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to_LocallyRingedSpace_glue_data : LocallyRingedSpace.glue_data | { f_open := D.f_open,
to_glue_data := 𝖣 .map_glue_data forget_to_LocallyRingedSpace } | abbreviation | algebraic_geometry.Scheme.glue_data.to_LocallyRingedSpace_glue_data | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The glue data of locally ringed spaces spaces associated to a family of glue data of schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued_Scheme : Scheme | begin
apply LocallyRingedSpace.is_open_immersion.Scheme
D.to_LocallyRingedSpace_glue_data.to_glue_data.glued,
intro x,
obtain ⟨i, y, rfl⟩ := D.to_LocallyRingedSpace_glue_data.ι_jointly_surjective x,
refine ⟨_, _ ≫ D.to_LocallyRingedSpace_glue_data.to_glue_data.ι i, _⟩,
swap, exact (D.U i).affine_cover.map... | def | algebraic_geometry.Scheme.glue_data.glued_Scheme | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"set.mem_image_of_mem",
"set.mem_range",
"set.range_comp"
] | (Implementation). The glued scheme of a glue data.
This should not be used outside this file. Use `Scheme.glue_data.glued` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued : Scheme | 𝖣 .glued | abbreviation | algebraic_geometry.Scheme.glue_data.glued | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The glued scheme of a glued space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι (i : D.J) : D.U i ⟶ D.glued | 𝖣 .ι i | abbreviation | algebraic_geometry.Scheme.glue_data.ι | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The immersion from `D.U i` into the glued space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_LocallyRingedSpace :
D.glued.to_LocallyRingedSpace ≅ D.to_LocallyRingedSpace_glue_data.to_glue_data.glued | 𝖣 .glued_iso forget_to_LocallyRingedSpace | abbreviation | algebraic_geometry.Scheme.glue_data.iso_LocallyRingedSpace | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The gluing as sheafed spaces is isomorphic to the gluing as presheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_iso_LocallyRingedSpace_inv (i : D.J) :
D.to_LocallyRingedSpace_glue_data.to_glue_data.ι i ≫ D.iso_LocallyRingedSpace.inv = 𝖣 .ι i | 𝖣 .ι_glued_iso_inv forget_to_LocallyRingedSpace i | lemma | algebraic_geometry.Scheme.glue_data.ι_iso_LocallyRingedSpace_inv | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | 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_LocallyRingedSpace_inv, apply_instance } | instance | algebraic_geometry.Scheme.glue_data.ι_is_open_immersion | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_jointly_surjective (x : 𝖣 .glued.carrier) :
∃ (i : D.J) (y : (D.U i).carrier), (D.ι i).1.base y = x | 𝖣 .ι_jointly_surjective (forget_to_Top ⋙ forget Top) x | lemma | algebraic_geometry.Scheme.glue_data.ι_jointly_surjective | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_condition (i j : D.J) :
D.t i j ≫ D.f j i ≫ D.ι j = D.f i j ≫ D.ι i | 𝖣 .glue_condition i j | lemma | algebraic_geometry.Scheme.glue_data.glue_condition | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
V_pullback_cone (i j : D.J) : pullback_cone (D.ι i) (D.ι j) | pullback_cone.mk (D.f i j) (D.t i j ≫ D.f j i) (by simp) | def | algebraic_geometry.Scheme.glue_data.V_pullback_cone | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The pullback cone spanned by `V i j ⟶ U i` and `V i j ⟶ U j`.
This is a pullback diagram (`V_pullback_cone_is_limit`). | 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_LocallyRingedSpace i j
(D.to_LocallyRingedSpace_glue_data.V_pullback_cone_is_limit _ _) | def | algebraic_geometry.Scheme.glue_data.V_pullback_cone_is_limit | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | 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 |
iso_carrier :
D.glued.carrier ≅ D.to_LocallyRingedSpace_glue_data.to_SheafedSpace_glue_data
.to_PresheafedSpace_glue_data.to_Top_glue_data.to_glue_data.glued | begin
refine (PresheafedSpace.forget _).map_iso _ ≪≫
glue_data.glued_iso _ (PresheafedSpace.forget _),
refine SheafedSpace.forget_to_PresheafedSpace.map_iso _ ≪≫
SheafedSpace.glue_data.iso_PresheafedSpace _,
refine LocallyRingedSpace.forget_to_SheafedSpace.map_iso _ ≪≫
LocallyRingedSpace.glue_data.iso... | def | algebraic_geometry.Scheme.glue_data.iso_carrier | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The underlying topological space of the glued scheme is isomorphic to the gluing of the
underlying spacess | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_iso_carrier_inv (i : D.J) :
D.to_LocallyRingedSpace_glue_data.to_SheafedSpace_glue_data
.to_PresheafedSpace_glue_data.to_Top_glue_data.to_glue_data.ι i ≫ D.iso_carrier.inv =
(D.ι i).1.base | begin
delta iso_carrier,
simp only [functor.map_iso_inv, iso.trans_inv, iso.trans_assoc,
glue_data.ι_glued_iso_inv_assoc, functor.map_iso_trans, category.assoc],
iterate 3 { erw ← comp_base },
simp_rw ← category.assoc,
rw D.to_LocallyRingedSpace_glue_data.to_SheafedSpace_glue_data.ι_iso_PresheafedSpace_in... | lemma | algebraic_geometry.Scheme.glue_data.ι_iso_carrier_inv | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel (a b : Σ i, ((D.U i).carrier : Type*)) : Prop | a = b ∨ ∃ (x : (D.V (a.1, b.1)).carrier),
(D.f _ _).1.base x = a.2 ∧ (D.t _ _ ≫ D.f _ _).1.base x = b.2 | def | algebraic_geometry.Scheme.glue_data.rel | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"rel"
] | An equivalence relation on `Σ i, D.U i` that holds iff `𝖣 .ι i x = 𝖣 .ι j y`.
