersion (to_SheafedSpace_hom Y f) | H | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_to_SheafedSpace {X Y : SheafedSpace.{v} C} (f : X ⟶ Y)
[is_open_immersion f] : to_SheafedSpace Y f = X | by unfreezingI { cases X, refl } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.SheafedSpace_to_SheafedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_LocallyRingedSpace : LocallyRingedSpace | { to_SheafedSpace := to_SheafedSpace Y.to_SheafedSpace f,
local_ring := λ x, begin
haveI : local_ring (Y.to_SheafedSpace.to_PresheafedSpace.stalk (f.base x)) := Y.local_ring _,
exact (as_iso (stalk_map f x)).CommRing_iso_to_ring_equiv.local_ring
end } | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"local_ring"
] | If `X ⟶ Y` is an open immersion, and `Y` is a LocallyRingedSpace, then so is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_LocallyRingedSpace_to_SheafedSpace :
(to_LocallyRingedSpace Y f).to_SheafedSpace = (to_SheafedSpace Y.1 f) | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_to_SheafedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_LocallyRingedSpace_hom : to_LocallyRingedSpace Y f ⟶ Y | ⟨f, λ x, infer_instance⟩ | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_hom | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | If `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a LocallyRingedSpace, we can
upgrade it into a morphism of LocallyRingedSpace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_LocallyRingedSpace_hom_val :
(to_LocallyRingedSpace_hom Y f).val = f | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_hom_val | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_LocallyRingedSpace_is_open_immersion :
LocallyRingedSpace.is_open_immersion (to_LocallyRingedSpace_hom Y f) | H | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LocallyRingedSpace_to_LocallyRingedSpace {X Y : LocallyRingedSpace} (f : X ⟶ Y)
[LocallyRingedSpace.is_open_immersion f] :
to_LocallyRingedSpace Y f.1 = X | by unfreezingI { cases X, delta to_LocallyRingedSpace, simp } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.LocallyRingedSpace_to_LocallyRingedSpace | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_subset {X Y : PresheafedSpace.{v} C} (f : X ⟶ Y)
[H : PresheafedSpace.is_open_immersion f] (U : opens Y.carrier)
(hU : (U : set Y.carrier) ⊆ set.range f.base) : is_iso (f.c.app $ op U) | begin
have : U = H.base_open.is_open_map.functor.obj ((opens.map f.base).obj U),
{ ext1,
exact (set.inter_eq_left_iff_subset.mpr hU).symm.trans set.image_preimage_eq_inter_range.symm },
convert PresheafedSpace.is_open_immersion.c_iso ((opens.map f.base).obj U),
end | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.is_iso_of_subset | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso {X Y : SheafedSpace.{v} C} (f : X ⟶ Y) [is_iso f] :
SheafedSpace.is_open_immersion f | @@PresheafedSpace.is_open_immersion.of_is_iso _ f
(SheafedSpace.forget_to_PresheafedSpace.map_is_iso _) | instance | algebraic_geometry.SheafedSpace.is_open_immersion.of_is_iso | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z)
[SheafedSpace.is_open_immersion f] [SheafedSpace.is_open_immersion g] :
SheafedSpace.is_open_immersion (f ≫ g) | PresheafedSpace.is_open_immersion.comp f g | instance | algebraic_geometry.SheafedSpace.is_open_immersion.comp | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_map_is_open_immersion :
PresheafedSpace.is_open_immersion (forget .map f) | ⟨H.base_open, H.c_iso⟩ | instance | algebraic_geometry.SheafedSpace.is_open_immersion.forget_map_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_cospan_forget_of_left : has_limit (cospan f g ⋙ forget) | begin
apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,
change has_limit (cospan (forget .map f) (forget .map g)),
apply_instance
end | instance | algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_cospan_forget_of_left' : has_limit (cospan ((cospan f g ⋙ forget).map hom.inl)
((cospan f g ⋙ forget).map hom.inr)) | show has_limit (cospan (forget .map f) (forget .map g)), from infer_instance | instance | algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_left' | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_cospan_forget_of_right : has_limit (cospan g f ⋙ forget) | begin
apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,
change has_limit (cospan (forget .map g) (forget .map f)),
