statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
quasi_separated_eq_diagonal_is_quasi_compact :
@quasi_separated = morphism_property.diagonal @quasi_compact | by { ext, exact quasi_separated_iff _ } | lemma | algebraic_geometry.quasi_separated_eq_diagonal_is_quasi_compact | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_compact_affine_property_diagonal_eq :
quasi_compact.affine_property.diagonal = quasi_separated.affine_property | by { ext, rw quasi_compact_affine_property_iff_quasi_separated_space, refl } | lemma | algebraic_geometry.quasi_compact_affine_property_diagonal_eq | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_eq_affine_property_diagonal :
@quasi_separated =
target_affine_locally quasi_compact.affine_property.diagonal | begin
rw [quasi_separated_eq_diagonal_is_quasi_compact, quasi_compact_eq_affine_property],
exact diagonal_target_affine_locally_eq_target_affine_locally
_ quasi_compact.affine_property_is_local
end | lemma | algebraic_geometry.quasi_separated_eq_affine_property_diagonal | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_eq_affine_property :
@quasi_separated =
target_affine_locally quasi_separated.affine_property | by rw [quasi_separated_eq_affine_property_diagonal, quasi_compact_affine_property_diagonal_eq] | lemma | algebraic_geometry.quasi_separated_eq_affine_property | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.affine_property_is_local :
quasi_separated.affine_property.is_local | quasi_compact_affine_property_diagonal_eq ▸
quasi_compact.affine_property_is_local.diagonal | lemma | algebraic_geometry.quasi_separated.affine_property_is_local | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_of_mono {X Y : Scheme} (f : X ⟶ Y) [mono f] : quasi_separated f | ⟨infer_instance⟩ | instance | algebraic_geometry.quasi_separated_of_mono | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_stable_under_composition :
morphism_property.stable_under_composition @quasi_separated | quasi_separated_eq_diagonal_is_quasi_compact.symm ▸
quasi_compact_stable_under_composition.diagonal
quasi_compact_respects_iso
quasi_compact_stable_under_base_change | lemma | algebraic_geometry.quasi_separated_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_stable_under_base_change :
morphism_property.stable_under_base_change @quasi_separated | quasi_separated_eq_diagonal_is_quasi_compact.symm ▸
quasi_compact_stable_under_base_change.diagonal
quasi_compact_respects_iso | lemma | algebraic_geometry.quasi_separated_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[quasi_separated f] [quasi_separated g] : quasi_separated (f ≫ g) | quasi_separated_stable_under_composition f g infer_instance infer_instance | instance | algebraic_geometry.quasi_separated_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_respects_iso : morphism_property.respects_iso @quasi_separated | quasi_separated_eq_diagonal_is_quasi_compact.symm ▸
quasi_compact_respects_iso.diagonal | lemma | algebraic_geometry.quasi_separated_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.affine_open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [quasi_separated f,
∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)],
∀ (i : 𝒰.J), quasi_separated_space (pullback f (𝒰.map i)).carrier,
∀ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)] (i : 𝒰.J),
... | begin
have := quasi_compact.affine_property_is_local.diagonal_affine_open_cover_tfae f,
simp_rw [← quasi_compact_eq_affine_property,
← quasi_separated_eq_diagonal_is_quasi_compact,
quasi_compact_affine_property_diagonal_eq] at this,
exact this
end | lemma | algebraic_geometry.quasi_separated.affine_open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"compact_space",
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.is_local_at_target :
property_is_local_at_target @quasi_separated | quasi_separated_eq_affine_property_diagonal.symm ▸
quasi_compact.affine_property_is_local.diagonal.target_affine_locally_is_local | lemma | algebraic_geometry.quasi_separated.is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) :
tfae [quasi_separated f,
∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J),
quasi_separated (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i),
∀ (𝒰 : Scheme.open_cover.{u} Y) (i : 𝒰.J),
quasi_separated (pullback.snd : (𝒰.pullba... | quasi_separated.is_local_at_target.open_cover_tfae f | lemma | algebraic_geometry.quasi_separated.open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_over_affine_iff {X Y : Scheme} (f : X ⟶ Y) [is_affine Y] :
quasi_separated f ↔ quasi_separated_space X.carrier | by rw [quasi_separated_eq_affine_property,
quasi_separated.affine_property_is_local.affine_target_iff f,
quasi_separated.affine_property] | lemma | algebraic_geometry.quasi_separated_over_affine_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_space_iff_quasi_separated (X : Scheme) :
quasi_separated_space X.carrier ↔ quasi_separated (terminal.from X) | (quasi_separated_over_affine_iff _).symm | lemma | algebraic_geometry.quasi_separated_space_iff_quasi_separated | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.affine_open_cover_iff {X Y : Scheme.{u}} (𝒰 : Scheme.open_cover.{u} Y)
[∀ i, is_affine (𝒰.obj i)] (f : X ⟶ Y) :
quasi_separated f ↔ ∀ i, quasi_separated_space (pullback f (𝒰.map i)).carrier | begin
rw [quasi_separated_eq_affine_property,
quasi_separated.affine_property_is_local.affine_open_cover_iff f 𝒰],
refl,
end | lemma | algebraic_geometry.quasi_separated.affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated.open_cover_iff {X Y : Scheme.{u}} (𝒰 : Scheme.open_cover.{u} Y)
(f : X ⟶ Y) :
quasi_separated f ↔ ∀ i, quasi_separated (pullback.snd : pullback f (𝒰.map i) ⟶ _) | quasi_separated.is_local_at_target.open_cover_iff f 𝒰 | lemma | algebraic_geometry.quasi_separated.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_space_of_quasi_separated {X Y : Scheme} (f : X ⟶ Y)