See `Scheme.gluing_data.ι_eq_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_eq_iff (i j : D.J) (x : (D.U i).carrier) (y : (D.U j).carrier) :
(𝖣 .ι i).1.base x = (𝖣 .ι j).1.base y ↔ D.rel ⟨i, x⟩ ⟨j, y⟩ | begin
refine iff.trans _ (D.to_LocallyRingedSpace_glue_data.to_SheafedSpace_glue_data
.to_PresheafedSpace_glue_data.to_Top_glue_data.ι_eq_iff_rel i j x y),
rw ← ((Top.mono_iff_injective D.iso_carrier.inv).mp infer_instance).eq_iff,
simp_rw [← comp_apply, D.ι_iso_carrier_inv]
end | lemma | algebraic_geometry.Scheme.glue_data.ι_eq_iff | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"Top.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff (U : set D.glued.carrier) : is_open U ↔ ∀ i, is_open ((D.ι i).1.base ⁻¹' U) | begin
rw ← (Top.homeo_of_iso D.iso_carrier.symm).is_open_preimage,
rw Top.glue_data.is_open_iff,
apply forall_congr,
intro i,
erw [← set.preimage_comp, ← coe_comp, ι_iso_carrier_inv]
end | lemma | algebraic_geometry.Scheme.glue_data.is_open_iff | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"Top.glue_data.is_open_iff",
"Top.homeo_of_iso",
"is_open",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover (D : Scheme.glue_data) : open_cover D.glued | { J := D.J,
obj := D.U,
map := D.ι,
f := λ x, (D.ι_jointly_surjective x).some,
covers := λ x, ⟨_, (D.ι_jointly_surjective x).some_spec.some_spec⟩ } | def | algebraic_geometry.Scheme.glue_data.open_cover | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The open cover of the glued space given by the glue data. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued_cover_t' (x y z : 𝒰.J) :
pullback (pullback.fst : pullback (𝒰.map x) (𝒰.map y) ⟶ _)
(pullback.fst : pullback (𝒰.map x) (𝒰.map z) ⟶ _) ⟶
pullback (pullback.fst : pullback (𝒰.map y) (𝒰.map z) ⟶ _)
(pullback.fst : pullback (𝒰.map y) (𝒰.map x) ⟶ _) | begin
refine (pullback_right_pullback_fst_iso _ _ _).hom ≫ _,
refine _ ≫ (pullback_symmetry _ _).hom,
refine _ ≫ (pullback_right_pullback_fst_iso _ _ _).inv,
refine pullback.map _ _ _ _ (pullback_symmetry _ _).hom (𝟙 _) (𝟙 _) _ _,
{ simp [pullback.condition] },
{ simp }
end | def | algebraic_geometry.Scheme.open_cover.glued_cover_t' | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | (Implementation) the transition maps in the glue data associated with an open cover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
glued_cover_t'_fst_fst (x y z : 𝒰.J) :
𝒰.glued_cover_t' x y z ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd | by { delta glued_cover_t', simp } | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_t'_fst_fst | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_t'_fst_snd (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.snd | by { delta glued_cover_t', simp } | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_t'_fst_snd | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_t'_snd_fst (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd | by { delta glued_cover_t', simp } | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_t'_snd_fst | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_t'_snd_snd (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst | by { delta glued_cover_t', simp } | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_t'_snd_snd | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_cocycle_fst (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ glued_cover_t' 𝒰 y z x ≫ glued_cover_t' 𝒰 z x y ≫ pullback.fst =
pullback.fst | by apply pullback.hom_ext; simp | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_cocycle_fst | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_cocycle_snd (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ glued_cover_t' 𝒰 y z x ≫ glued_cover_t' 𝒰 z x y ≫ pullback.snd =
pullback.snd | by apply pullback.hom_ext; simp [pullback.condition] | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_cocycle_snd | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover_cocycle (x y z : 𝒰.J) :
glued_cover_t' 𝒰 x y z ≫ glued_cover_t' 𝒰 y z x ≫ glued_cover_t' 𝒰 z x y = 𝟙 _ | begin
apply pullback.hom_ext; simp_rw [category.id_comp, category.assoc],
apply glued_cover_cocycle_fst,
apply glued_cover_cocycle_snd,
end | lemma | algebraic_geometry.Scheme.open_cover.glued_cover_cocycle | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glued_cover : Scheme.glue_data.{u} | { J := 𝒰.J,
U := 𝒰.obj,
V := λ ⟨x, y⟩, pullback (𝒰.map x) (𝒰.map y),
f := λ x y, pullback.fst,
f_id := λ x, infer_instance,
t := λ x y, (pullback_symmetry _ _).hom,
t_id := λ x, by simpa,
t' := λ x y z, glued_cover_t' 𝒰 x y z,
t_fac := λ x y z, by apply pullback.hom_ext; simp,
-- The `cocycle` fi... | def | algebraic_geometry.Scheme.open_cover.glued_cover | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The glue data associated with an open cover.