apply_instance
end | instance | algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | 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.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_right' | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | 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_SheafedSpace Y
(@pullback.snd (PresheafedSpace C) _ _ _ _ f g _))
(eq_to_iso (show pullback _ _ = pullback _ _, by congr)
≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm) | instance | algebraic_geometry.SheafedSpace.is_open_immersion.forget_creates_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | 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_SheafedSpace Y
(@pullback.fst (PresheafedSpace C) _ _ _ _ g f _))
(eq_to_iso (show pullback _ _ = pullback _ _, by congr)
≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm) | instance | algebraic_geometry.SheafedSpace.is_open_immersion.forget_creates_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_forget_preserves_of_left :
preserves_limit (cospan f g) (SheafedSpace.forget C) | @@limits.comp_preserves_limit _ _ _ _ forget (PresheafedSpace.forget C) _
begin
apply_with (preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{v} _).symm) { instances := tt },
dsimp,
apply_instance
end | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_forget_preserves_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_forget_preserves_of_right :
preserves_limit (cospan g f) (SheafedSpace.forget C) | preserves_pullback_symmetry _ _ _ | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_forget_preserves_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_has_pullback_of_left : has_pullback f g | has_limit_of_created (cospan f g) forget | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_has_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_has_pullback_of_right : has_pullback g f | has_limit_of_created (cospan g f) forget | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_has_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_pullback_snd_of_left :
SheafedSpace.is_open_immersion (pullback.snd : pullback f g ⟶ _) | begin
delta pullback.snd,
have : _ = limit.π (cospan f g) right := preserves_limits_iso_hom_π
forget (cospan f g) right,
rw ← this,
have := has_limit.iso_of_nat_iso_hom_π
(diagram_iso_cospan.{v} (cospan f g ⋙ forget))
right,
erw category.comp_id at this,
rw ← this,
dsimp,
apply_instance
en... | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_snd_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | Open immersions are stable under base-change. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
SheafedSpace_pullback_fst_of_right :
SheafedSpace.is_open_immersion (pullback.fst : pullback g f ⟶ _) | begin
delta pullback.fst,
have : _ = limit.π (cospan g f) left := preserves_limits_iso_hom_π
forget (cospan g f) left,
rw ← this,
have := has_limit.iso_of_nat_iso_hom_π
(diagram_iso_cospan.{v} (cospan g f ⋙ forget)) left,
erw category.comp_id at this,
rw ← this,
dsimp,
apply_instance
end | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_fst_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace_pullback_to_base_is_open_immersion [SheafedSpace.is_open_immersion g] :
SheafedSpace.is_open_immersion (limit.π (cospan f g) one : pullback f g ⟶ Z) | begin
rw [←limit.w (cospan f g) hom.inl, cospan_map_inl],
apply_instance
end | instance | algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_to_base_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_stalk_iso {X Y : SheafedSpace C} (f : X ⟶ Y)
(hf : open_embedding f.base) [H : ∀ x : X, is_iso (PresheafedSpace.stalk_map f x)] :
SheafedSpace.is_open_immersion f | { base_open := hf,
c_iso := λ U, begin
apply_with (Top.presheaf.app_is_iso_of_stalk_functor_map_iso
(show Y.sheaf ⟶ (Top.sheaf.pushforward f.base).obj X.sheaf, from ⟨f.c⟩)) { instances := ff },
rintros ⟨_, y, hy, rfl⟩,
specialize H y,
delta PresheafedSpace.stalk_map at H,
haveI H' := Top.pre... | lemma | algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.presheaf.app_is_iso_of_stalk_functor_map_iso",
"Top.presheaf.stalk_pushforward.stalk_pushforward_iso_of_open_embedding",
"Top.sheaf.pushforward",
"open_embedding"
] | Suppose `X Y : SheafedSpace C`, where `C` is a concrete category,
whose forgetful functor reflects isomorphisms, preserves limits and filtered colimits.
