statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_affine_AffineScheme (X : AffineScheme.{u}) : is_affine X.obj | ⟨functor.ess_image.unit_is_iso X.property⟩ | instance | algebraic_geometry.is_affine_AffineScheme | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_is_affine (R : CommRingᵒᵖ) : is_affine (Scheme.Spec.obj R) | algebraic_geometry.is_affine_AffineScheme ⟨_, Scheme.Spec.obj_mem_ess_image R⟩ | instance | algebraic_geometry.Spec_is_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"algebraic_geometry.is_affine_AffineScheme"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_of_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] [h : is_affine Y] :
is_affine X | by { rw [← mem_Spec_ess_image] at h ⊢, exact functor.ess_image.of_iso (as_iso f).symm h } | lemma | algebraic_geometry.is_affine_of_iso | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec : CommRingᵒᵖ ⥤ AffineScheme | Scheme.Spec.to_ess_image | def | algebraic_geometry.AffineScheme.Spec | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The `Spec` functor into the category of affine schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_Scheme : AffineScheme ⥤ Scheme | Scheme.Spec.ess_image_inclusion | def | algebraic_geometry.AffineScheme.forget_to_Scheme | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The forgetful functor `AffineScheme ⥤ Scheme`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : AffineSchemeᵒᵖ ⥤ CommRing | forget_to_Scheme.op ⋙ Scheme.Γ | def | algebraic_geometry.AffineScheme.Γ | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"CommRing"
] | The global section functor of an affine scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_CommRing : AffineScheme ≌ CommRingᵒᵖ | equiv_ess_image_of_reflective.symm | def | algebraic_geometry.AffineScheme.equiv_CommRing | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The category of affine schemes is equivalent to the category of commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ_is_equiv : is_equivalence Γ.{u} | begin
haveI : is_equivalence Γ.{u}.right_op.op := is_equivalence.of_equivalence equiv_CommRing.op,
exact (functor.is_equivalence_trans Γ.{u}.right_op.op (op_op_equivalence _).functor : _),
end | instance | algebraic_geometry.AffineScheme.Γ_is_equiv | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open {X : Scheme} (U : opens X.carrier) : Prop | is_affine (X.restrict U.open_embedding) | def | algebraic_geometry.is_affine_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | An open subset of a scheme is affine if the open subscheme is affine. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.affine_opens (X : Scheme) : set (opens X.carrier) | { U : opens X.carrier | is_affine_open U } | def | algebraic_geometry.Scheme.affine_opens | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The set of affine opens as a subset of `opens X.carrier`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_is_affine_open_of_open_immersion {X Y : Scheme} [is_affine X] (f : X ⟶ Y)
[H : is_open_immersion f] : is_affine_open f.opens_range | begin
refine is_affine_of_iso (is_open_immersion.iso_of_range_eq f (Y.of_restrict _) _).inv,
exact subtype.range_coe.symm,
apply_instance
end | lemma | algebraic_geometry.range_is_affine_open_of_open_immersion | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_is_affine_open (X : Scheme) [is_affine X] : is_affine_open (⊤ : opens X.carrier) | begin
convert range_is_affine_open_of_open_immersion (𝟙 X),
ext1,
exact set.range_id.symm
end | lemma | algebraic_geometry.top_is_affine_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.affine_cover_is_affine (X : Scheme) (i : X.affine_cover.J) :
is_affine (X.affine_cover.obj i) | algebraic_geometry.Spec_is_affine _ | instance | algebraic_geometry.Scheme.affine_cover_is_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"algebraic_geometry.Spec_is_affine"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.affine_basis_cover_is_affine (X : Scheme) (i : X.affine_basis_cover.J) :
is_affine (X.affine_basis_cover.obj i) | algebraic_geometry.Spec_is_affine _ | instance | algebraic_geometry.Scheme.affine_basis_cover_is_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"algebraic_geometry.Spec_is_affine"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis_affine_open (X : Scheme) :
opens.is_basis X.affine_opens | begin
rw opens.is_basis_iff_nbhd,
rintros U x (hU : x ∈ (U : set X.carrier)),
obtain ⟨S, hS, hxS, hSU⟩ := X.affine_basis_cover_is_basis.exists_subset_of_mem_open hU U.is_open,
refine ⟨⟨S, X.affine_basis_cover_is_basis.is_open hS⟩, _, hxS, hSU⟩,
rcases hS with ⟨i, rfl⟩,
exact range_is_affine_open_of_open_imm... | lemma | algebraic_geometry.is_basis_affine_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
Scheme.Spec.obj (op $ X.presheaf.obj $ op U) ⟶ X | begin
haveI : is_affine (X.restrict U.open_embedding) := hU,
have : U.open_embedding.is_open_map.functor.obj ⊤ = U,
{ ext1, exact set.image_univ.trans subtype.range_coe },
exact Scheme.Spec.map (X.presheaf.map (eq_to_hom this.symm).op).op ≫
(X.restrict U.open_embedding).iso_Spec.inv ≫ X.of_restrict _
end | def | algebraic_geometry.is_affine_open.from_Spec | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"subtype.range_coe"