The canonical isomorphism `𝒰.glued_cover.glued ⟶ X` is provided by `𝒰.from_glued`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_glued : 𝒰.glued_cover.glued ⟶ X | begin
fapply multicoequalizer.desc,
exact λ x, (𝒰.map x),
rintro ⟨x, y⟩,
change pullback.fst ≫ _ = ((pullback_symmetry _ _).hom ≫ pullback.fst) ≫ _,
simpa using pullback.condition
end | def | algebraic_geometry.Scheme.open_cover.from_glued | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | The canonical morphism from the gluing of an open cover of `X` into `X`.
This is an isomorphism, as witnessed by an `is_iso` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_from_glued (x : 𝒰.J) :
𝒰.glued_cover.ι x ≫ 𝒰.from_glued = 𝒰.map x | multicoequalizer.π_desc _ _ _ _ _ | lemma | algebraic_geometry.Scheme.open_cover.ι_from_glued | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_glued_injective : function.injective 𝒰.from_glued.1.base | begin
intros x y h,
obtain ⟨i, x, rfl⟩ := 𝒰.glued_cover.ι_jointly_surjective x,
obtain ⟨j, y, rfl⟩ := 𝒰.glued_cover.ι_jointly_surjective y,
simp_rw [← comp_apply, ← SheafedSpace.comp_base, ← LocallyRingedSpace.comp_val] at h,
erw [ι_from_glued, ι_from_glued] at h,
let e := (Top.pullback_cone_is_limit _ _... | lemma | algebraic_geometry.Scheme.open_cover.from_glued_injective | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"Top.pullback_cone_is_limit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_glued_stalk_iso (x : 𝒰.glued_cover.glued.carrier) :
is_iso (PresheafedSpace.stalk_map 𝒰.from_glued.val x) | begin
obtain ⟨i, x, rfl⟩ := 𝒰.glued_cover.ι_jointly_surjective x,
have := PresheafedSpace.stalk_map.congr_hom _ _
(congr_arg LocallyRingedSpace.hom.val $ 𝒰.ι_from_glued i) x,
erw PresheafedSpace.stalk_map.comp at this,
rw ← is_iso.eq_comp_inv at this,
rw this,
apply_instance,
end | instance | algebraic_geometry.Scheme.open_cover.from_glued_stalk_iso | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_glued_open_map : is_open_map 𝒰.from_glued.1.base | begin
intros U hU,
rw is_open_iff_forall_mem_open,
intros x hx,
rw 𝒰.glued_cover.is_open_iff at hU,
use 𝒰.from_glued.val.base '' U ∩ set.range (𝒰.map (𝒰.f x)).1.base,
use set.inter_subset_left _ _,
split,
{ rw ← set.image_preimage_eq_inter_range,
apply (show is_open_immersion (𝒰.map (𝒰.f x)), ... | lemma | algebraic_geometry.Scheme.open_cover.from_glued_open_map | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"is_open_iff_forall_mem_open",
"is_open_map",
"set.image_preimage_eq_inter_range",
"set.inter_subset_left",
"set.preimage_comp",
"set.preimage_image_eq",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_glued_open_embedding : open_embedding 𝒰.from_glued.1.base | open_embedding_of_continuous_injective_open (by continuity) 𝒰.from_glued_injective
𝒰.from_glued_open_map | lemma | algebraic_geometry.Scheme.open_cover.from_glued_open_embedding | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"continuity",
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_glued_open_immersion : is_open_immersion 𝒰.from_glued | SheafedSpace.is_open_immersion.of_stalk_iso _ 𝒰.from_glued_open_embedding | instance | algebraic_geometry.Scheme.open_cover.from_glued_open_immersion | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_morphisms {Y : Scheme} (f : ∀ x, 𝒰.obj x ⟶ Y)
(hf : ∀ x y, (pullback.fst : pullback (𝒰.map x) (𝒰.map y) ⟶ _) ≫ f x = pullback.snd ≫ f y) :
X ⟶ Y | begin
refine inv 𝒰.from_glued ≫ _,
fapply multicoequalizer.desc,
exact f,
rintro ⟨i, j⟩,
change pullback.fst ≫ f i = (_ ≫ _) ≫ f j,
erw pullback_symmetry_hom_comp_fst,
exact hf i j
end | def | algebraic_geometry.Scheme.open_cover.glue_morphisms | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | Given an open cover of `X`, and a morphism `𝒰.obj x ⟶ Y` for each open subscheme in the cover,
such that these morphisms are compatible in the intersection (pullback), we may glue the morphisms
together into a morphism `X ⟶ Y`.