Then a morphism `X ⟶ Y` that is a topological open embedding
is an open immersion iff every stalk map is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_ι_open_embedding : open_embedding (colimit.ι F i).base | begin
rw ← (show _ = (colimit.ι F i).base,
from ι_preserves_colimits_iso_inv (SheafedSpace.forget C) F i),
have : _ = _ ≫ colimit.ι (discrete.functor ((F ⋙ SheafedSpace.forget C).obj ∘ discrete.mk)) i :=
has_colimit.iso_of_nat_iso_ι_hom discrete.nat_iso_functor i,
rw ← iso.eq_comp_inv at this,
rw this,
... | lemma | algebraic_geometry.SheafedSpace.is_open_immersion.sigma_ι_open_embedding | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.open_embedding_iff_comp_is_iso",
"Top.open_embedding_iff_is_iso_comp",
"open_embedding",
"open_embedding_sigma_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_preimage_is_empty (j : discrete ι) (h : i ≠ j) (U : opens (F.obj i)) :
(opens.map (colimit.ι (F ⋙ SheafedSpace.forget_to_PresheafedSpace) j).base).obj
((opens.map (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj
((sigma_ι_open_embedding F i).is_open_map.functor.obj U)) = ⊥ | begin
ext,
apply iff_false_intro,
rintro ⟨y, hy, eq⟩,
replace eq := concrete_category.congr_arg
(preserves_colimit_iso (SheafedSpace.forget C) F ≪≫
has_colimit.iso_of_nat_iso discrete.nat_iso_functor ≪≫ Top.sigma_iso_sigma.{v} _).hom eq,
simp_rw [category_theory.iso.trans_hom, ← Top.comp_app, ← Pres... | lemma | algebraic_geometry.SheafedSpace.is_open_immersion.image_preimage_is_empty | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.comp_app"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_ι_is_open_immersion [has_strict_terminal_objects C] :
SheafedSpace.is_open_immersion (colimit.ι F i) | { base_open := sigma_ι_open_embedding F i,
c_iso := λ U, begin
have e : colimit.ι F i = _ :=
(ι_preserves_colimits_iso_inv SheafedSpace.forget_to_PresheafedSpace F i).symm,
have H : open_embedding (colimit.ι (F ⋙ SheafedSpace.forget_to_PresheafedSpace) i ≫
(preserves_colimit_iso SheafedSpace.forge... | instance | algebraic_geometry.SheafedSpace.is_open_immersion.sigma_ι_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"open_embedding",
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso [is_iso g] :
LocallyRingedSpace.is_open_immersion g | @@PresheafedSpace.is_open_immersion.of_is_iso _ g.1 ⟨⟨(inv g).1,
by { erw ← LocallyRingedSpace.comp_val, rw is_iso.hom_inv_id,
erw ← LocallyRingedSpace.comp_val, rw is_iso.inv_hom_id, split; simpa }⟩⟩ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.of_is_iso | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : Z ⟶ Y) [LocallyRingedSpace.is_open_immersion g] :
LocallyRingedSpace.is_open_immersion (f ≫ g) | PresheafedSpace.is_open_immersion.comp f.1 g.1 | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.comp | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono : mono f | LocallyRingedSpace.forget_to_SheafedSpace.mono_of_mono_map (show mono f.1, by apply_instance) | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.mono | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_cone_of_left : pullback_cone f g | begin
refine pullback_cone.mk _
(Y.of_restrict (Top.snd_open_embedding_of_left_open_embedding H.base_open g.1.base)) _,
{ use PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst f.1 g.1,
intro x,
have := PresheafedSpace.stalk_map.congr_hom _ _
(PresheafedSpace.is_open_immersion.pullback_co... | def | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_cone_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.snd_open_embedding_of_left_open_embedding"
] | An explicit pullback cone over `cospan f g` if `f` is an open immersion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_left_is_limit : is_limit (pullback_cone_of_left f g) | pullback_cone.is_limit_aux' _ $ λ s,
begin
use PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift f.1 g.1
(pullback_cone.mk s.fst.1 s.snd.1 (congr_arg LocallyRingedSpace.hom.val s.condition)),
{ intro x,
have := PresheafedSpace.stalk_map.congr_hom _ _
(PresheafedSpace.is_open_immersion.pullb... | def | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_cone_of_left_is_limit | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | The constructed `pullback_cone_of_left` is indeed limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_snd_of_left :
LocallyRingedSpace.is_open_immersion (pullback.snd : pullback f g ⟶ _) | begin
delta pullback.snd,
rw ← limit.iso_limit_cone_hom_π ⟨_, pullback_cone_of_left_is_limit f g⟩ walking_cospan.right,
apply_instance
end | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_snd_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | Open immersions are stable under base-change. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_fst_of_right :
LocallyRingedSpace.is_open_immersion (pullback.fst : pullback g f ⟶ _) | begin
rw ← pullback_symmetry_hom_comp_snd,