] | The open immersion `Spec 𝒪ₓ(U) ⟶ X` for an affine `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine_open.is_open_immersion_from_Spec {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
is_open_immersion hU.from_Spec | by { delta is_affine_open.from_Spec, apply_instance } | instance | algebraic_geometry.is_affine_open.is_open_immersion_from_Spec | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec_range {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
set.range hU.from_Spec.1.base = (U : set X.carrier) | begin
delta is_affine_open.from_Spec,
erw [← category.assoc, Scheme.comp_val_base],
rw [coe_comp, set.range_comp, set.range_iff_surjective.mpr, set.image_univ],
exact subtype.range_coe,
rw ← Top.epi_iff_surjective,
apply_instance
end | lemma | algebraic_geometry.is_affine_open.from_Spec_range | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"Top.epi_iff_surjective",
"set.image_univ",
"set.range",
"set.range_comp",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec_image_top {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
hU.is_open_immersion_from_Spec.base_open.is_open_map.functor.obj ⊤ = U | by { ext1, exact set.image_univ.trans hU.from_Spec_range } | lemma | algebraic_geometry.is_affine_open.from_Spec_image_top | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.is_compact {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
is_compact (U : set X.carrier) | begin
convert @is_compact.image _ _ _ _ set.univ hU.from_Spec.1.base
prime_spectrum.compact_space.1 (by continuity),
convert hU.from_Spec_range.symm,
exact set.image_univ
end | lemma | algebraic_geometry.is_affine_open.is_compact | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"continuity",
"is_compact",
"is_compact.image",
"set.image_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.image_is_open_immersion {X Y : Scheme} {U : opens X.carrier}
(hU : is_affine_open U)
(f : X ⟶ Y) [H : is_open_immersion f] : is_affine_open (f.opens_functor.obj U) | begin
haveI : is_affine _ := hU,
convert range_is_affine_open_of_open_immersion (X.of_restrict U.open_embedding ≫ f),
ext1,
exact set.image_eq_range _ _
end | lemma | algebraic_geometry.is_affine_open.image_is_open_immersion | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"set.image_eq_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open_iff_of_is_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]
(U : opens X.carrier) :
is_affine_open (H.open_functor.obj U) ↔ is_affine_open U | begin
refine ⟨λ hU, @@is_affine_of_iso _ _ hU, λ hU, hU.image_is_open_immersion f⟩,
refine (is_open_immersion.iso_of_range_eq (X.of_restrict _ ≫ f) (Y.of_restrict _) _).hom,
{ rw [Scheme.comp_val_base, coe_comp, set.range_comp],
dsimp [opens.inclusion],
rw [subtype.range_coe, subtype.range_coe],
refl ... | lemma | algebraic_geometry.is_affine_open_iff_of_is_open_immersion | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"set.range_comp",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.quasi_compact_of_affine (X : Scheme) [is_affine X] : compact_space X.carrier | ⟨(top_is_affine_open X).is_compact⟩ | instance | algebraic_geometry.Scheme.quasi_compact_of_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec_base_preimage
{X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :
(opens.map hU.from_Spec.val.base).obj U = ⊤ | begin
ext1,
change hU.from_Spec.1.base ⁻¹' (U : set X.carrier) = set.univ,
rw [← hU.from_Spec_range, ← set.image_univ],
exact set.preimage_image_eq _ PresheafedSpace.is_open_immersion.base_open.inj
end | lemma | algebraic_geometry.is_affine_open.from_Spec_base_preimage | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"set.image_univ",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.Spec_map_presheaf_map_eq_to_hom {X : Scheme} {U V : opens X.carrier} (h : U = V) (W) :
(Scheme.Spec.map (X.presheaf.map (eq_to_hom h).op).op).val.c.app W =
eq_to_hom (by { cases h, induction W using opposite.rec, dsimp, simp, }) | begin
have : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 _,
{ rw [X.presheaf.map_id, op_id, Scheme.Spec.map_id] },
cases h,
refine (Scheme.congr_app this _).trans _,
erw category.id_comp,
simpa [eq_to_hom_map],
end | lemma | algebraic_geometry.Scheme.Spec_map_presheaf_map_eq_to_hom | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.Spec_Γ_identity_hom_app_from_Spec {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
(Spec_Γ_identity.hom.app (X.presheaf.obj $ op U)) ≫ hU.from_Spec.1.c.app (op U) =
(Scheme.Spec.obj _).presheaf.map (eq_to_hom hU.from_Spec_base_preimage).op | begin
haveI : is_affine _ := hU,
have e₁ :=
Spec_Γ_identity.hom.naturality (X.presheaf.map (eq_to_hom U.open_embedding_obj_top).op),
rw ← is_iso.comp_inv_eq at e₁,
have e₂ := Γ_Spec.adjunction_unit_app_app_top (X.restrict U.open_embedding),
erw ← e₂ at e₁,
simp only [functor.id_map, quiver.hom.unop_op, ... | lemma | algebraic_geometry.is_affine_open.Spec_Γ_identity_hom_app_from_Spec | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec_app_eq {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) :
hU.from_Spec.1.c.app (op U) = Spec_Γ_identity.inv.app (X.presheaf.obj $ op U) ≫
(Scheme.Spec.obj _).presheaf.map (eq_to_hom hU.from_Spec_base_preimage).op | by rw [← hU.Spec_Γ_identity_hom_app_from_Spec, iso.inv_hom_id_app_assoc] | lemma | algebraic_geometry.is_affine_open.from_Spec_app_eq | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.basic_open_is_affine {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) : is_affine_open (X.basic_open f) | begin