[hY : quasi_separated_space Y.carrier] [quasi_separated f] : quasi_separated_space X.carrier | begin
rw quasi_separated_space_iff_quasi_separated at hY ⊢,
have : f ≫ terminal.from Y = terminal.from X := terminal_is_terminal.hom_ext _ _,
rw ← this,
resetI, apply_instance
end | lemma | algebraic_geometry.quasi_separated_space_of_quasi_separated | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_space_of_is_affine (X : Scheme) [is_affine X] :
quasi_separated_space X.carrier | begin
constructor,
intros U V hU hU' hV hV',
obtain ⟨s, hs, e⟩ := (is_compact_open_iff_eq_basic_open_union _).mp ⟨hU', hU⟩,
obtain ⟨s', hs', e'⟩ := (is_compact_open_iff_eq_basic_open_union _).mp ⟨hV', hV⟩,
rw [e, e', set.Union₂_inter],
simp_rw [set.inter_Union₂],
apply hs.is_compact_bUnion,
{ intros i h... | instance | algebraic_geometry.quasi_separated_space_of_is_affine | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"is_compact",
"quasi_separated_space",
"set.Union₂_inter",
"set.inter_Union₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.is_quasi_separated {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
is_quasi_separated (U : set X.carrier) | begin
rw is_quasi_separated_iff_quasi_separated_space,
exacts [@@algebraic_geometry.quasi_separated_space_of_is_affine _ hU, U.is_open],
end | lemma | algebraic_geometry.is_affine_open.is_quasi_separated | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"algebraic_geometry.quasi_separated_space_of_is_affine",
"is_quasi_separated",
"is_quasi_separated_iff_quasi_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_separated_of_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[H : quasi_separated (f ≫ g)] : quasi_separated f | begin
rw (quasi_separated.affine_open_cover_tfae f).out 0 1,
rw (quasi_separated.affine_open_cover_tfae (f ≫ g)).out 0 2 at H,
use (Z.affine_cover.pullback_cover g).bind (λ x, Scheme.affine_cover _),
split, { intro i, dsimp, apply_instance },
rintro ⟨i, j⟩, dsimp at *,
specialize H _ i,
refine @@quasi_sep... | lemma | algebraic_geometry.quasi_separated_of_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"algebraic_geometry.quasi_separated_of_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_mul_of_is_affine_open (X : Scheme) (U : opens X.carrier) (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) (x : X.presheaf.obj (op $ X.basic_open f)) :
∃ (n : ℕ) (y : X.presheaf.obj (op U)),
y |_ X.basic_open f = (f |_ X.basic_open f) ^ n * x | begin
have := (is_localization_basic_open hU f).2,
obtain ⟨⟨y, _, n, rfl⟩, d⟩ := this x,
use [n, y],
delta Top.presheaf.restrict_open Top.presheaf.restrict,
simpa [mul_comm x] using d.symm,
end | lemma | algebraic_geometry.exists_eq_pow_mul_of_is_affine_open | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"Top.presheaf.restrict",
"Top.presheaf.restrict_open",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux (X : Scheme)
(S : X.affine_opens) (U₁ U₂ : opens X.carrier)
{n₁ n₂ : ℕ} {y₁ : X.presheaf.obj (op U₁)}
{y₂ : X.presheaf.obj (op U₂)} {f : X.presheaf.obj (op $ U₁ ⊔ U₂)}
{x : X.presheaf.obj (op $ X.basic_open f)}
(h₁ : S.1 ≤ U₁) (h₂ : S.1 ≤ U₂)
(e₁ ... | begin
have := (is_localization_basic_open S.2
(X.presheaf.map (hom_of_le $ le_trans h₁ le_sup_left).op f)),
obtain ⟨⟨_, n, rfl⟩, e⟩ :=
(@is_localization.eq_iff_exists _ _ _ _ _ _ this (X.presheaf.map (hom_of_le $ h₁).op
((X.presheaf.map (hom_of_le le_sup_left).op f) ^ n₂ * y₁))
(X.presheaf.map (ho... | lemma | algebraic_geometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"inf_le_left",
"inf_le_right",
"le_sup_left",
"le_sup_right",
"map_mul",
"map_pow",
"mul_assoc",
"pow_add",
"ring_hom.algebra_map_to_algebra",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_mul_of_is_compact_of_is_quasi_separated (X : Scheme)
(U : opens X.carrier) (hU : is_compact U.1) (hU' : is_quasi_separated U.1)
(f : X.presheaf.obj (op U)) (x : X.presheaf.obj (op $ X.basic_open f)) :
∃ (n : ℕ) (y : X.presheaf.obj (op U)), y |_ X.basic_open f = (f |_ X.basic_open f) ^ n * x | begin
delta Top.presheaf.restrict_open Top.presheaf.restrict,
revert hU' f x,
apply compact_open_induction_on U hU,
{ intros hU' f x,
use [0, f],
refine @@subsingleton.elim (CommRing.subsingleton_of_is_terminal
(X.sheaf.is_terminal_of_eq_empty _)) _ _,
erw eq_bot_iff,
exact X.basic_open_le... | lemma | algebraic_geometry.exists_eq_pow_mul_of_is_compact_of_is_quasi_separated | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"CommRing.subsingleton_of_is_terminal",
"Top.presheaf.restrict",
"Top.presheaf.restrict_open",
"eq_bot_iff",
"finset.le_sup",
"finset.mem_univ",
"inf_le_left",
"inf_le_right",
"inf_sup_right",
"is_compact",
"is_quasi_separated",
"le_sup_left",
"le_sup_right",
"le_supr",
"map_mul",
"map... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_basic_open_of_qcqs {X : Scheme} {U : opens X.carrier}
(hU : is_compact U.1) (hU' : is_quasi_separated U.1)
(f : X.presheaf.obj (op U)) :
is_localization.away f (X.presheaf.obj (op $ X.basic_open f)) | begin
constructor,
{ rintro ⟨_, n, rfl⟩,
simp only [map_pow, subtype.coe_mk, ring_hom.algebra_map_to_algebra],
exact is_unit.pow _ (RingedSpace.is_unit_res_basic_open _ f), },
{ intro z,
obtain ⟨n, y, e⟩ := exists_eq_pow_mul_of_is_compact_of_is_quasi_separated X U hU hU' f z,
refine ⟨⟨y, _, n, rfl... | lemma | algebraic_geometry.is_localization_basic_open_of_qcqs | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/quasi_separated.lean | [
"algebraic_geometry.morphisms.quasi_compact",
"topology.quasi_separated"
] | [
"is_compact",
"is_localization.away",
"is_quasi_separated",
"is_unit.pow",
"map_mul",
"map_pow",
"mul_comm",
"mul_right_inj",
"mul_zero",
"ring_hom.algebra_map_to_algebra",
"subtype.coe_mk"
] | If `U` is qcqs, then `Γ(X, D(f)) ≃ Γ(X, U)_f` for every `f : Γ(X, U)`.