apply_instance
end | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_fst_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | Open immersions are stable under base-change. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_to_base_is_open_immersion [LocallyRingedSpace.is_open_immersion g] :
LocallyRingedSpace.is_open_immersion (limit.π (cospan f g) walking_cospan.one) | begin
rw [←limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],
apply_instance
end | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_to_base_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_pullback_of_left :
preserves_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace | preserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)
begin
apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,
apply is_limit_of_is_limit_pullback_cone_map SheafedSpace.forget_to_PresheafedSpace,
exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1
end | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_preserves_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_PresheafedSpace_preserves_pullback_of_left :
preserves_limit (cospan f g)
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) | preserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)
begin
apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,
exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1
end | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_PresheafedSpace_preserves_open_immersion :
PresheafedSpace.is_open_immersion ((LocallyRingedSpace.forget_to_SheafedSpace ⋙
SheafedSpace.forget_to_PresheafedSpace).map f) | H | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Top_preserves_pullback_of_left :
preserves_limit (cospan f g)
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _) | begin
change preserves_limit _
((LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace)
⋙ PresheafedSpace.forget _),
apply_with limits.comp_preserves_limit { instances := ff },
apply_instance,
apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,
dsimp ... | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_Top_preserves_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_reflects_pullback_of_left :
reflects_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace | reflects_limit_of_reflects_isomorphisms _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_reflects_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_pullback_of_right :
preserves_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace | preserves_pullback_symmetry _ _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_preserves_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_PresheafedSpace_preserves_pullback_of_right :
preserves_limit (cospan g f) (LocallyRingedSpace.forget_to_SheafedSpace ⋙
SheafedSpace.forget_to_PresheafedSpace) | preserves_pullback_symmetry _ _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_reflects_pullback_of_right :
reflects_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace | reflects_limit_of_reflects_isomorphisms _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_reflects_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_PresheafedSpace_reflects_pullback_of_left :
reflects_limit (cospan f g)
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) | reflects_limit_of_reflects_isomorphisms _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_reflects_pullback_of_left | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_PresheafedSpace_reflects_pullback_of_right :
reflects_limit (cospan g f)
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) | reflects_limit_of_reflects_isomorphisms _ _ | instance | algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_reflects_pullback_of_right | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_snd_is_iso_of_range_subset (H' : set.range g.1.base ⊆ set.range f.1.base) :
is_iso (pullback.snd : pullback f g ⟶ _) | begin
apply_with (reflects_isomorphisms.reflects LocallyRingedSpace.forget_to_SheafedSpace)
{ instances := ff },
apply_with (reflects_isomorphisms.reflects SheafedSpace.forget_to_PresheafedSpace)
{ instances := ff },
erw ← preserves_pullback.iso_hom_snd
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ She... | lemma | algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (H' : set.range g.1.base ⊆ set.range f.1.base) : Y ⟶ X | begin
haveI := pullback_snd_is_iso_of_range_subset f g H',
exact inv (pullback.snd : pullback f g ⟶ _) ≫ pullback.fst,
end | def | algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"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 | by { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition } | lemma | algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_fac | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"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' | by rw [← cancel_mono f, hl, lift_fac] | lemma | algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_uniq | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"lift",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_range (H' : set.range g.1.base ⊆ set.range f.1.base) :