convert range_is_affine_open_of_open_immersion (Scheme.Spec.map (CommRing.of_hom
(algebra_map (X.presheaf.obj (op U)) (localization.away f))).op ≫ hU.from_Spec),
ext1,
have : hU.from_Spec.val.base '' (hU.from_Spec.val.base ⁻¹' (X.basic_open f : set X.carrier)) =
(X.basic_open f : set X.carrier),
{... | lemma | algebraic_geometry.is_affine_open.basic_open_is_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"CommRing.of_hom",
"algebra_map",
"localization.away",
"prime_spectrum.localization_away_comap_range",
"set.image_preimage_eq_inter_range",
"set.image_univ",
"set.inter_eq_left_iff_subset",
"set.preimage_image_eq",
"set.range_comp",
"top_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.map_restrict_basic_open {X : Scheme} (r : X.presheaf.obj (op ⊤))
{U : opens X.carrier} (hU : is_affine_open U) :
is_affine_open ((opens.map (X.of_restrict (X.basic_open r).open_embedding).1.base).obj U) | begin
apply (is_affine_open_iff_of_is_open_immersion
(X.of_restrict (X.basic_open r).open_embedding) _).mp,
delta PresheafedSpace.is_open_immersion.open_functor,
dsimp,
erw [opens.functor_obj_map_obj, opens.open_embedding_obj_top, inf_comm,
← Scheme.basic_open_res _ _ (hom_of_le le_top).op],
exact hU.... | lemma | algebraic_geometry.is_affine_open.map_restrict_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"inf_comm",
"le_top",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.map_prime_spectrum_basic_open_of_affine (X : Scheme) [is_affine X]
(f : Scheme.Γ.obj (op X)) :
(opens.map X.iso_Spec.hom.1.base).obj (prime_spectrum.basic_open f) = X.basic_open f | begin
rw ← basic_open_eq_of_affine,
transitivity (opens.map X.iso_Spec.hom.1.base).obj ((Scheme.Spec.obj
(op (Scheme.Γ.obj (op X)))).basic_open ((inv (X.iso_Spec.hom.1.c.app
(op ((opens.map (inv X.iso_Spec.hom).val.base).obj ⊤)))) ((X.presheaf.map (eq_to_hom _)) f))),
congr,
{ rw [← is_iso.inv_eq_inv,... | lemma | algebraic_geometry.Scheme.map_prime_spectrum_basic_open_of_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"prime_spectrum.basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis_basic_open (X : Scheme) [is_affine X] :
opens.is_basis (set.range (X.basic_open : X.presheaf.obj (op ⊤) → opens X.carrier)) | begin
delta opens.is_basis,
convert prime_spectrum.is_basis_basic_opens.inducing
(Top.homeo_of_iso (Scheme.forget_to_Top.map_iso X.iso_Spec)).inducing using 1,
ext,
simp only [set.mem_image, exists_exists_eq_and],
split,
{ rintro ⟨_, ⟨x, rfl⟩, rfl⟩,
refine ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, _⟩,
exact congr... | lemma | algebraic_geometry.is_basis_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"Top.homeo_of_iso",
"exists_exists_eq_and",
"inducing",
"set.mem_image",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.exists_basic_open_le {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) {V : opens X.carrier} (x : V) (h : ↑x ∈ U) :
∃ f : X.presheaf.obj (op U), X.basic_open f ≤ V ∧ ↑x ∈ X.basic_open f | begin
haveI : is_affine _ := hU,
obtain ⟨_, ⟨_, ⟨r, rfl⟩, rfl⟩, h₁, h₂⟩ := (is_basis_basic_open (X.restrict U.open_embedding))
.exists_subset_of_mem_open _ ((opens.map U.inclusion).obj V).is_open,
swap, exact ⟨x, h⟩,
have : U.open_embedding.is_open_map.functor.obj ((X.restrict U.open_embedding).basic_open r... | lemma | algebraic_geometry.is_affine_open.exists_basic_open_le | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"is_open",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.opens_map_from_Spec_basic_open {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) :
(opens.map hU.from_Spec.val.base).obj (X.basic_open f) =
RingedSpace.basic_open _ (Spec_Γ_identity.inv.app (X.presheaf.obj $ op U) f) | begin
erw LocallyRingedSpace.preimage_basic_open,
refine eq.trans _ (RingedSpace.basic_open_res_eq (Scheme.Spec.obj $ op $ X.presheaf.obj (op U))
.to_LocallyRingedSpace.to_RingedSpace (eq_to_hom hU.from_Spec_base_preimage).op _),
congr,
rw ← comp_apply,
congr,
erw ← hU.Spec_Γ_identity_hom_app_from_Spec,... | lemma | algebraic_geometry.is_affine_open.opens_map_from_Spec_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_sections_to_affine {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) : X.presheaf.obj (op $ X.basic_open f) ⟶
(Scheme.Spec.obj $ op $ X.presheaf.obj (op U)).presheaf.obj
(op $ Scheme.basic_open _ $ Spec_Γ_identity.inv.app (X.presheaf.obj (op U)) f) | hU.from_Spec.1.c.app (op $ X.basic_open f) ≫ (Scheme.Spec.obj $ op $ X.presheaf.obj (op U))
.presheaf.map (eq_to_hom $ (hU.opens_map_from_Spec_basic_open f).symm).op | def | algebraic_geometry.basic_open_sections_to_affine | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The canonical map `Γ(𝒪ₓ, D(f)) ⟶ Γ(Spec 𝒪ₓ(U), D(Spec_Γ_identity.inv f))`
This is an isomorphism, as witnessed by an `is_iso` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_basic_open {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U)
(f : X.presheaf.obj (op U)) :
is_localization.away f (X.presheaf.obj (op $ X.basic_open f)) | begin
apply (is_localization.is_localization_iff_of_ring_equiv (submonoid.powers f)
(as_iso $ basic_open_sections_to_affine hU f ≫ (Scheme.Spec.obj _).presheaf.map
(eq_to_hom (basic_open_eq_of_affine _).symm).op).CommRing_iso_to_ring_equiv).mpr,
convert structure_sheaf.is_localization.to_basic_open _ f,