This is known as the **Qcqs lemma** in [R. Vakil, *The rising sea*][RisingSea]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
respects_iso.basic_open_iff (hP : respects_iso @P) {X Y : Scheme}
[is_affine X] [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basic_open r).op) ↔
P (@is_localization.away.map (Y.presheaf.obj (opposite.op ⊤)) _
(Y.presheaf.obj (opposite.op $ Y.basic_open r)) _ _
... | begin
rw [Γ_map_morphism_restrict, hP.cancel_left_is_iso, hP.cancel_right_is_iso,
← (hP.cancel_right_is_iso (f.val.c.app (opposite.op (Y.basic_open r))) (X.presheaf.map
(eq_to_hom (Scheme.preimage_basic_open f r).symm).op)), ← eq_iff_iff],
congr,
delta is_localization.away.map,
refine is_localization.... | lemma | ring_hom.respects_iso.basic_open_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"eq_iff_iff",
"is_localization.away.map",
"is_localization.map_comp",
"is_localization.ring_hom_ext",
"opposite.op",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.basic_open_iff_localization (hP : respects_iso @P)
{X Y : Scheme} [is_affine X] [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basic_open r).op) ↔
P (localization.away_map (Scheme.Γ.map f.op) r) | (hP.basic_open_iff _ _).trans (hP.is_localization_away_iff _ _ _ _).symm | lemma | ring_hom.respects_iso.basic_open_iff_localization | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"localization.away_map",
"opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.of_restrict_morphism_restrict_iff (hP : ring_hom.respects_iso @P)
{X Y : Scheme} [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (opposite.op ⊤))
(U : opens X.carrier) (hU : is_affine_open U) {V : opens _}
(e : V = (opens.map (X.of_restrict ((opens.map f.1.base).obj _).open_embedding).1.base).obj U) :
... | begin
subst e,
convert (hP.is_localization_away_iff _ _ _ _).symm,
rotate,
{ apply_instance },
{ apply ring_hom.to_algebra,
refine X.presheaf.map
(@hom_of_le _ _ ((is_open_map.functor _).obj _) ((is_open_map.functor _).obj _) _).op,
rw [← set_like.coe_subset_coe],
dsimp,
simp only [set.i... | lemma | ring_hom.respects_iso.of_restrict_morphism_restrict_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"CommRing",
"algebraic_geometry.is_localization_of_eq_basic_open",
"algebraic_geometry.Γ_restrict_is_localization",
"inf_comm",
"is_localization.away.map",
"is_localization.map_comp",
"is_localization.ring_hom_ext",
"is_open_map.functor",
"le_top",
"localization.away_map",
"open_embedding",
"o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.Γ_pullback_fst
(hP : stable_under_base_change @P) (hP' : respects_iso @P) {X Y S : Scheme}
[is_affine X] [is_affine Y] [is_affine S]
(f : X ⟶ S) (g : Y ⟶ S) (H : P (Scheme.Γ.map g.op)) :
P (Scheme.Γ.map (pullback.fst : pullback f g ⟶ _).op) | begin
rw [← preserves_pullback.iso_inv_fst AffineScheme.forget_to_Scheme
(AffineScheme.of_hom f) (AffineScheme.of_hom g), op_comp, functor.map_comp,
hP'.cancel_right_is_iso, AffineScheme.forget_to_Scheme_map],
have := _root_.congr_arg quiver.hom.unop (preserves_pullback.iso_hom_fst AffineScheme.Γ.right_op
... | lemma | ring_hom.stable_under_base_change.Γ_pullback_fst | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"quiver.hom.unop",
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_locally : affine_target_morphism_property | λ X Y f hY, ∀ (U : X.affine_opens), P (Scheme.Γ.map (X.of_restrict U.1.open_embedding ≫ f).op) | def | algebraic_geometry.source_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | For `P` a property of ring homomorphisms, `source_affine_locally P` holds for `f : X ⟶ Y`
whenever `P` holds for the restriction of `f` on every affine open subset of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
affine_locally : morphism_property Scheme | target_affine_locally (source_affine_locally @P) | abbreviation | algebraic_geometry.affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | For `P` a property of ring homomorphisms, `affine_locally P` holds for `f : X ⟶ Y` if for each
affine open `U = Spec A ⊆ Y` and `V = Spec B ⊆ f ⁻¹' U`, the ring hom `A ⟶ B` satisfies `P`.
Also see `affine_locally_iff_affine_opens_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
source_affine_locally_respects_iso (h₁ : ring_hom.respects_iso @P) :
(source_affine_locally @P).to_property.respects_iso | begin
apply affine_target_morphism_property.respects_iso_mk,
{ introv H U,
rw [← h₁.cancel_right_is_iso _ (Scheme.Γ.map (Scheme.restrict_map_iso e.inv U.1).hom.op),
← functor.map_comp, ← op_comp],
convert H ⟨_, U.prop.map_is_iso e.inv⟩ using 3,
rw [is_open_immersion.iso_of_range_eq_hom, is_open_im... | lemma | algebraic_geometry.source_affine_locally_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_locally_respects_iso (h : ring_hom.respects_iso @P) :
(affine_locally @P).respects_iso | target_affine_locally_respects_iso (source_affine_locally_respects_iso h) | lemma | algebraic_geometry.affine_locally_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_locally_iff_affine_opens_le
(hP : ring_hom.respects_iso @P) {X Y : Scheme} (f : X ⟶ Y) :
affine_locally @P f ↔
(∀ (U : Y.affine_opens) (V : X.affine_opens) (e : V.1 ≤ (opens.map f.1.base).obj U.1),
P (f.app_le e)) | begin
apply forall_congr,
intro U,
delta source_affine_locally,
simp_rw [op_comp, Scheme.Γ.map_comp, Γ_map_morphism_restrict, category.assoc, Scheme.Γ_map_op,
hP.cancel_left_is_iso],
split,
{ intros H V e,
let U' := (opens.map f.val.base).obj U.1,
have e' : U'.open_embedding.is_open_map.functor.... | lemma | algebraic_geometry.affine_locally_iff_affine_opens_le | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.respects_iso",
"subtype.coe_image_subset",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme_restrict_basic_open_of_localization_preserves
(h₁ : ring_hom.respects_iso @P)
(h₂ : ring_hom.localization_preserves @P)
{X Y : Scheme} [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (op ⊤))
(H : source_affine_locally @P f)
(U : (X.restrict ((opens.map f.1.base).obj $ Y.basic_open r).open_embedding).affi... | begin
specialize H ⟨_, U.2.image_is_open_immersion (X.of_restrict _)⟩,