set.range (lift f g H').1.base = f.1.base ⁻¹' (set.range g.1.base) | begin
haveI := pullback_snd_is_iso_of_range_subset f g H',
dsimp only [lift],
have : _ = (pullback.fst : pullback f g ⟶ _).val.base := preserves_pullback.iso_hom_fst
(LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _) f g,
rw [LocallyRingedSpace.comp_val, SheafedSpace.comp_base, ← this, ← ca... | lemma | algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_range | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.epi_iff_surjective",
"Top.pullback_fst_range",
"lift",
"map_inv",
"set.image_univ",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_restrict {X Y : LocallyRingedSpace} {f : X ⟶ Y}
(H : LocallyRingedSpace.is_open_immersion f) : X ≅ Y.restrict H.base_open | begin
apply LocallyRingedSpace.iso_of_SheafedSpace_iso,
refine SheafedSpace.forget_to_PresheafedSpace.preimage_iso _,
exact H.iso_restrict
end | def | algebraic_geometry.LocallyRingedSpace.is_open_immersion.iso_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | An open immersion is isomorphic to the induced open subscheme on its image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_immersion {X Y : Scheme} (f : X ⟶ Y) : Prop | LocallyRingedSpace.is_open_immersion f | abbreviation | algebraic_geometry.is_open_immersion | 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"
] | [] | A morphism of Schemes is an open immersion if it is an open immersion as a morphism
of LocallyRingedSpaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme (X : LocallyRingedSpace)
(h : ∀ (x : X), ∃ (R : CommRing) (f : Spec.to_LocallyRingedSpace.obj (op R) ⟶ X),
(x ∈ set.range f.1.base : _) ∧ LocallyRingedSpace.is_open_immersion f) : Scheme | { to_LocallyRingedSpace := X,
local_affine :=
begin
intro x,
obtain ⟨R, f, h₁, h₂⟩ := h x,
refine ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩,
apply LocallyRingedSpace.iso_of_SheafedSpace_iso,
refine SheafedSpace.forget_to_PresheafedSpace.preimage_iso _,
resetI,
apply PresheafedSpace.is... | def | algebraic_geometry.LocallyRingedSpace.is_open_immersion.Scheme | 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"
] | [
"CommRing",
"set.range",
"subtype.range_coe_subtype"
] | To show that a locally ringed space is a scheme, it suffices to show that it has a jointly
surjective family of open immersions from affine schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_immersion.open_range {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f] :
is_open (set.range f.1.base) | H.base_open.open_range | lemma | algebraic_geometry.is_open_immersion.open_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"
] | [
"is_open",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover (X : Scheme.{u}) | (J : Type v)
(obj : Π (j : J), Scheme)
(map : Π (j : J), obj j ⟶ X)
(f : X.carrier → J)
(covers : ∀ x, x ∈ set.range ((map (f x)).1.base))
(is_open : ∀ x, is_open_immersion (map x) . tactic.apply_instance) | structure | algebraic_geometry.Scheme.open_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"
] | [
"is_open",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_cover (X : Scheme) : open_cover X | { J := X.carrier,
obj := λ x, Spec.obj $ opposite.op (X.local_affine x).some_spec.some,
map := λ x, ((X.local_affine x).some_spec.some_spec.some.inv ≫
X.to_LocallyRingedSpace.of_restrict _ : _),
f := λ x, x,
is_open := λ x, begin
apply_with PresheafedSpace.is_open_immersion.comp { instances := ff },
... | def | algebraic_geometry.Scheme.affine_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",
"is_open",
"opposite.op",
"set.image_univ",
"set.range_comp",
"subtype.range_coe_subtype"
] | The affine cover of a scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover.bind (f : Π (x : 𝒰.J), open_cover (𝒰.obj x)) : open_cover X | { J := Σ (i : 𝒰.J), (f i).J,
obj := λ x, (f x.1).obj x.2,
map := λ x, (f x.1).map x.2 ≫ 𝒰.map x.1,
f := λ x, ⟨_, (f _).f (𝒰.covers x).some⟩,
covers := λ x,
begin
let y := (𝒰.covers x).some,
have hy : (𝒰.map (𝒰.f x)).val.base y = x := (𝒰.covers x).some_spec,
rcases (f (𝒰.f x)).covers y with... | def | algebraic_geometry.Scheme.open_cover.bind | 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"
] | Given an open cover `{ Uᵢ }` of `X`, and for each `Uᵢ` an open cover, we may combine these
open covers to form an open cover of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover_of_is_iso {X Y : Scheme.{u}} (f : X ⟶ Y) [is_iso f] :
open_cover Y | { J := punit.{v+1},
obj := λ _, X,
map := λ _, f,
f := λ _, punit.star,
covers := λ x, by { rw set.range_iff_surjective.mpr, { trivial }, rw ← Top.epi_iff_surjective,