... | lemma | algebraic_geometry.is_localization_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"is_localization.away",
"is_localization.is_localization_iff_of_ring_equiv",
"ring_hom.algebra_map_to_algebra",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_of_eq_basic_open {X : Scheme} {U V : opens X.carrier} (i : V ⟶ U)
(hU : is_affine_open U) (r : X.presheaf.obj (op U)) (e : V = X.basic_open r) :
@@is_localization.away _ r (X.presheaf.obj (op V)) _ (X.presheaf.map i.op).to_algebra | by { subst e, convert is_localization_basic_open hU r using 3 } | lemma | algebraic_geometry.is_localization_of_eq_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"is_localization.away"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_restrict_algebra
{X : Scheme} {Y : Top} {f : Y ⟶ X.carrier} (hf : open_embedding f) :
algebra (Scheme.Γ.obj (op X)) (Scheme.Γ.obj (op $ X.restrict hf)) | (Scheme.Γ.map (X.of_restrict hf).op).to_algebra | instance | algebraic_geometry.Γ_restrict_algebra | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"Top",
"algebra",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_restrict_is_localization (X : Scheme.{u}) [is_affine X] (r : Scheme.Γ.obj (op X)) :
is_localization.away r (Scheme.Γ.obj (op $ X.restrict (X.basic_open r).open_embedding)) | is_localization_of_eq_basic_open _ (top_is_affine_open X) r (opens.open_embedding_obj_top _) | instance | algebraic_geometry.Γ_restrict_is_localization | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"is_localization.away",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_basic_open_is_basic_open {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) (g : X.presheaf.obj (op $ X.basic_open f)) :
∃ f' : X.presheaf.obj (op U), X.basic_open f' = X.basic_open g | begin
haveI := is_localization_basic_open hU f,
obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := is_localization.surj' (submonoid.powers f) g,
use f * x,
rw [algebra.smul_def, Scheme.basic_open_mul, Scheme.basic_open_mul],
erw Scheme.basic_open_res,
refine (inf_eq_left.mpr _).symm,
convert inf_le_left using 1,
apply Sch... | lemma | algebraic_geometry.basic_open_basic_open_is_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"algebra.smul_def",
"inf_le_left",
"is_localization.surj'",
"is_localization.to_inv_submonoid",
"submonoid.left_inv_le_is_unit",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_basic_open_le_affine_inter {X : Scheme} {U V : opens X.carrier}
(hU : is_affine_open U) (hV : is_affine_open V) (x : X.carrier) (hx : x ∈ U ⊓ V) :
∃ (f : X.presheaf.obj $ op U) (g : X.presheaf.obj $ op V),
X.basic_open f = X.basic_open g ∧ x ∈ X.basic_open f | begin
obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basic_open_le ⟨x, hx.2⟩ hx.1,
obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basic_open_le ⟨x, hf₂⟩ hx.2,
obtain ⟨f', hf'⟩ := basic_open_basic_open_is_basic_open hU f
(X.presheaf.map (hom_of_le hf₁ : _ ⟶ V).op g),
replace hf' := (hf'.trans (RingedSpace.basic_open_res _ _ _)).trans... | lemma | algebraic_geometry.exists_basic_open_le_affine_inter | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.prime_ideal_of {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
prime_spectrum (X.presheaf.obj $ op U) | ((Scheme.Spec.map (X.presheaf.map (eq_to_hom $
show U.open_embedding.is_open_map.functor.obj ⊤ = U, from
opens.ext (set.image_univ.trans subtype.range_coe)).op).op).1.base
((@@Scheme.iso_Spec (X.restrict U.open_embedding) hU).hom.1.base x)) | def | algebraic_geometry.is_affine_open.prime_ideal_of | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"prime_spectrum",
"subtype.range_coe"
] | The prime ideal of `𝒪ₓ(U)` corresponding to a point `x : U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine_open.from_Spec_prime_ideal_of {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
hU.from_Spec.val.base (hU.prime_ideal_of x) = x.1 | begin
dsimp only [is_affine_open.from_Spec, subtype.coe_mk],
erw [← Scheme.comp_val_base_apply, ← Scheme.comp_val_base_apply],
simpa only [← functor.map_comp_assoc, ← functor.map_comp, ← op_comp, eq_to_hom_trans, op_id,
eq_to_hom_refl, category_theory.functor.map_id, category.id_comp, iso.hom_inv_id_assoc]
en... | lemma | algebraic_geometry.is_affine_open.from_Spec_prime_ideal_of | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.is_localization_stalk_aux {X : Scheme} (U : opens X.carrier)
[is_affine (X.restrict U.open_embedding)] :
(inv (Γ_Spec.adjunction.unit.app (X.restrict U.open_embedding))).1.c.app
(op ((opens.map U.inclusion).obj U)) =
X.presheaf.map (eq_to_hom $ by rw opens.inclusion_map_eq_top :
U.o... | begin
have e : (opens.map (inv (Γ_Spec.adjunction.unit.app (X.restrict U.open_embedding))).1.base).obj
((opens.map U.inclusion).obj U) = ⊤,
by { rw [opens.inclusion_map_eq_top], refl },
rw [Scheme.inv_val_c_app, is_iso.comp_inv_eq, Scheme.app_eq _ e,
Γ_Spec.adjunction_unit_app_app_top],
simp only [categ... | lemma | algebraic_geometry.is_affine_open.is_localization_stalk_aux | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.is_localization_stalk {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (x : U) :
is_localization.at_prime (X.presheaf.stalk x) (hU.prime_ideal_of x).as_ideal | begin
haveI : is_affine _ := hU,
haveI : nonempty U := ⟨x⟩,
rcases x with ⟨x, hx⟩,
let y := hU.prime_ideal_of ⟨x, hx⟩,