convert (h₁.of_restrict_morphism_restrict_iff _ _ _ _ _).mpr _ using 1,
swap 5,
{ exact h₂.away r H },
{ apply_instance },
{ exact U.2.image_is_open_immersion _},
{ ext1, exact (set.preimage_image_eq _ subtype.coe_injective).symm }
end | lemma | algebraic_geometry.Scheme_restrict_basic_open_of_localization_preserves | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"open_embedding",
"ring_hom.localization_preserves",
"ring_hom.respects_iso",
"set.preimage_image_eq",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_locally_is_local
(h₁ : ring_hom.respects_iso @P)
(h₂ : ring_hom.localization_preserves @P)
(h₃ : ring_hom.of_localization_span @P) : (source_affine_locally @P).is_local | begin
constructor,
{ exact source_affine_locally_respects_iso h₁ },
{ introv H U,
apply Scheme_restrict_basic_open_of_localization_preserves h₁ h₂; assumption },
{ introv hs hs' U,
resetI,
apply h₃ _ _ hs,
intro r,
have := hs' r ⟨(opens.map (X.of_restrict _).1.base).obj U.1, _⟩,
rwa h₁.o... | lemma | algebraic_geometry.source_affine_locally_is_local | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.localization_preserves",
"ring_hom.of_localization_span",
"ring_hom.respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_locally_of_source_open_cover_aux
(h₁ : ring_hom.respects_iso @P)
(h₃ : ring_hom.of_localization_span_target @P)
{X Y : Scheme} (f : X ⟶ Y) (U : X.affine_opens)
(s : set (X.presheaf.obj (op U.1))) (hs : ideal.span s = ⊤)
(hs' : ∀ (r : s), P (Scheme.Γ.map (X.of_restrict (X.basic_open r.1).open_emb... | begin
apply_fun ideal.map (X.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op) at hs,
rw [ideal.map_span, ideal.map_top] at hs,
apply h₃ _ _ hs,
rintro ⟨s, r, hr, hs⟩,
have := (@@localization.alg_equiv _ _ _ _ _ (@@algebraic_geometry.Γ_restrict_is_localization
_ U.2 s)).to_ring_equiv.to_CommRing_iso... | lemma | algebraic_geometry.source_affine_locally_of_source_open_cover_aux | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"CommRing.comp_eq_ring_hom_comp",
"algebraic_geometry.Γ_restrict_is_localization",
"ideal.map",
"ideal.map_span",
"ideal.map_top",
"ideal.span",
"is_localization.map_comp",
"localization.alg_equiv",
"open_embedding",
"ring_hom.algebra_map_to_algebra",
"ring_hom.comp_assoc",
"ring_hom.comp_id",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_immersion_comp_of_source_affine_locally (h₁ : ring_hom.respects_iso @P)
{X Y Z : Scheme} [is_affine X] [is_affine Z] (f : X ⟶ Y) [is_open_immersion f] (g : Y ⟶ Z)
(h₂ : source_affine_locally @P g) :
P (Scheme.Γ.map (f ≫ g).op) | begin
rw [← h₁.cancel_right_is_iso _ (Scheme.Γ.map (is_open_immersion.iso_of_range_eq
(Y.of_restrict _) f _).hom.op), ← functor.map_comp, ← op_comp],
convert h₂ ⟨_, range_is_affine_open_of_open_immersion f⟩ using 3,
{ rw [is_open_immersion.iso_of_range_eq_hom, is_open_immersion.lift_fac_assoc] },
{ apply_in... | lemma | algebraic_geometry.is_open_immersion_comp_of_source_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.respects_iso",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_locally_of_source_open_cover
{X Y : Scheme} (f : X ⟶ Y) [is_affine Y]
(𝒰 : X.open_cover) [∀ i, is_affine (𝒰.obj i)] (H : ∀ i, P (Scheme.Γ.map (𝒰.map i ≫ f).op)) :
source_affine_locally @P f | begin
let S := λ i, (⟨⟨set.range (𝒰.map i).1.base, (𝒰.is_open i).base_open.open_range⟩,
range_is_affine_open_of_open_immersion (𝒰.map i)⟩ : X.affine_opens),
intros U,
apply of_affine_open_cover U,
swap 5, { exact set.range S },
{ intros U r H,
convert hP.stable_under_composition _ _ H _ using 1,
... | lemma | ring_hom.property_is_local.source_affine_locally_of_source_open_cover | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"is_open_map.functor",
"open_embedding",
"ring_hom.algebra_map_to_algebra",
"set.eq_univ_iff_forall",
"set.mem_Union",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_open_cover_tfae {X Y : Scheme.{u}}
[is_affine Y] (f : X ⟶ Y) :
tfae [source_affine_locally @P f,
∃ (𝒰 : Scheme.open_cover.{u} X) [∀ i, is_affine (𝒰.obj i)],
∀ (i : 𝒰.J), P (Scheme.Γ.map (𝒰.map i ≫ f).op),
∀ (𝒰 : Scheme.open_cover.{u} X) [∀ i, is_affine (𝒰.obj i)] (i : 𝒰.J),
P (Sche... | begin
tfae_have : 1 → 4,
{ intros H U g _ hg,
resetI,
specialize H ⟨⟨_, hg.base_open.open_range⟩,
range_is_affine_open_of_open_immersion g⟩,
rw [← hP.respects_iso.cancel_right_is_iso _ (Scheme.Γ.map (is_open_immersion.iso_of_range_eq
g (X.of_restrict (opens.open_embedding ⟨_, hg.base_open.op... | lemma | ring_hom.property_is_local.affine_open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_cover_tfae {X Y : Scheme.{u}} [is_affine Y] (f : X ⟶ Y) :
tfae [source_affine_locally @P f,
∃ (𝒰 : Scheme.open_cover.{u} X), ∀ (i : 𝒰.J), source_affine_locally @P (𝒰.map i ≫ f),
∀ (𝒰 : Scheme.open_cover.{u} X) (i : 𝒰.J), source_affine_locally @P (𝒰.map i ≫ f),
∀ {U : Scheme} (g : U ⟶ X) [is_ope... | begin
tfae_have : 1 → 4,
{ intros H U g hg V,
resetI,
rw (hP.affine_open_cover_tfae f).out 0 3 at H,
haveI : is_affine _ := V.2,
rw ← category.assoc,
apply H },
tfae_have : 4 → 3,
{ intros H 𝒰 _ i, resetI, apply H },
tfae_have : 3 → 2,
{ intro H, refine ⟨X.affine_cover, H _⟩ },
tfae_h... | lemma | ring_hom.property_is_local.open_cover_tfae | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_locally_comp_of_is_open_immersion
{X Y Z : Scheme.{u}} [is_affine Z] (f : X ⟶ Y) (g : Y ⟶ Z) [is_open_immersion f]
(H : source_affine_locally @P g) : source_affine_locally @P (f ≫ g) | by apply ((hP.open_cover_tfae g).out 0 3).mp H | lemma | ring_hom.property_is_local.source_affine_locally_comp_of_is_open_immersion | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_affine_open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
[is_affine Y] (𝒰 : Scheme.open_cover.{u} X) [∀ i, is_affine (𝒰.obj i)] :
source_affine_locally @P f ↔ (∀ i, P (Scheme.Γ.map (𝒰.map i ≫ f).op)) | ⟨λ H, let h := ((hP.affine_open_cover_tfae f).out 0 2).mp H in h 𝒰,
λ H, let h := ((hP.affine_open_cover_tfae f).out 1 0).mp in h ⟨𝒰, infer_instance, H⟩⟩ | lemma | ring_hom.property_is_local.source_affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_source_affine_locally :