apply_instance } } | def | algebraic_geometry.Scheme.open_cover_of_is_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"
] | An isomorphism `X ⟶ Y` is an open cover of `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover.copy {X : Scheme} (𝒰 : open_cover X)
(J : Type*) (obj : J → Scheme) (map : ∀ i, obj i ⟶ X)
(e₁ : J ≃ 𝒰.J) (e₂ : ∀ i, obj i ≅ 𝒰.obj (e₁ i))
(e₂ : ∀ i, map i = (e₂ i).hom ≫ 𝒰.map (e₁ i)) : open_cover X | { J := J,
obj := obj,
map := map,
f := λ x, e₁.symm (𝒰.f x),
covers := λ x, begin
rw [e₂, Scheme.comp_val_base, coe_comp, set.range_comp, set.range_iff_surjective.mpr,
set.image_univ, e₁.right_inverse_symm],
{ exact 𝒰.covers x },
{ rw ← Top.epi_iff_surjective, apply_instance }
end,
is_o... | def | algebraic_geometry.Scheme.open_cover.copy | 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",
"is_open",
"set.image_univ",
"set.range_comp"
] | We construct an open cover from another, by providing the needed fields and showing that the
provided fields are isomorphic with the original open cover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover.pushforward_iso {X Y : Scheme} (𝒰 : open_cover X)
(f : X ⟶ Y) [is_iso f] :
open_cover Y | ((open_cover_of_is_iso f).bind (λ _, 𝒰)).copy 𝒰.J _ _
((equiv.punit_prod _).symm.trans (equiv.sigma_equiv_prod punit 𝒰.J).symm)
(λ _, iso.refl _)
(λ _, (category.id_comp _).symm) | def | algebraic_geometry.Scheme.open_cover.pushforward_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"
] | [
"equiv.punit_prod",
"equiv.sigma_equiv_prod"
] | The pushforward of an open cover along an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_cover.add {X : Scheme} (𝒰 : X.open_cover) {Y : Scheme} (f : Y ⟶ X)
[is_open_immersion f] : X.open_cover | { J := option 𝒰.J,
obj := λ i, option.rec Y 𝒰.obj i,
map := λ i, option.rec f 𝒰.map i,
f := λ x, some (𝒰.f x),
covers := 𝒰.covers,
is_open := by rintro (_|_); dsimp; apply_instance } | def | algebraic_geometry.Scheme.open_cover.add | 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"
] | Adding an open immersion into an open cover gives another open cover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_base_is_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] : is_iso f.1.base | Scheme.forget_to_Top.map_is_iso f | instance | algebraic_geometry.Scheme.val_base_is_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_is_open_immersion {R : CommRing} (f : R) :
algebraic_geometry.is_open_immersion (Scheme.Spec.map (CommRing.of_hom
(algebra_map R (localization.away f))).op) | begin
apply_with SheafedSpace.is_open_immersion.of_stalk_iso { instances := ff },
any_goals { apply_instance },
any_goals { apply_instance },
exact (prime_spectrum.localization_away_open_embedding (localization.away f) f : _),
intro x,
exact Spec_map_localization_is_iso R (submonoid.powers f) x,
end | instance | algebraic_geometry.Scheme.basic_open_is_open_immersion | 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"
] | [
"CommRing",
"CommRing.of_hom",
"algebra_map",
"algebraic_geometry.is_open_immersion",
"localization.away",
"prime_spectrum.localization_away_open_embedding",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_basis_cover_of_affine (R : CommRing) : open_cover (Spec.obj (opposite.op R)) | { J := R,
obj := λ r, Spec.obj (opposite.op $ CommRing.of $ localization.away r),
map := λ r, Spec.map (quiver.hom.op (algebra_map R (localization.away r) : _)),
f := λ x, 1,
covers := λ r,
begin
rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp _),
{ exact trivial },
{ apply_instance... | def | algebraic_geometry.Scheme.affine_basis_cover_of_affine | 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"
] | [
"CommRing",
"CommRing.of",
"Top.epi_iff_surjective",
"algebra_map",
"algebraic_geometry.Scheme.basic_open_is_open_immersion",
"is_open",
"localization.away",
"opposite.op",
"quiver.hom.op"
] | The basic open sets form an affine open cover of `Spec R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_basis_cover (X : Scheme) : open_cover X | X.affine_cover.bind (λ x, affine_basis_cover_of_affine _) | def | algebraic_geometry.Scheme.affine_basis_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"
] | [] | We may bind the basic open sets of an open affine cover to form a affine cover that is also
a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_basis_cover_ring (X : Scheme) (i : X.affine_basis_cover.J) : CommRing | CommRing.of $ @localization.away (X.local_affine i.1).some_spec.some _ i.2 | def | algebraic_geometry.Scheme.affine_basis_cover_ring | 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"
] | [
"CommRing",
"CommRing.of",
"localization.away"
] | The coordinate ring of a component in the `affine_basis_cover`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_basis_cover_obj (X : Scheme) (i : X.affine_basis_cover.J) :