have : hU.from_Spec.val.base y = x := hU.from_Spec_prime_ideal_of ⟨x, hx⟩,
change is_localization y.as_ideal.prime_compl _,
clear_value y,
subst this,
apply (is_localization.is_locali... | lemma | algebraic_geometry.is_affine_open.is_localization_stalk | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"is_localization",
"is_localization.at_prime",
"is_localization.is_localization_iff_of_ring_equiv",
"le_top",
"quiver.hom.op",
"ring_hom.algebra_map_to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.affine_basic_open (X : Scheme) {U : X.affine_opens}
(f : X.presheaf.obj $ op U) : X.affine_opens | ⟨X.basic_open f, U.prop.basic_open_is_affine f⟩ | def | algebraic_geometry.Scheme.affine_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [] | The basic open set of a section `f` on an an affine open as an `X.affine_opens`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_affine_open.basic_open_from_Spec_app {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) :
@Scheme.basic_open (Scheme.Spec.obj $ op (X.presheaf.obj $ op U))
((opens.map hU.from_Spec.1.base).obj U)
(hU.from_Spec.1.c.app (op U) f) = prime_spectrum.basic_open f | begin
rw [← Scheme.basic_open_res_eq _ _ (eq_to_hom hU.from_Spec_base_preimage.symm).op,
basic_open_eq_of_affine', is_affine_open.from_Spec_app_eq],
congr,
rw [← comp_apply, ← comp_apply, category.assoc, ← functor.map_comp_assoc,
eq_to_hom_op, eq_to_hom_op, eq_to_hom_trans, eq_to_hom_refl, category_theory... | lemma | algebraic_geometry.is_affine_open.basic_open_from_Spec_app | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"prime_spectrum.basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.from_Spec_map_basic_open {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (f : X.presheaf.obj (op U)) :
(opens.map hU.from_Spec.val.base).obj (X.basic_open f) = prime_spectrum.basic_open f | by simp | lemma | algebraic_geometry.is_affine_open.from_Spec_map_basic_open | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"prime_spectrum.basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.basic_open_union_eq_self_iff {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (s : set (X.presheaf.obj $ op U)) :
(⨆ (f : s), X.basic_open (f : X.presheaf.obj $ op U)) = U ↔ ideal.span s = ⊤ | begin
transitivity (⋃ (i : s), (prime_spectrum.basic_open i.1).1) = set.univ,
transitivity hU.from_Spec.1.base ⁻¹' (⨆ (f : s), X.basic_open (f : X.presheaf.obj $ op U)).1 =
hU.from_Spec.1.base ⁻¹' U.1,
{ refine ⟨λ h, by rw h, _⟩,
intro h,
apply_fun set.image hU.from_Spec.1.base at h,
rw [set.image... | lemma | algebraic_geometry.is_affine_open.basic_open_union_eq_self_iff | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"ideal.span",
"prime_spectrum.basic_open",
"prime_spectrum.basic_open_eq_zero_locus_compl",
"prime_spectrum.zero_locus_Union",
"prime_spectrum.zero_locus_empty_iff_eq_top",
"prime_spectrum.zero_locus_span",
"set.Union_singleton_eq_range",
"set.Union_subset_iff",
"set.compl_Inter",
"set.compl_univ_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.self_le_basic_open_union_iff {X : Scheme} {U : opens X.carrier}
(hU : is_affine_open U) (s : set (X.presheaf.obj $ op U)) :
U ≤ (⨆ (f : s), X.basic_open (f : X.presheaf.obj $ op U)) ↔ ideal.span s = ⊤ | begin
rw [← hU.basic_open_union_eq_self_iff, @comm _ eq],
refine ⟨λ h, le_antisymm h _, le_of_eq⟩,
simp only [supr_le_iff, set_coe.forall],
intros x hx,
exact X.basic_open_le x
end | lemma | algebraic_geometry.is_affine_open.self_le_basic_open_union_iff | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"comm",
"ideal.span",
"set_coe.forall",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_affine_open_cover {X : Scheme} (V : X.affine_opens) (S : set X.affine_opens)
{P : X.affine_opens → Prop}
(hP₁ : ∀ (U : X.affine_opens) (f : X.presheaf.obj $ op U.1), P U →
P (X.affine_basic_open f))
(hP₂ : ∀ (U : X.affine_opens) (s : finset (X.presheaf.obj $ op U))
(hs : ideal.span (s : set (X.presheaf... | begin
classical,
have : ∀ (x : V), ∃ (f : X.presheaf.obj $ op V.1),
↑x ∈ (X.basic_open f) ∧ P (X.affine_basic_open f),
{ intro x,
have : ↑x ∈ (set.univ : set X.carrier) := trivial,
rw ← hS at this,
obtain ⟨W, hW⟩ := set.mem_Union.mp this,
obtain ⟨f, g, e, hf⟩ := exists_basic_open_le_affine_int... | lemma | algebraic_geometry.of_affine_open_cover | algebraic_geometry | src/algebraic_geometry/AffineScheme.lean | [
"algebraic_geometry.Gamma_Spec_adjunction",
"algebraic_geometry.open_immersion.Scheme",
"category_theory.limits.opposites",
"ring_theory.localization.inv_submonoid"
] | [
"finset",
"ideal.span",
"ideal.span_eq_top_iff_finite",
"set.range",
"supr_range'"
] | Let `P` be a predicate on the affine open sets of `X` satisfying
1. If `P` holds on `U`, then `P` holds on the basic open set of every section on `U`.
2. If `P` holds for a family of basic open sets covering `U`, then `P` holds for `U`.
3. There exists an affine open cover of `X` each satisfying `P`.
Then `P` holds fo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.function_field [irreducible_space X.carrier] : CommRing | X.presheaf.stalk (generic_point X.carrier) | abbreviation | algebraic_geometry.Scheme.function_field | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"CommRing",
"generic_point",
"irreducible_space"
] | The function field of an irreducible scheme is the local ring at its generic point.