Note:
If `X` is exactly (defeq to) the gluing of `U i`, then using `multicoequalizer.desc`... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_glue_morphisms {Y : Scheme} (f : ∀ x, 𝒰.obj x ⟶ Y)
(hf : ∀ x y, (pullback.fst : pullback (𝒰.map x) (𝒰.map y) ⟶ _) ≫ f x = pullback.snd ≫ f y)
(x : 𝒰.J) : (𝒰.map x) ≫ 𝒰.glue_morphisms f hf = f x | begin
rw [← ι_from_glued, category.assoc],
erw [is_iso.hom_inv_id_assoc, multicoequalizer.π_desc],
end | lemma | algebraic_geometry.Scheme.open_cover.ι_glue_morphisms | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {Y : Scheme} (f₁ f₂ : X ⟶ Y) (h : ∀ x, 𝒰.map x ≫ f₁ = 𝒰.map x ≫ f₂) : f₁ = f₂ | begin
rw ← cancel_epi 𝒰.from_glued,
apply multicoequalizer.hom_ext,
intro x,
erw multicoequalizer.π_desc_assoc,
erw multicoequalizer.π_desc_assoc,
exact h x,
end | lemma | algebraic_geometry.Scheme.open_cover.hom_ext | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_Z_is_terminal : is_terminal (Scheme.Spec.obj (op $ CommRing.of ℤ)) | @@is_terminal.is_terminal_obj _ _ Scheme.Spec _ infer_instance
(terminal_op_of_initial CommRing.Z_is_initial) | def | algebraic_geometry.Spec_Z_is_terminal | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"CommRing.Z_is_initial",
"CommRing.of"
] | `Spec ℤ` is the terminal object in the category of schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.empty_to (X : Scheme.{u}) : ∅ ⟶ X | ⟨{ base := ⟨λ x, pempty.elim x, by continuity⟩,
c := { app := λ U, CommRing.punit_is_terminal.from _ } }, λ x, pempty.elim x⟩ | def | algebraic_geometry.Scheme.empty_to | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"pempty.elim"
] | The map from the empty scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.empty_ext {X : Scheme.{u}} (f g : ∅ ⟶ X) : f = g | by { ext a, exact pempty.elim a } | lemma | algebraic_geometry.Scheme.empty_ext | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"pempty.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.eq_empty_to {X : Scheme.{u}} (f : ∅ ⟶ X) : f = Scheme.empty_to X | Scheme.empty_ext f (Scheme.empty_to X) | lemma | algebraic_geometry.Scheme.eq_empty_to | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
empty_is_initial : is_initial (∅ : Scheme.{u}) | is_initial.of_unique _ | def | algebraic_geometry.empty_is_initial | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [] | The empty scheme is the initial object in the category of schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_is_initial_to : empty_is_initial.to = Scheme.empty_to | rfl | lemma | algebraic_geometry.empty_is_initial_to | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_punit_is_empty : is_empty (Scheme.Spec.obj (op $ CommRing.of punit)).carrier | ⟨prime_spectrum.punit⟩ | instance | algebraic_geometry.Spec_punit_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"CommRing.of",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty X.carrier] :
is_open_immersion f | begin
apply_with is_open_immersion.of_stalk_iso { instances := ff },
{ apply open_embedding_of_continuous_injective_open,
{ continuity },
{ rintro (i : X.carrier), exact is_empty_elim i },
{ intros U hU, convert is_open_empty, ext, apply (iff_false _).mpr,
exact λ x, is_empty_elim (show X.carrier,... | instance | algebraic_geometry.is_open_immersion_of_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"continuity",
"is_empty",
"is_empty_elim",
"is_open_empty",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty Y.carrier] : is_iso f | begin
haveI : is_empty X.carrier := ⟨λ x, is_empty_elim (show Y.carrier, from f.1.base x)⟩,
haveI : epi f.1.base,
{ rw Top.epi_iff_surjective, rintro (x : Y.carrier), exact is_empty_elim x },
apply is_open_immersion.to_iso
end | instance | algebraic_geometry.is_iso_of_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"Top.epi_iff_surjective",
"is_empty",
"is_empty_elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_initial_of_is_empty {X : Scheme} [is_empty X.carrier] : is_initial X | empty_is_initial.of_iso (as_iso $ empty_is_initial.to _) | def | algebraic_geometry.is_initial_of_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"is_empty"
] | A scheme is initial if its underlying space is empty . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec_punit_is_initial : is_initial (Scheme.Spec.obj (op $ CommRing.of punit)) | empty_is_initial.of_iso (as_iso $ empty_is_initial.to _) | def | algebraic_geometry.Spec_punit_is_initial | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"CommRing.of"
] | `Spec 0` is the initial object in the category of schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine_of_is_empty {X : Scheme} [is_empty X.carrier] : is_affine X | is_affine_of_iso (inv (empty_is_initial.to X) ≫
empty_is_initial.to (Scheme.Spec.obj (op $ CommRing.of punit))) | instance | algebraic_geometry.is_affine_of_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"CommRing.of",
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_is_empty : is_empty (⊥_ Scheme).carrier | ⟨λ x, ((initial.to Scheme.empty : _).1.base x).elim⟩ | instance | algebraic_geometry.initial_is_empty | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_is_affine_open (X : Scheme) : is_affine_open (⊥ : opens X.carrier) | begin
convert range_is_affine_open_of_open_immersion (initial.to X),