X.affine_basis_cover.obj i = Spec.obj (op $ X.affine_basis_cover_ring i) | rfl | lemma | algebraic_geometry.Scheme.affine_basis_cover_obj | 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 | |
affine_basis_cover_map_range (X : Scheme)
(x : X.carrier) (r : (X.local_affine x).some_spec.some) :
set.range (X.affine_basis_cover.map ⟨x, r⟩).1.base =
(X.affine_cover.map x).1.base '' (prime_spectrum.basic_open r).1 | begin
erw [coe_comp, set.range_comp],
congr,
exact (prime_spectrum.localization_away_comap_range (localization.away r) r : _)
end | lemma | algebraic_geometry.Scheme.affine_basis_cover_map_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"
] | [
"localization.away",
"prime_spectrum.basic_open",
"prime_spectrum.localization_away_comap_range",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_basis_cover_is_basis (X : Scheme) :
topological_space.is_topological_basis
{ x : set X.carrier | ∃ a : X.affine_basis_cover.J, x =
set.range ((X.affine_basis_cover.map a).1.base) } | begin
apply topological_space.is_topological_basis_of_open_of_nhds,
{ rintros _ ⟨a, rfl⟩,
exact is_open_immersion.open_range (X.affine_basis_cover.map a) },
{ rintros a U haU hU,
rcases X.affine_cover.covers a with ⟨x, e⟩,
let U' := (X.affine_cover.map (X.affine_cover.f a)).1.base ⁻¹' U,
have hxU'... | lemma | algebraic_geometry.Scheme.affine_basis_cover_is_basis | 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.range",
"topological_space.is_topological_basis",
"topological_space.is_topological_basis_of_open_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover.finite_subcover {X : Scheme} (𝒰 : open_cover X) [H : compact_space X.carrier] :
open_cover X | begin
have := @@compact_space.elim_nhds_subcover _ H
(λ (x : X.carrier), set.range ((𝒰.map (𝒰.f x)).1.base))
(λ x, (is_open_immersion.open_range (𝒰.map (𝒰.f x))).mem_nhds (𝒰.covers x)),
let t := this.some,
have h : ∀ (x : X.carrier), ∃ (y : t), x ∈ set.range ((𝒰.map (𝒰.f y)).1.base),
{ intro x,
... | def | algebraic_geometry.Scheme.open_cover.finite_subcover | 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"
] | [
"compact_space",
"compact_space.elim_nhds_subcover",
"set.mem_Union",
"set.range"
] | Every open cover of a quasi-compact scheme can be refined into a finite subcover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Scheme : Scheme | begin
apply LocallyRingedSpace.is_open_immersion.Scheme (to_LocallyRingedSpace _ f),
intro x,
obtain ⟨_,⟨i,rfl⟩,hx,hi⟩ := Y.affine_basis_cover_is_basis.exists_subset_of_mem_open
(set.mem_range_self x) H.base_open.open_range,
use Y.affine_basis_cover_ring i,
use LocallyRingedSpace.is_open_immersion.lift ... | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme | 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.mem_range_self"
] | If `X ⟶ Y` is an open immersion, and `Y` is a scheme, then so is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Scheme_to_LocallyRingedSpace :
(to_Scheme Y f).to_LocallyRingedSpace = (to_LocallyRingedSpace Y.1 f) | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_to_LocallyRingedSpace | 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 | |
to_Scheme_hom : to_Scheme Y f ⟶ Y | to_LocallyRingedSpace_hom _ f | def | algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom | 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"
] | [] | If `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a Scheme, we can
upgrade it into a morphism of Schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Scheme_hom_val :
(to_Scheme_hom Y f).val = f | rfl | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom_val | 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 | |
to_Scheme_hom_is_open_immersion :
is_open_immersion (to_Scheme_hom Y f) | H | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom_is_open_immersion | 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_eq_of_LocallyRingedSpace_eq {X Y : Scheme}
(H : X.to_LocallyRingedSpace = Y.to_LocallyRingedSpace) : X = Y | by { cases X, cases Y, congr, exact H } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.Scheme_eq_of_LocallyRingedSpace_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme_to_Scheme {X Y : Scheme} (f : X ⟶ Y) [is_open_immersion f] :
to_Scheme Y f.1 = X | begin
apply Scheme_eq_of_LocallyRingedSpace_eq,
exact LocallyRingedSpace_to_LocallyRingedSpace f
end | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.Scheme_to_Scheme | 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 {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier} (h : open_embedding f) :
Scheme | { to_PresheafedSpace := X.to_PresheafedSpace.restrict h,
..(PresheafedSpace.is_open_immersion.to_Scheme X (X.to_PresheafedSpace.of_restrict h)) } | def | algebraic_geometry.Scheme.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"
] | [
"Top",
"Top.of",
"open_embedding"