Despite the name, this is a field only when the scheme is integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scheme.germ_to_function_field [irreducible_space X.carrier] (U : opens X.carrier)
[h : nonempty U] : X.presheaf.obj (op U) ⟶ X.function_field | X.presheaf.germ ⟨generic_point X.carrier,
((generic_point_spec X.carrier).mem_open_set_iff U.is_open).mpr (by simpa using h)⟩ | abbreviation | algebraic_geometry.Scheme.germ_to_function_field | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"generic_point_spec",
"irreducible_space"
] | The restriction map from a component to the function field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_injective_of_is_integral [is_integral X] {U : opens X.carrier} (x : U) :
function.injective (X.presheaf.germ x) | begin
rw injective_iff_map_eq_zero,
intros y hy,
rw ← (X.presheaf.germ x).map_zero at hy,
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy,
cases (show iU = iV, from subsingleton.elim _ _),
haveI : nonempty W := ⟨⟨_, hW⟩⟩,
exact map_injective_of_is_integral X iU e
end | lemma | algebraic_geometry.germ_injective_of_is_integral | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme.germ_to_function_field_injective [is_integral X] (U : opens X.carrier)
[nonempty U] : function.injective (X.germ_to_function_field U) | germ_injective_of_is_integral _ _ | lemma | algebraic_geometry.Scheme.germ_to_function_field_injective | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generic_point_eq_of_is_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]
[hX : irreducible_space X.carrier] [irreducible_space Y.carrier] :
f.1.base (generic_point X.carrier : _) = (generic_point Y.carrier : _) | begin
apply ((generic_point_spec _).eq _).symm,
show t0_space Y.carrier, by apply_instance,
convert (generic_point_spec X.carrier).image (show continuous f.1.base, by continuity),
symmetry,
rw [eq_top_iff, set.top_eq_univ, set.top_eq_univ],
convert subset_closure_inter_of_is_preirreducible_of_is_open _ H.ba... | lemma | algebraic_geometry.generic_point_eq_of_is_open_immersion | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"continuity",
"continuous",
"eq_top_iff",
"generic_point",
"generic_point_spec",
"irreducible_space",
"preirreducible_space",
"set.image_univ",
"set.mem_range_self",
"set.top_eq_univ",
"set.univ_inter",
"subset_closure_inter_of_is_preirreducible_of_is_open",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_function_field_algebra [irreducible_space X.carrier] (x : X.carrier) :
algebra (X.presheaf.stalk x) X.function_field | begin
apply ring_hom.to_algebra,
exact X.presheaf.stalk_specializes ((generic_point_spec X.carrier).specializes trivial)
end | instance | algebraic_geometry.stalk_function_field_algebra | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"algebra",
"generic_point_spec",
"irreducible_space",
"ring_hom.to_algebra",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_field_is_scalar_tower [irreducible_space X.carrier] (U : opens X.carrier) (x : U)
[nonempty U] :
is_scalar_tower (X.presheaf.obj $ op U) (X.presheaf.stalk x) X.function_field | begin
apply is_scalar_tower.of_algebra_map_eq',
simp_rw [ring_hom.algebra_map_to_algebra],
change _ = X.presheaf.germ x ≫ _,
rw X.presheaf.germ_stalk_specializes,
refl
end | instance | algebraic_geometry.function_field_is_scalar_tower | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"irreducible_space",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq'",
"ring_hom.algebra_map_to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generic_point_eq_bot_of_affine (R : CommRing) [is_domain R] :
generic_point (Scheme.Spec.obj $ op R).carrier = (⟨0, ideal.bot_prime⟩ : prime_spectrum R) | begin
apply (generic_point_spec (Scheme.Spec.obj $ op R).carrier).eq,
simp [is_generic_point_def, ← prime_spectrum.zero_locus_vanishing_ideal_eq_closure]
end | lemma | algebraic_geometry.generic_point_eq_bot_of_affine | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"CommRing",
"generic_point",
"generic_point_spec",
"is_domain",
"is_generic_point_def",
"prime_spectrum",
"prime_spectrum.zero_locus_vanishing_ideal_eq_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_field_is_fraction_ring_of_affine (R : CommRing.{u}) [is_domain R] :
is_fraction_ring R (Scheme.Spec.obj $ op R).function_field | begin
convert structure_sheaf.is_localization.to_stalk R _,
delta is_fraction_ring is_localization.at_prime,
congr' 1,
rw generic_point_eq_bot_of_affine,
ext,
exact mem_non_zero_divisors_iff_ne_zero
end | instance | algebraic_geometry.function_field_is_fraction_ring_of_affine | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"function_field",
"is_domain",
"is_fraction_ring",
"is_localization.at_prime",
"mem_non_zero_divisors_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_affine_open.prime_ideal_of_generic_point {X : Scheme} [is_integral X]
{U : opens X.carrier} (hU : is_affine_open U) [h : nonempty U] :
hU.prime_ideal_of ⟨generic_point X.carrier,
((generic_point_spec X.carrier).mem_open_set_iff U.is_open).mpr (by simpa using h)⟩ =
generic_point (Scheme.Spec.obj $ op $ X.pr... | begin
haveI : is_affine _ := hU,
have e : U.open_embedding.is_open_map.functor.obj ⊤ = U,
{ ext1, exact set.image_univ.trans subtype.range_coe },
delta is_affine_open.prime_ideal_of,
rw ← Scheme.comp_val_base_apply,
convert (generic_point_eq_of_is_open_immersion ((X.restrict U.open_embedding).iso_Spec.hom ≫... | lemma | algebraic_geometry.is_affine_open.prime_ideal_of_generic_point | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"generic_point",
"generic_point_spec",
"is_integral",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function_field_is_fraction_ring_of_is_affine_open [is_integral X] (U : opens X.carrier)
(hU : is_affine_open U) [hU' : nonempty U] :
is_fraction_ring (X.presheaf.obj $ op U) X.function_field | begin
haveI : is_affine _ := hU,
haveI : nonempty (X.restrict U.open_embedding).carrier := hU',
haveI : is_integral (X.restrict U.open_embedding) := @@is_integral_of_is_affine_is_domain _ _ _