ext,
exact (false_iff _).mpr (λ x, is_empty_elim (show (⊥_ Scheme).carrier, from x.some)),
end | lemma | algebraic_geometry.bot_is_affine_open | algebraic_geometry | src/algebraic_geometry/limits.lean | [
"algebraic_geometry.pullbacks",
"algebraic_geometry.AffineScheme"
] | [
"is_empty_elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LocallyRingedSpace extends SheafedSpace CommRing | (local_ring : ∀ x, local_ring (presheaf.stalk x)) | structure | algebraic_geometry.LocallyRingedSpace | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"CommRing",
"local_ring"
] | A `LocallyRingedSpace` is a topological space equipped with a sheaf of commutative rings
such that all the stalks are local rings.
A morphism of locally ringed spaces is a morphism of ringed spaces
such that the morphisms induced on stalks are local ring homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_RingedSpace : RingedSpace | X.to_SheafedSpace | def | algebraic_geometry.LocallyRingedSpace.to_RingedSpace | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | An alias for `to_SheafedSpace`, where the result type is a `RingedSpace`.
This allows us to use dot-notation for the `RingedSpace` namespace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Top : Top | X.1.carrier | def | algebraic_geometry.LocallyRingedSpace.to_Top | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"Top"
] | The underlying topological space of a locally ringed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
𝒪 : sheaf CommRing X.to_Top | X.to_SheafedSpace.sheaf | def | algebraic_geometry.LocallyRingedSpace.𝒪 | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"CommRing"
] | The structure sheaf of a locally ringed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (X Y : LocallyRingedSpace.{u}) : Type u | (val : X.to_SheafedSpace ⟶ Y.to_SheafedSpace)
(prop : ∀ x, is_local_ring_hom (PresheafedSpace.stalk_map val x)) | structure | algebraic_geometry.LocallyRingedSpace.hom | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_local_ring_hom"
] | A morphism of locally ringed spaces is a morphism of ringed spaces
such that the morphims induced on stalks are local ring homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk (X : LocallyRingedSpace) (x : X) : CommRing | X.presheaf.stalk x | def | algebraic_geometry.LocallyRingedSpace.stalk | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_map {X Y : LocallyRingedSpace} (f : X ⟶ Y) (x : X) :
Y.stalk (f.1.1 x) ⟶ X.stalk x | PresheafedSpace.stalk_map f.1 x | def | algebraic_geometry.LocallyRingedSpace.stalk_map | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | A morphism of locally ringed spaces `f : X ⟶ Y` induces
a local ring homomorphism from `Y.stalk (f x)` to `X.stalk x` for any `x : X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (X : LocallyRingedSpace) : hom X X | ⟨𝟙 _, λ x, by { erw PresheafedSpace.stalk_map.id, apply is_local_ring_hom_id, }⟩ | def | algebraic_geometry.LocallyRingedSpace.id | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_local_ring_hom_id"
] | The identity morphism on a locally ringed space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {X Y Z : LocallyRingedSpace} (f : hom X Y) (g : hom Y Z) : hom X Z | ⟨f.val ≫ g.val, λ x,
begin
erw PresheafedSpace.stalk_map.comp,
exact @is_local_ring_hom_comp _ _ _ _ _ _ _ _ (f.2 _) (g.2 _),
end⟩ | def | algebraic_geometry.LocallyRingedSpace.comp | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_local_ring_hom_comp"
] | Composition of morphisms of locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_SheafedSpace : LocallyRingedSpace ⥤ SheafedSpace CommRing | { obj := λ X, X.to_SheafedSpace,
map := λ X Y f, f.1, } | def | algebraic_geometry.LocallyRingedSpace.forget_to_SheafedSpace | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"CommRing"
] | The forgetful functor from `LocallyRingedSpace` to `SheafedSpace CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_Top : LocallyRingedSpace ⥤ Top | forget_to_SheafedSpace ⋙ SheafedSpace.forget _ | def | algebraic_geometry.LocallyRingedSpace.forget_to_Top | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"Top"
] | The forgetful functor from `LocallyRingedSpace` to `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_val {X Y Z : LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val = f.val ≫ g.val | rfl | lemma | algebraic_geometry.LocallyRingedSpace.comp_val | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val_c {X Y Z : LocallyRingedSpace.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.c = g.val.c ≫ (presheaf.pushforward _ g.val.base).map f.val.c | rfl | lemma | algebraic_geometry.LocallyRingedSpace.comp_val_c | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val_c_app {X Y Z : LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z) (U : (opens Z)ᵒᵖ) :
(f ≫ g).val.c.app U = g.val.c.app U ≫ f.val.c.app (op $ (opens.map g.val.base).obj U.unop) | rfl | lemma | algebraic_geometry.LocallyRingedSpace.comp_val_c_app | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_SheafedSpace_hom_of_is_iso {X Y : LocallyRingedSpace}
(f : X.to_SheafedSpace ⟶ Y.to_SheafedSpace) [is_iso f] : X ⟶ Y | hom.mk f $ λ x,
-- Here we need to see that the stalk maps are really local ring homomorphisms.