] | The restriction of a Scheme along an open embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.of_restrict {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier} (h : open_embedding f) :
X.restrict h ⟶ X | X.to_LocallyRingedSpace.of_restrict h | def | algebraic_geometry.Scheme.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"
] | [
"Top",
"Top.of",
"open_embedding"
] | The canonical map from the restriction to the supspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_immersion.of_restrict {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier}
(h : open_embedding f) : is_open_immersion (X.of_restrict h) | show PresheafedSpace.is_open_immersion (X.to_PresheafedSpace.of_restrict h), by apply_instance | instance | algebraic_geometry.is_open_immersion.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"
] | [
"Top",
"Top.of",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso [is_iso g] :
is_open_immersion g | @@LocallyRingedSpace.is_open_immersion.of_is_iso _
(show is_iso ((induced_functor _).map g), by apply_instance) | instance | algebraic_geometry.is_open_immersion.of_is_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_iso {X Y : Scheme} (f : X ⟶ Y) [h : is_open_immersion f]
[epi f.1.base] : is_iso f | @@is_iso_of_reflects_iso _ _ f (Scheme.forget_to_LocallyRingedSpace ⋙
LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace)
(@@PresheafedSpace.is_open_immersion.to_iso _ f.1 h _) _ | lemma | algebraic_geometry.is_open_immersion.to_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_stalk_iso {X Y : Scheme} (f : X ⟶ Y) (hf : open_embedding f.1.base)
[∀ x, is_iso (PresheafedSpace.stalk_map f.1 x)] : is_open_immersion f | SheafedSpace.is_open_immersion.of_stalk_iso f.1 hf | lemma | algebraic_geometry.is_open_immersion.of_stalk_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"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_stalk_iso {X Y : Scheme} (f : X ⟶ Y) :
is_open_immersion f ↔ open_embedding f.1.base ∧ ∀ x, is_iso (PresheafedSpace.stalk_map f.1 x) | ⟨λ H, ⟨H.1, by exactI infer_instance⟩, λ ⟨h₁, h₂⟩, @@is_open_immersion.of_stalk_iso f h₁ h₂⟩ | lemma | algebraic_geometry.is_open_immersion.iff_stalk_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"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.algebraic_geometry.is_iso_iff_is_open_immersion {X Y : Scheme} (f : X ⟶ Y) :
is_iso f ↔ is_open_immersion f ∧ epi f.1.base | ⟨λ H, by exactI ⟨infer_instance, infer_instance⟩, λ ⟨h₁, h₂⟩, @@is_open_immersion.to_iso f h₁ h₂⟩ | lemma | algebraic_geometry.is_iso_iff_is_open_immersion | 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 | |
_root_.algebraic_geometry.is_iso_iff_stalk_iso {X Y : Scheme} (f : X ⟶ Y) :
is_iso f ↔ is_iso f.1.base ∧ ∀ x, is_iso (PresheafedSpace.stalk_map f.1 x) | begin
rw [is_iso_iff_is_open_immersion, is_open_immersion.iff_stalk_iso, and_comm, ← and_assoc],
refine and_congr ⟨_, _⟩ iff.rfl,
{ rintro ⟨h₁, h₂⟩,
convert_to is_iso (Top.iso_of_homeo (homeomorph.homeomorph_of_continuous_open
(equiv.of_bijective _ ⟨h₂.inj, (Top.epi_iff_surjective _).mp h₁⟩)
h₂.co... | lemma | algebraic_geometry.is_iso_iff_stalk_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",
"Top.homeo_of_iso",
"Top.iso_of_homeo",
"equiv.of_bijective",
"homeomorph.homeomorph_of_continuous_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_restrict : X ≅ (Z.restrict H.base_open : _) | ⟨H.iso_restrict.hom, H.iso_restrict.inv, H.iso_restrict.hom_inv_id, H.iso_restrict.inv_hom_id⟩ | def | algebraic_geometry.is_open_immersion.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"
] | [] | A open immersion induces an isomorphism from the domain onto the image | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono : mono f | (induced_functor _).mono_of_mono_map (show @mono LocallyRingedSpace _ _ _ f, by apply_instance) | instance | algebraic_geometry.is_open_immersion.mono | 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_map_is_open_immersion : LocallyRingedSpace.is_open_immersion (forget .map f) | ⟨H.base_open, H.c_iso⟩ | instance | algebraic_geometry.is_open_immersion.forget_map_is_open_immersion | 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_left :
has_limit (cospan f g ⋙ Scheme.forget_to_LocallyRingedSpace) | begin
apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,
change has_limit (cospan (forget .map f) (forget .map g)),
apply_instance
end | instance | algebraic_geometry.is_open_immersion.has_limit_cospan_forget_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_limit_cospan_forget_of_left' :
has_limit (cospan ((cospan f g ⋙ forget).map hom.inl)
((cospan f g ⋙ forget).map hom.inr)) | show has_limit (cospan (forget .map f) (forget .map g)), from infer_instance | instance | algebraic_geometry.is_open_immersion.has_limit_cospan_forget_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 |
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