(by { dsimp, rw opens.open_embedding_obj_top, apply_instance }),
have e : U.open_embedding.is_open_map.functor.obj ⊤... | lemma | algebraic_geometry.function_field_is_fraction_ring_of_is_affine_open | algebraic_geometry | src/algebraic_geometry/function_field.lean | [
"algebraic_geometry.properties"
] | [
"is_fraction_ring",
"is_integral",
"mem_non_zero_divisors_iff_ne_zero",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_to_stalk (x : X) : Γ.obj (op X) ⟶ X.presheaf.stalk x | X.presheaf.germ (⟨x,trivial⟩ : (⊤ : opens X)) | def | algebraic_geometry.LocallyRingedSpace.Γ_to_stalk | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The map from the global sections to a stalk. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_fun : X → prime_spectrum (Γ.obj (op X)) | λ x, comap (X.Γ_to_stalk x) (local_ring.closed_point (X.presheaf.stalk x)) | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_fun | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"local_ring.closed_point",
"prime_spectrum"
] | The canonical map from the underlying set to the prime spectrum of `Γ(X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_mem_prime_iff_unit_in_stalk (r : Γ.obj (op X)) (x : X) :
r ∉ (X.to_Γ_Spec_fun x).as_ideal ↔ is_unit (X.Γ_to_stalk x r) | by erw [local_ring.mem_maximal_ideal, not_not] | lemma | algebraic_geometry.LocallyRingedSpace.not_mem_prime_iff_unit_in_stalk | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"is_unit",
"local_ring.mem_maximal_ideal",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Γ_Spec_preim_basic_open_eq (r : Γ.obj (op X)) :
X.to_Γ_Spec_fun⁻¹' (basic_open r).1 = (X.to_RingedSpace.basic_open r).1 | by { ext, erw X.to_RingedSpace.mem_top_basic_open, apply not_mem_prime_iff_unit_in_stalk } | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_preim_basic_open_eq | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The preimage of a basic open in `Spec Γ(X)` under the unit is the basic
open in `X` defined by the same element (they are equal as sets). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_continuous : continuous X.to_Γ_Spec_fun | begin
apply is_topological_basis_basic_opens.continuous,
rintro _ ⟨r, rfl⟩,
erw X.to_Γ_Spec_preim_basic_open_eq r,
exact (X.to_RingedSpace.basic_open r).2,
end | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_continuous | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"continuous"
] | `to_Γ_Spec_fun` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_base : X.to_Top ⟶ Spec.Top_obj (Γ.obj (op X)) | { to_fun := X.to_Γ_Spec_fun,
continuous_to_fun := X.to_Γ_Spec_continuous } | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_base | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The canonical (bundled) continuous map from the underlying topological
space of `X` to the prime spectrum of its global sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_map_basic_open : opens X | (opens.map X.to_Γ_Spec_base).obj (basic_open r) | abbreviation | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_map_basic_open | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The preimage in `X` of a basic open in `Spec Γ(X)` (as an open set). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_map_basic_open_eq : X.to_Γ_Spec_map_basic_open r = X.to_RingedSpace.basic_open r | opens.ext (X.to_Γ_Spec_preim_basic_open_eq r) | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_map_basic_open_eq | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The preimage is the basic open in `X` defined by the same element `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_to_Γ_Spec_map_basic_open :
X.presheaf.obj (op ⊤) ⟶ X.presheaf.obj (op $ X.to_Γ_Spec_map_basic_open r) | X.presheaf.map (X.to_Γ_Spec_map_basic_open r).le_top.op | abbreviation | algebraic_geometry.LocallyRingedSpace.to_to_Γ_Spec_map_basic_open | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The map from the global sections `Γ(X)` to the sections on the (preimage of) a basic open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_res_to_Γ_Spec_map_basic_open :
is_unit (X.to_to_Γ_Spec_map_basic_open r r) | begin
convert (X.presheaf.map $ (eq_to_hom $ X.to_Γ_Spec_map_basic_open_eq r).op)
.is_unit_map (X.to_RingedSpace.is_unit_res_basic_open r),
rw ← comp_apply,
erw ← functor.map_comp,
congr
end | lemma | algebraic_geometry.LocallyRingedSpace.is_unit_res_to_Γ_Spec_map_basic_open | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"is_unit"
] | `r` is a unit as a section on the basic open defined by `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_c_app :
(structure_sheaf $ Γ.obj $ op X).val.obj (op $ basic_open r) ⟶
X.presheaf.obj (op $ X.to_Γ_Spec_map_basic_open r) | is_localization.away.lift r (is_unit_res_to_Γ_Spec_map_basic_open _ r) | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_c_app | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"is_localization.away.lift"
] | Define the sheaf hom on individual basic opens for the unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_c_app_iff
(f : (structure_sheaf $ Γ.obj $ op X).val.obj (op $ basic_open r) ⟶
X.presheaf.obj (op $ X.to_Γ_Spec_map_basic_open r)) :
to_open _ (basic_open r) ≫ f = X.to_to_Γ_Spec_map_basic_open r ↔ f = X.to_Γ_Spec_c_app r | begin
rw ← (is_localization.away.away_map.lift_comp r
(X.is_unit_res_to_Γ_Spec_map_basic_open r)),
swap 5, exact is_localization.to_basic_open _ r,
split,
{ intro h, refine is_localization.ring_hom_ext _ _,
swap 5, exact is_localization.to_basic_open _ r, exact h },
apply congr_arg,
end | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_c_app_iff | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"is_localization.away.away_map.lift_comp",
"is_localization.ring_hom_ext"
] | Characterization of the sheaf hom on basic opens,
direction ← (next lemma) is used at various places, but → is not used in this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_c_app_spec :