(source_affine_locally @P).is_local | source_affine_locally_is_local hP.respects_iso hP.localization_preserves
(@ring_hom.property_is_local.of_localization_span _ hP) | lemma | ring_hom.property_is_local.is_local_source_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"ring_hom.property_is_local.of_localization_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_affine_locally :
property_is_local_at_target (affine_locally @P) | hP.is_local_source_affine_locally.target_affine_locally_is_local | lemma | ring_hom.property_is_local.is_local_affine_locally | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)]
(𝒰' : ∀ i, Scheme.open_cover.{u} ((𝒰.pullback_cover f).obj i)) [∀ i j, is_affine ((𝒰' i).obj j)] :
affine_locally @P f ↔
(∀ i j, P (Scheme.Γ.map ((𝒰' i).map j ≫ pullback.snd).op)) | (hP.is_local_source_affine_locally.affine_open_cover_iff f 𝒰).trans
(forall_congr (λ i, hP.source_affine_open_cover_iff _ (𝒰' i))) | lemma | ring_hom.property_is_local.affine_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} X) :
affine_locally @P f ↔ ∀ i, affine_locally @P (𝒰.map i ≫ f) | begin
split,
{ intros H i U,
rw morphism_restrict_comp,
delta morphism_restrict,
apply hP.source_affine_locally_comp_of_is_open_immersion,
apply H },
{ intros H U,
haveI : is_affine _ := U.2,
apply ((hP.open_cover_tfae (f ∣_ U.1)).out 1 0).mp,
use 𝒰.pullback_cover (X.of_restrict _),
... | lemma | ring_hom.property_is_local.source_open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_locally_of_is_open_immersion (hP : ring_hom.property_is_local @P) {X Y : Scheme}
(f : X ⟶ Y) [hf : is_open_immersion f] : affine_locally @P f | begin
intro U,
haveI H : is_affine _ := U.2,
rw ← category.comp_id (f ∣_ U),
apply hP.source_affine_locally_comp_of_is_open_immersion,
rw hP.source_affine_open_cover_iff _ (Scheme.open_cover_of_is_iso (𝟙 _)),
{ intro i, erw [category.id_comp, op_id, Scheme.Γ.map_id],
convert hP.holds_for_localization_a... | lemma | ring_hom.property_is_local.affine_locally_of_is_open_immersion | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"is_localization.away_of_is_unit_of_bijective",
"is_unit_one",
"ring_hom.algebra_map_to_algebra",
"ring_hom.property_is_local"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_locally_of_comp
(H : ∀ {R S T : Type.{u}} [comm_ring R] [comm_ring S] [comm_ring T], by exactI
∀ (f : R →+* S) (g : S →+* T), P (g.comp f) → P g)
{X Y Z : Scheme} {f : X ⟶ Y} {g : Y ⟶ Z} (h : affine_locally @P (f ≫ g)) :
affine_locally @P f | begin
let 𝒰 : ∀ i, ((Z.affine_cover.pullback_cover (f ≫ g)).obj i).open_cover,
{ intro i,
refine Scheme.open_cover.bind _ (λ i, Scheme.affine_cover _),
apply Scheme.open_cover.pushforward_iso _
(pullback_right_pullback_fst_iso g (Z.affine_cover.map i) f).hom,
apply Scheme.pullback.open_cover_of_rig... | lemma | ring_hom.property_is_local.affine_locally_of_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_locally_stable_under_composition :
(affine_locally @P).stable_under_composition | begin
intros X Y S f g hf hg,
let 𝒰 : ∀ i, ((S.affine_cover.pullback_cover (f ≫ g)).obj i).open_cover,
{ intro i,
refine Scheme.open_cover.bind _ (λ i, Scheme.affine_cover _),
apply Scheme.open_cover.pushforward_iso _
(pullback_right_pullback_fst_iso g (S.affine_cover.map i) f).hom,
apply Scheme.... | lemma | ring_hom.property_is_local.affine_locally_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/ring_hom_properties.lean | [
"algebraic_geometry.morphisms.basic",
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed (f : X ⟶ Y) : Prop | (out : universally (topologically @is_closed_map) f) | class | algebraic_geometry.universally_closed | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"is_closed_map"
] | A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`
along any morphism `Y' ⟶ Y` is (topologically) a closed map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
universally_closed_eq :
@universally_closed = universally (topologically @is_closed_map) | begin
ext X Y f, rw universally_closed_iff
end | lemma | algebraic_geometry.universally_closed_eq | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_respects_iso :
respects_iso @universally_closed | universally_closed_eq.symm ▸ universally_respects_iso (topologically @is_closed_map) | lemma | algebraic_geometry.universally_closed_respects_iso | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_stable_under_base_change :
stable_under_base_change @universally_closed | universally_closed_eq.symm ▸ universally_stable_under_base_change (topologically @is_closed_map) | lemma | algebraic_geometry.universally_closed_stable_under_base_change | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_stable_under_composition :
stable_under_composition @universally_closed | begin
rw universally_closed_eq,
exact stable_under_composition.universally (λ X Y Z f g hf hg, is_closed_map.comp hg hf),
end | lemma | algebraic_geometry.universally_closed_stable_under_composition | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"is_closed_map.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_type_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : universally_closed f] [hg : universally_closed g] :
universally_closed (f ≫ g) | universally_closed_stable_under_composition f g hf hg | instance | algebraic_geometry.universally_closed_type_comp | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_fst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z)
[hg : universally_closed g] :
universally_closed (pullback.fst : pullback f g ⟶ _) | universally_closed_stable_under_base_change.fst f g hg | instance | algebraic_geometry.universally_closed_fst | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z)
[hf : universally_closed f] :
universally_closed (pullback.snd : pullback f g ⟶ _) | universally_closed_stable_under_base_change.snd f g hf | instance | algebraic_geometry.universally_closed_snd | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
morphism_restrict_base {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) :