-- This can be solved by type class inference, because stalk maps of isomorphisms are isomorphisms
-- and isomorphisms are local ring homomorphisms.
show is_local_ring_hom (PresheafedSpace.stalk_map
(SheafedSpace.forget_to... | def | algebraic_geometry.LocallyRingedSpace.hom_of_SheafedSpace_hom_of_is_iso | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_local_ring_hom"
] | Given two locally ringed spaces `X` and `Y`, an isomorphism between `X` and `Y` as _sheafed_
spaces can be lifted to a morphism `X ⟶ Y` as locally ringed spaces.
See also `iso_of_SheafedSpace_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_SheafedSpace_iso {X Y : LocallyRingedSpace}
(f : X.to_SheafedSpace ≅ Y.to_SheafedSpace) : X ≅ Y | { hom := hom_of_SheafedSpace_hom_of_is_iso f.hom,
inv := hom_of_SheafedSpace_hom_of_is_iso f.inv,
hom_inv_id' := hom.ext _ _ f.hom_inv_id,
inv_hom_id' := hom.ext _ _ f.inv_hom_id } | def | algebraic_geometry.LocallyRingedSpace.iso_of_SheafedSpace_iso | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | Given two locally ringed spaces `X` and `Y`, an isomorphism between `X` and `Y` as _sheafed_
spaces can be lifted to an isomorphism `X ⟶ Y` as locally ringed spaces.
This is related to the property that the functor `forget_to_SheafedSpace` reflects isomorphisms.
In fact, it is slightly stronger as we do not require `f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_SheafedSpace_iso {X Y : LocallyRingedSpace} (f : X ⟶ Y) [is_iso f] :
is_iso f.1 | LocallyRingedSpace.forget_to_SheafedSpace.map_is_iso f | instance | algebraic_geometry.LocallyRingedSpace.is_SheafedSpace_iso | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict {U : Top} (X : LocallyRingedSpace) {f : U ⟶ X.to_Top}
(h : open_embedding f) : LocallyRingedSpace | { local_ring :=
begin
intro x,
dsimp at *,
-- We show that the stalk of the restriction is isomorphic to the original stalk,
apply @ring_equiv.local_ring _ _ _ (X.local_ring (f x)),
exact (X.to_PresheafedSpace.restrict_stalk_iso h x).symm.CommRing_iso_to_ring_equiv,
end,
to_SheafedSpace := X.t... | def | algebraic_geometry.LocallyRingedSpace.restrict | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"Top",
"local_ring",
"open_embedding",
"ring_equiv.local_ring"
] | The restriction of a locally ringed space along an open embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_restrict {U : Top} (X : LocallyRingedSpace) {f : U ⟶ X.to_Top}
(h : open_embedding f) : X.restrict h ⟶ X | ⟨X.to_PresheafedSpace.of_restrict h, λ x, infer_instance⟩ | def | algebraic_geometry.LocallyRingedSpace.of_restrict | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"Top",
"open_embedding"
] | The canonical map from the restriction to the supspace. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_top_iso (X : LocallyRingedSpace) :
X.restrict (opens.open_embedding ⊤) ≅ X | @iso_of_SheafedSpace_iso (X.restrict (opens.open_embedding ⊤)) X
X.to_SheafedSpace.restrict_top_iso | def | algebraic_geometry.LocallyRingedSpace.restrict_top_iso | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | The restriction of a locally ringed space `X` to the top subspace is isomorphic to `X` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : LocallyRingedSpaceᵒᵖ ⥤ CommRing | forget_to_SheafedSpace.op ⋙ SheafedSpace.Γ | def | algebraic_geometry.LocallyRingedSpace.Γ | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"CommRing"
] | The global sections, notated Gamma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ_def : Γ = forget_to_SheafedSpace.op ⋙ SheafedSpace.Γ | rfl | lemma | algebraic_geometry.LocallyRingedSpace.Γ_def | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj (X : LocallyRingedSpaceᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.LocallyRingedSpace.Γ_obj | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj_op (X : LocallyRingedSpace) : Γ.obj (op X) = X.presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.LocallyRingedSpace.Γ_obj_op | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map {X Y : LocallyRingedSpaceᵒᵖ} (f : X ⟶ Y) :