⇑(f ∣_ U).1.base = U.1.restrict_preimage f.1 | funext (λ x, subtype.ext $ morphism_restrict_base_coe f U x) | lemma | algebraic_geometry.morphism_restrict_base | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed_is_local_at_target :
property_is_local_at_target @universally_closed | begin
rw universally_closed_eq,
apply universally_is_local_at_target_of_morphism_restrict,
{ exact stable_under_composition.respects_iso (λ X Y Z f g hf hg, is_closed_map.comp hg hf)
(λ X Y f, (Top.homeo_of_iso (Scheme.forget_to_Top.map_iso f)).is_closed_map) },
{ intros X Y f ι U hU H,
simp_rw [topol... | lemma | algebraic_geometry.universally_closed_is_local_at_target | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [
"Top.homeo_of_iso",
"is_closed_map",
"is_closed_map.comp",
"is_closed_map_iff_is_closed_map_of_supr_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_closed.open_cover_iff {X Y : Scheme.{u}} (f : X ⟶ Y)
(𝒰 : Scheme.open_cover.{u} Y) :
universally_closed f ↔
(∀ i, universally_closed (pullback.snd : pullback f (𝒰.map i) ⟶ _)) | universally_closed_is_local_at_target.open_cover_iff f 𝒰 | lemma | algebraic_geometry.universally_closed.open_cover_iff | algebraic_geometry.morphisms | src/algebraic_geometry/morphisms/universally_closed.lean | [
"algebraic_geometry.morphisms.basic",
"topology.local_at_target"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
PresheafedSpace.is_open_immersion {X Y : PresheafedSpace.{v} C} (f : X ⟶ Y) : Prop | (base_open : open_embedding f.base)
(c_iso : ∀ U : opens X, is_iso (f.c.app (op (base_open.is_open_map.functor.obj U)))) | class | algebraic_geometry.PresheafedSpace.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"
] | An open immersion of PresheafedSpaces is an open embedding `f : X ⟶ U ⊆ Y` of the underlying
spaces, such that the sheaf map `Y(V) ⟶ f _* X(V)` is an iso for each `V ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
SheafedSpace.is_open_immersion {X Y : SheafedSpace.{v} C} (f : X ⟶ Y) : Prop | PresheafedSpace.is_open_immersion f | abbreviation | algebraic_geometry.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"
] | [] | A morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism
of PresheafedSpaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LocallyRingedSpace.is_open_immersion {X Y : LocallyRingedSpace} (f : X ⟶ Y) : Prop | SheafedSpace.is_open_immersion f.1 | abbreviation | algebraic_geometry.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"
] | [] | A morphism of LocallyRingedSpaces is an open immersion if it is an open immersion as a morphism
of SheafedSpaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_functor | H.base_open.is_open_map.functor | abbreviation | algebraic_geometry.PresheafedSpace.is_open_immersion.open_functor | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [] | The functor `opens X ⥤ opens Y` associated with an open immersion `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_restrict : X ≅ Y.restrict H.base_open | PresheafedSpace.iso_of_components (iso.refl _)
begin
symmetry,
fapply nat_iso.of_components,
intro U,
refine as_iso (f.c.app (op (H.open_functor.obj (unop U)))) ≪≫ X.presheaf.map_iso (eq_to_iso _),
{ induction U using opposite.rec,
cases U,
dsimp only [is_open_map.functor, functor.op, opens.map],
... | def | algebraic_geometry.PresheafedSpace.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"
] | [
"Top.presheaf.pushforward_obj_map",
"category_theory.eq_to_iso.hom",
"is_open_map.functor",
"opposite.rec",
"set.preimage_image_eq"
] | An open immersion `f : X ⟶ Y` induces an isomorphism `X ≅ Y|_{f(X)}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_restrict_hom_of_restrict : H.iso_restrict.hom ≫ Y.of_restrict _ = f | begin
ext,
{ simp only [comp_c_app, iso_restrict_hom_c_app, nat_trans.comp_app,
eq_to_hom_refl, of_restrict_c_app, category.assoc, whisker_right_id'],
erw [category.comp_id, f.c.naturality_assoc, ←X.presheaf.map_comp],
transitivity f.c.app x ≫ X.presheaf.map (𝟙 _),
{ congr },
{ erw [X.preshea... | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict_hom_of_restrict | 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 | |
iso_restrict_inv_of_restrict : H.iso_restrict.inv ≫ f = Y.of_restrict _ | by { rw [iso.inv_comp_eq, iso_restrict_hom_of_restrict] } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict_inv_of_restrict | 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 [H : is_open_immersion f] : mono f | by { rw ← H.iso_restrict_hom_of_restrict, apply mono_comp } | instance | algebraic_geometry.PresheafedSpace.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 | |
comp {Z : PresheafedSpace C} (f : X ⟶ Y) [hf : is_open_immersion f] (g : Y ⟶ Z)
[hg : is_open_immersion g] :
is_open_immersion (f ≫ g) | { base_open := hg.base_open.comp hf.base_open,
c_iso := λ U,
begin
generalize_proofs h,
dsimp only [algebraic_geometry.PresheafedSpace.comp_c_app, unop_op, functor.op, comp_base,
Top.presheaf.pushforward_obj_obj, opens.map_comp_obj],
apply_with is_iso.comp_is_iso { instances := ff },
swap,
... | instance | algebraic_geometry.PresheafedSpace.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"
] | [
"Top.presheaf.pushforward_obj_obj",
"algebraic_geometry.PresheafedSpace.comp_c_app",
"set.image_image",
"set.preimage_image_eq"
] | The composition of two open immersions is an open immersion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_app (U : opens X) : X.presheaf.obj (op U) ⟶ Y.presheaf.obj (op (H.open_functor.obj U)) | X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) ≫
inv (f.c.app (op (H.open_functor.obj U))) | def | algebraic_geometry.PresheafedSpace.is_open_immersion.inv_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.preimage_image_eq"
] | For an open immersion `f : X ⟶ Y` and an open set `U ⊆ X`, we have the map `X(U) ⟶ Y(U)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_naturality {U V : (opens X)ᵒᵖ} (i : U ⟶ V) :
X.presheaf.map i ≫ H.inv_app (unop V) = H.inv_app (unop U) ≫
Y.presheaf.map (H.open_functor.op.map i) | begin
simp only [inv_app, ←category.assoc],
rw [is_iso.comp_inv_eq],
simp only [category.assoc, f.c.naturality, is_iso.inv_hom_id_assoc, ← X.presheaf.map_comp],
erw ← X.presheaf.map_comp,
congr
end | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.inv_naturality | 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 | |
inv_inv_app (U : opens X) :
inv (H.inv_app U) = f.c.app (op (H.open_functor.obj U)) ≫
X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) | begin
rw ← cancel_epi (H.inv_app U),
rw is_iso.hom_inv_id,
delta inv_app,
simp [← functor.map_comp]
end | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.inv_inv_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_app_app (U : opens X) :
H.inv_app U ≫ f.c.app (op (H.open_functor.obj U)) =
X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) | by rw [inv_app, category.assoc, is_iso.inv_hom_id, category.comp_id] | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.inv_app_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_inv_app (U : opens Y) :
f.c.app (op U) ≫ H.inv_app ((opens.map f.base).obj U) =
Y.presheaf.map ((hom_of_le (by exact set.image_preimage_subset f.base U)).op :
op U ⟶ op (H.open_functor.obj ((opens.map f.base).obj U))) | by { erw ← category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.app_inv_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.image_preimage_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_inv_app' (U : opens Y) (hU : (U : set Y) ⊆ set.range f.base) :
f.c.app (op U) ≫ H.inv_app ((opens.map f.base).obj U) =
Y.presheaf.map (eq_to_hom (by
{ apply le_antisymm,
{ exact set.image_preimage_subset f.base U.1 },
{ rw [← set_like.coe_subset_coe],
refine has_le.le.trans_eq _ (@set.im... | by { erw ← category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr } | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.app_inv_app' | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"set.image_preimage_eq_inter_range",
"set.image_preimage_subset",
"set.range",
"set_like.coe_subset_coe"
] | A variant of `app_inv_app` that gives an `eq_to_hom` instead of `hom_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso {X Y : PresheafedSpace.{v} C} (H : X ≅ Y) : is_open_immersion H.hom | { base_open := (Top.homeo_of_iso ((forget C).map_iso H)).open_embedding,
c_iso := λ _, infer_instance } | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.homeo_of_iso",
"open_embedding"
] | An isomorphism is an open immersion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_iso {X Y : PresheafedSpace.{v} C} (f : X ⟶ Y) [is_iso f] : is_open_immersion f | algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso (as_iso f) | instance | algebraic_geometry.PresheafedSpace.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"
] | [
"algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict {X : Top} (Y : PresheafedSpace C) {f : X ⟶ Y.carrier}
(hf : open_embedding f) : is_open_immersion (Y.of_restrict hf) | { base_open := hf,
c_iso := λ U,
begin
dsimp,
have : (opens.map f).obj (hf.is_open_map.functor.obj U) = U,
{ ext1,
exact set.preimage_image_eq _ hf.inj },
convert (show is_iso (Y.presheaf.map (𝟙 _)), from infer_instance),
{ apply subsingleton.helim,
rw this },
{ rw Y.presheaf.ma... | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.of_restrict | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top",
"open_embedding",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_inv_app {C : Type*} [category C] (X : PresheafedSpace C) {Y : Top}
{f : Y ⟶ Top.of X.carrier}
(h : open_embedding f) (U : opens (X.restrict h).carrier) :
(PresheafedSpace.is_open_immersion.of_restrict X h).inv_app U = 𝟙 _ | begin
delta PresheafedSpace.is_open_immersion.inv_app,
rw [is_iso.comp_inv_eq, category.id_comp],
change X.presheaf.map _ = X.presheaf.map _,
congr
end | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.of_restrict_inv_app | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top",
"Top.of",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_iso (f : X ⟶ Y) [h : is_open_immersion f] [h' : epi f.base] : is_iso f | begin
apply_with is_iso_of_components { instances := ff },
{ let : X ≃ₜ Y := (homeomorph.of_embedding _ h.base_open.to_embedding).trans
{ to_fun := subtype.val, inv_fun := λ x, ⟨x,
by { rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp h'), trivial }⟩,
left_inv := λ ⟨_,_⟩, rfl, right_in... | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.to_iso | 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.iso_of_homeo",
"homeomorph.of_embedding",
"inv_fun",
"opposite.rec",
"set.image_preimage_eq"
] | An open immersion is an iso if the underlying continuous map is epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_iso [has_colimits C] [H : is_open_immersion f] (x : X) : is_iso (stalk_map f x) | begin
rw ← H.iso_restrict_hom_of_restrict,
rw PresheafedSpace.stalk_map.comp,
apply_instance
end | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.stalk_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 | |
pullback_cone_of_left_fst :
Y.restrict (Top.snd_open_embedding_of_left_open_embedding hf.base_open g.base) ⟶ X | { base := pullback.fst,
c :=
{ app := λ U, hf.inv_app (unop U) ≫
g.c.app (op (hf.base_open.is_open_map.functor.obj (unop U))) ≫
Y.presheaf.map (eq_to_hom
(begin
simp only [is_open_map.functor, subtype.mk_eq_mk, unop_op, op_inj_iff, opens.map,
subtype.coe_mk, functor.op_obj, subtype... | def | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.presheaf.pushforward_obj_map",
"Top.pullback_iso_prod_subtype",
"Top.snd_open_embedding_of_left_open_embedding",
"is_open_map.functor",
"opposite.rec",
"quiver.hom.unop_op",
"subtype.coe_mk",
"subtype.mk_eq_mk",
"subtype.val_eq_coe"
] | (Implementation.) The projection map when constructing the pullback along an open immersion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_left_condition :
pullback_cone_of_left_fst f g ≫ f = Y.of_restrict _ ≫ g | begin
ext U,
{ induction U using opposite.rec,
dsimp only [comp_c_app, nat_trans.comp_app, unop_op,
whisker_right_app, pullback_cone_of_left_fst],
simp only [quiver.hom.unop_op, Top.presheaf.pushforward_obj_map, app_inv_app_assoc,