Γ.map f = f.unop.1.c.app (op ⊤) | rfl | lemma | algebraic_geometry.LocallyRingedSpace.Γ_map | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map_op {X Y : LocallyRingedSpace} (f : X ⟶ Y) :
Γ.map f.op = f.1.c.app (op ⊤) | rfl | lemma | algebraic_geometry.LocallyRingedSpace.Γ_map_op | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_basic_open {X Y : LocallyRingedSpace} (f : X ⟶ Y) {U : opens Y}
(s : Y.presheaf.obj (op U)) :
(opens.map f.1.base).obj (Y.to_RingedSpace.basic_open s) =
@RingedSpace.basic_open X.to_RingedSpace ((opens.map f.1.base).obj U) (f.1.c.app _ s) | begin
ext,
split,
{ rintros ⟨⟨y, hyU⟩, (hy : is_unit _), (rfl : y = _)⟩,
erw RingedSpace.mem_basic_open _ _ ⟨x, show x ∈ (opens.map f.1.base).obj U, from hyU⟩,
rw ← PresheafedSpace.stalk_map_germ_apply,
exact (PresheafedSpace.stalk_map f.1 _).is_unit_map hy },
{ rintros ⟨y, (hy : is_unit _), rfl⟩,
... | lemma | algebraic_geometry.LocallyRingedSpace.preimage_basic_open | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_unit",
"is_unit_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_zero (X : LocallyRingedSpace) (U : opens X.carrier) :
X.to_RingedSpace.basic_open (0 : X.presheaf.obj $ op U) = ⊥ | begin
simp only [RingedSpace.basic_open, is_unit_zero_iff, map_zero,
zero_ne_one' (X.presheaf.stalk _), set.set_of_false, set.image_empty],
refl
end | lemma | algebraic_geometry.LocallyRingedSpace.basic_open_zero | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"is_unit_zero_iff",
"set.image_empty",
"set.set_of_false",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
component_nontrivial (X : LocallyRingedSpace) (U : opens X.carrier)
[hU : nonempty U] : nontrivial (X.presheaf.obj $ op U) | (X.to_PresheafedSpace.presheaf.germ hU.some).domain_nontrivial | instance | algebraic_geometry.LocallyRingedSpace.component_nontrivial | algebraic_geometry | src/algebraic_geometry/locally_ringed_space.lean | [
"algebraic_geometry.ringed_space",
"algebraic_geometry.stalks"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
PresheafedSpace | (carrier : Top.{w})
(presheaf : carrier.presheaf C) | structure | algebraic_geometry.PresheafedSpace | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | A `PresheafedSpace C` is a topological space equipped with a presheaf of `C`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_carrier : has_coe (PresheafedSpace.{w v u} C) Top.{w} | { coe := λ X, X.carrier } | instance | algebraic_geometry.PresheafedSpace.coe_carrier | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_coe (X : PresheafedSpace.{w v u} C) : X.carrier = (X : Top.{w}) | rfl | lemma | algebraic_geometry.PresheafedSpace.as_coe | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (carrier) (presheaf) : (({ carrier := carrier, presheaf := presheaf } :
PresheafedSpace.{v} C) : Top.{v}) = carrier | rfl | lemma | algebraic_geometry.PresheafedSpace.mk_coe | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (X : Top) (Z : C) : PresheafedSpace C | { carrier := X,
presheaf :=
{ obj := λ U, Z,
map := λ U V f, 𝟙 Z, } } | def | algebraic_geometry.PresheafedSpace.const | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"Top"
] | The constant presheaf on `X` with value `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (X Y : PresheafedSpace.{w v u} C) | (base : (X : Top.{w}) ⟶ (Y : Top.{w}))
(c : Y.presheaf ⟶ base _* X.presheaf) | structure | algebraic_geometry.PresheafedSpace.hom | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | A morphism between presheafed spaces `X` and `Y` consists of a continuous map
`f` between the underlying topological spaces, and a (notice contravariant!) map
from the presheaf on `Y` to the pushforward of the presheaf on `X` via `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {X Y : PresheafedSpace C} (α β : hom X Y)
(w : α.base = β.base)
(h : α.c ≫ (whisker_right (eq_to_hom (by rw w)) _) = β.c) :
α = β | begin
cases α, cases β,
dsimp [presheaf.pushforward_obj] at *,
tidy, -- TODO including `injections` would make tidy work earlier.