eq_to_hom_app, eq_to_hom_unop, category.assoc, nat_trans.naturali... | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"Top.presheaf.pushforward_obj_map",
"opposite.rec",
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_cone_of_left : pullback_cone f g | pullback_cone.mk (pullback_cone_of_left_fst f g) (Y.of_restrict _)
(pullback_cone_of_left_condition f g) | def | algebraic_geometry.PresheafedSpace.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"
] | [] | We construct the pullback along an open immersion via restricting along the pullback of the
maps of underlying spaces (which is also an open embedding). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_left_lift : s.X ⟶ (pullback_cone_of_left f g).X | { base := pullback.lift s.fst.base s.snd.base
(congr_arg (λ x, PresheafedSpace.hom.base x) s.condition),
c :=
{ app := λ U, s.snd.c.app _ ≫ s.X.presheaf.map (eq_to_hom (begin
dsimp only [opens.map, is_open_map.functor, functor.op],
congr' 2,
let s' : pullback_cone f.base g.base := pullback_con... | def | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift | 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",
"is_open_map.functor",
"set.preimage_image_eq",
"set.preimage_preimage"
] | (Implementation.) Any cone over `cospan f g` indeed factors through the constructed cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_left_lift_fst :
pullback_cone_of_left_lift f g s ≫ (pullback_cone_of_left f g).fst = s.fst | begin
ext x,
{ induction x using opposite.rec,
change ((_ ≫ _) ≫ _ ≫ _) ≫ _ = _,
simp_rw [category.assoc],
erw ← s.X.presheaf.map_comp,
erw s.snd.c.naturality_assoc,
have := congr_app s.condition (op (hf.open_functor.obj x)),
dsimp only [comp_c_app, unop_op] at this,
rw ← is_iso.comp_inv... | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_fst | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_cone_of_left_lift_snd :
pullback_cone_of_left_lift f g s ≫ (pullback_cone_of_left f g).snd = s.snd | begin
ext x,
{ change (_ ≫ _ ≫ _) ≫ _ = _,
simp_rw category.assoc,
erw s.snd.c.naturality_assoc,
erw [← s.X.presheaf.map_comp, ← s.X.presheaf.map_comp],
transitivity s.snd.c.app x ≫ s.X.presheaf.map (𝟙 _),
{ congr },
{ rw s.X.presheaf.map_id, erw category.comp_id } },
{ change pullback.li... | lemma | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd | 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_snd_is_open_immersion :
is_open_immersion (pullback_cone_of_left f g).snd | begin
erw category_theory.limits.pullback_cone.mk_snd,
apply_instance
end | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_snd_is_open_immersion | algebraic_geometry.open_immersion | src/algebraic_geometry/open_immersion/basic.lean | [
"topology.category.Top.limits.pullbacks",
"algebraic_geometry.locally_ringed_space"
] | [
"category_theory.limits.pullback_cone.mk_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_cone_of_left_is_limit :
is_limit (pullback_cone_of_left f g) | begin
apply pullback_cone.is_limit_aux',
intro s,
use pullback_cone_of_left_lift f g s,
use pullback_cone_of_left_lift_fst f g s,
use pullback_cone_of_left_lift_snd f g s,
intros m h₁ h₂,
rw ← cancel_mono (pullback_cone_of_left f g).snd,
exact (h₂.trans (pullback_cone_of_left_lift_snd f g s).symm)
end | def | algebraic_geometry.PresheafedSpace.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 is indeed the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_left :
has_pullback f g | ⟨⟨⟨_, pullback_cone_of_left_is_limit f g⟩⟩⟩ | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.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 | |
has_pullback_of_right :
has_pullback g f | has_pullback_symmetry f g | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.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 | |
pullback_snd_of_left :
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.PresheafedSpace.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 :
is_open_immersion (pullback.fst : pullback g f ⟶ _) | begin
rw ← pullback_symmetry_hom_comp_snd,
apply_instance
end | instance | algebraic_geometry.PresheafedSpace.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 [is_open_immersion g] :
is_open_immersion (limit.π (cospan f g) walking_cospan.one) | begin
rw [←limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],
apply_instance
end | instance | algebraic_geometry.PresheafedSpace.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_limits_of_left : preserves_limit (cospan f g) (forget C) | preserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)
begin
apply (is_limit.postcompose_hom_equiv (diagram_iso_cospan.{v} _) _).to_fun,
refine (is_limit.equiv_iso_limit _).to_fun (limit.is_limit (cospan f.base g.base)),
fapply cones.ext,
exact (iso.refl _),
change ∀ j, _ = 𝟙 _ ≫ _ ≫ _,... | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.forget_preserves_limits_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_limits_of_right : preserves_limit (cospan g f) (forget C) | preserves_pullback_symmetry (forget C) f g | instance | algebraic_geometry.PresheafedSpace.is_open_immersion.forget_preserves_limits_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.base ⊆ set.range f.base) :
is_iso (pullback.snd : pullback f g ⟶ _) | begin
haveI := Top.snd_iso_of_left_embedding_range_subset hf.base_open.to_embedding g.base H,
haveI : is_iso (pullback.snd : pullback f g ⟶ _).base,
{ delta pullback.snd,
rw ← limit.iso_limit_cone_hom_π ⟨_, pullback_cone_of_left_is_limit f g⟩ walking_cospan.right,
change is_iso (_ ≫ pullback.snd),
app... | lemma | algebraic_geometry.PresheafedSpace.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"
] | [
"Top.snd_iso_of_left_embedding_range_subset",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (H : set.range g.base ⊆ set.range f.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.PresheafedSpace.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.base ⊆ set.range f.base) :
lift f g H ≫ f = g | by { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition } | lemma | algebraic_geometry.PresheafedSpace.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 |
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