end | lemma | algebraic_geometry.PresheafedSpace.ext | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hext {X Y : PresheafedSpace C} (α β : hom X Y)
(w : α.base = β.base)
(h : α.c == β.c) :
α = β | by { cases α, cases β, congr, exacts [w,h] } | lemma | algebraic_geometry.PresheafedSpace.hext | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (X : PresheafedSpace.{w v u} C) : hom X X | { base := 𝟙 (X : Top.{w}),
c := eq_to_hom (presheaf.pushforward.id_eq X.presheaf).symm } | def | algebraic_geometry.PresheafedSpace.id | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | The identity morphism of a `PresheafedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_inhabited (X : PresheafedSpace C) : inhabited (hom X X) | ⟨id X⟩ | instance | algebraic_geometry.PresheafedSpace.hom_inhabited | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {X Y Z : PresheafedSpace C} (α : hom X Y) (β : hom Y Z) : hom X Z | { base := α.base ≫ β.base,
c := β.c ≫ (presheaf.pushforward _ β.base).map α.c } | def | algebraic_geometry.PresheafedSpace.comp | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | Composition of morphisms of `PresheafedSpace`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_c {X Y Z : PresheafedSpace C} (α : hom X Y) (β : hom Y Z) :
(comp α β).c = β.c ≫ (presheaf.pushforward _ β.base).map α.c | rfl | lemma | algebraic_geometry.PresheafedSpace.comp_c | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category_of_PresheafedSpaces : category (PresheafedSpace.{v v u} C) | { hom := hom,
id := id,
comp := λ X Y Z f g, comp f g,
id_comp' := λ X Y f, begin
ext1,
{ rw comp_c,
erw eq_to_hom_map,
simp only [eq_to_hom_refl, assoc, whisker_right_id'],
erw [comp_id, comp_id] },
apply id_comp
end,
comp_id' := λ X Y f, begin
ext1,
{ rw comp_c,
e... | instance | algebraic_geometry.PresheafedSpace.category_of_PresheafedSpaces | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map
from the presheaf on the target to the pushforward of the presheaf on the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_base (X : PresheafedSpace.{v v u} C) :
((𝟙 X) : X ⟶ X).base = 𝟙 (X : Top.{v}) | rfl | lemma | algebraic_geometry.PresheafedSpace.id_base | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_c (X : PresheafedSpace.{v v u} C) :
((𝟙 X) : X ⟶ X).c = eq_to_hom (presheaf.pushforward.id_eq X.presheaf).symm | rfl | lemma | algebraic_geometry.PresheafedSpace.id_c | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_c_app (X : PresheafedSpace.{v v u} C) (U) :
((𝟙 X) : X ⟶ X).c.app U = X.presheaf.map
(eq_to_hom (by { induction U using opposite.rec, cases U, refl })) | by { induction U using opposite.rec, cases U, simp only [id_c], dsimp, simp, } | lemma | algebraic_geometry.PresheafedSpace.id_c_app | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_base {X Y Z : PresheafedSpace.{v v u} C} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).base = f.base ≫ g.base | rfl | lemma | algebraic_geometry.PresheafedSpace.comp_base | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_fun_eq {X Y : PresheafedSpace.{v v u} C} (f : X ⟶ Y) : (f : X → Y) = f.base | rfl | lemma | algebraic_geometry.PresheafedSpace.coe_to_fun_eq | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_c_app {X Y Z : PresheafedSpace.{v v u} C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :
(α ≫ β).c.app U = (β.c).app U ≫ (α.c).app (op ((opens.map (β.base)).obj (unop U))) | rfl | lemma | algebraic_geometry.PresheafedSpace.comp_c_app | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | Sometimes rewriting with `comp_c_app` doesn't work because of dependent type issues.
In that case, `erw comp_c_app_assoc` might make progress.
The lemma `comp_c_app_assoc` is also better suited for rewrites in the opposite direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_app {X Y : PresheafedSpace.{v v u} C} {α β : X ⟶ Y} (h : α = β) (U) :
α.c.app U = β.c.app U ≫ X.presheaf.map (eq_to_hom (by subst h)) | by { subst h, dsimp, simp, } | lemma | algebraic_geometry.PresheafedSpace.congr_app | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : PresheafedSpace.{v v u} C ⥤ Top | { obj := λ X, (X : Top.{v}),
map := λ X Y f, f.base } | def | algebraic_geometry.PresheafedSpace.forget | algebraic_geometry | src/algebraic_geometry/presheafed_space.lean | [
"topology.sheaves.presheaf",
"category_theory.adjunction.fully_faithful"
] | [
"Top"
] | The forgetful functor from `PresheafedSpace` to `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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