to_open _ (basic_open r) ≫ X.to_Γ_Spec_c_app r = X.to_to_Γ_Spec_map_basic_open r | (X.to_Γ_Spec_c_app_iff r _).2 rfl | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_c_app_spec | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Γ_Spec_c_basic_opens :
(induced_functor basic_open).op ⋙ (structure_sheaf (Γ.obj (op X))).1 ⟶
(induced_functor basic_open).op ⋙ ((Top.sheaf.pushforward X.to_Γ_Spec_base).obj X.𝒪).1 | { app := λ r, X.to_Γ_Spec_c_app r.unop,
naturality' := λ r s f, begin
apply (structure_sheaf.to_basic_open_epi (Γ.obj (op X)) r.unop).1,
simp only [← category.assoc],
erw X.to_Γ_Spec_c_app_spec r.unop,
convert X.to_Γ_Spec_c_app_spec s.unop,
symmetry,
apply X.presheaf.map_comp
end } | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_c_basic_opens | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"Top.sheaf.pushforward"
] | The sheaf hom on all basic opens, commuting with restrictions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_SheafedSpace : X.to_SheafedSpace ⟶ Spec.to_SheafedSpace.obj (op (Γ.obj (op X))) | { base := X.to_Γ_Spec_base,
c := Top.sheaf.restrict_hom_equiv_hom (structure_sheaf (Γ.obj (op X))).1 _
is_basis_basic_opens X.to_Γ_Spec_c_basic_opens } | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_SheafedSpace | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"Top.sheaf.restrict_hom_equiv_hom"
] | The canonical morphism of sheafed spaces from `X` to the spectrum of its global sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec_SheafedSpace_app_eq :
X.to_Γ_Spec_SheafedSpace.c.app (op (basic_open r)) = X.to_Γ_Spec_c_app r | Top.sheaf.extend_hom_app _ _ _ _ _ | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_SheafedSpace_app_eq | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"Top.sheaf.extend_hom_app"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Γ_Spec_SheafedSpace_app_spec (r : Γ.obj (op X)) :
to_open _ (basic_open r) ≫ X.to_Γ_Spec_SheafedSpace.c.app (op (basic_open r)) =
X.to_to_Γ_Spec_map_basic_open r | (X.to_Γ_Spec_SheafedSpace_app_eq r).symm ▸ X.to_Γ_Spec_c_app_spec r | lemma | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec_SheafedSpace_app_spec | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_stalk_stalk_map_to_Γ_Spec (x : X) : to_stalk _ _ ≫
PresheafedSpace.stalk_map X.to_Γ_Spec_SheafedSpace x = X.Γ_to_stalk x | begin
rw PresheafedSpace.stalk_map,
erw ← to_open_germ _ (basic_open (1 : Γ.obj (op X)))
⟨X.to_Γ_Spec_fun x, by rw basic_open_one; trivial⟩,
rw [← category.assoc, category.assoc (to_open _ _)],
erw stalk_functor_map_germ,
rw [← category.assoc (to_open _ _), X.to_Γ_Spec_SheafedSpace_app_spec 1],
unfold Γ... | lemma | algebraic_geometry.LocallyRingedSpace.to_stalk_stalk_map_to_Γ_Spec | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The map on stalks induced by the unit commutes with maps from `Γ(X)` to
stalks (in `Spec Γ(X)` and in `X`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Γ_Spec : X ⟶ Spec.LocallyRingedSpace_obj (Γ.obj (op X)) | { val := X.to_Γ_Spec_SheafedSpace,
prop :=
begin
intro x,
let p : prime_spectrum (Γ.obj (op X)) := X.to_Γ_Spec_fun x,
constructor, /- show stalk map is local hom ↓ -/
let S := (structure_sheaf _).presheaf.stalk p,
rintros (t : S) ht,
obtain ⟨⟨r, s⟩, he⟩ := is_localization.surj p.as_ideal.pri... | def | algebraic_geometry.LocallyRingedSpace.to_Γ_Spec | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"is_unit_of_mul_is_unit_left",
"prime_spectrum",
"ring_hom.map_mul"
] | The canonical morphism from `X` to the spectrum of its global sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_ring_hom_ext {X : LocallyRingedSpace} {R : CommRing}
{f : R ⟶ Γ.obj (op X)} {β : X ⟶ Spec.LocallyRingedSpace_obj R}
(w : X.to_Γ_Spec.1.base ≫ (Spec.LocallyRingedSpace_map f).1.base = β.1.base)
(h : ∀ r : R,
f ≫ X.presheaf.map (hom_of_le le_top : (opens.map β.1.base).obj (basic_open r) ⟶ _).op =
to_... | begin
ext1,
apply Spec.basic_open_hom_ext,
{ intros r _,
rw LocallyRingedSpace.comp_val_c_app,
erw to_open_comp_comap_assoc,
rw category.assoc,
erw [to_Γ_Spec_SheafedSpace_app_spec, ← X.presheaf.map_comp],
convert h r },
exact w,
end | lemma | algebraic_geometry.LocallyRingedSpace.comp_ring_hom_ext | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"CommRing",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_Spec_left_triangle : to_Spec_Γ (Γ.obj (op X)) ≫ X.to_Γ_Spec.1.c.app (op ⊤) = 𝟙 _ | begin
unfold to_Spec_Γ,
rw ← to_open_res _ (basic_open (1 : Γ.obj (op X))) ⊤ (eq_to_hom basic_open_one.symm),
erw category.assoc,
rw [nat_trans.naturality, ← category.assoc],
erw [X.to_Γ_Spec_SheafedSpace_app_spec 1, ← functor.map_comp],
convert eq_to_hom_map X.presheaf _, refl,
end | lemma | algebraic_geometry.LocallyRingedSpace.Γ_Spec_left_triangle | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | `to_Spec_Γ _` is an isomorphism so these are mutually two-sided inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
identity_to_Γ_Spec : 𝟭 LocallyRingedSpace.{u} ⟶ Γ.right_op ⋙ Spec.to_LocallyRingedSpace | { app := LocallyRingedSpace.to_Γ_Spec,
naturality' := λ X Y f, begin
symmetry,
apply LocallyRingedSpace.comp_ring_hom_ext,
{ ext1 x,
dsimp [Spec.Top_map, LocallyRingedSpace.to_Γ_Spec_fun],
rw [← local_ring.comap_closed_point (PresheafedSpace.stalk_map _ x),
← prime_spectrum.comap_comp_... | def | algebraic_geometry.identity_to_Γ_Spec | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"local_ring.comap_closed_point",
"prime_spectrum.comap_comp_apply"
] | The unit as a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_triangle (X : LocallyRingedSpace) :
Spec_Γ_identity.inv.app (Γ.obj (op X)) ≫ (identity_to_Γ_Spec.app X).val.c.app (op ⊤) = 𝟙 _ | X.Γ_Spec_left_triangle | lemma | algebraic_geometry.Γ_Spec.left_triangle | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_triangle (R : CommRing) :
identity_to_Γ_Spec.app (Spec.to_LocallyRingedSpace.obj $ op R) ≫
Spec.to_LocallyRingedSpace.map (Spec_Γ_identity.inv.app R).op = 𝟙 _ | begin
apply LocallyRingedSpace.comp_ring_hom_ext,
{ ext (p : prime_spectrum R) x,
erw ← is_localization.at_prime.to_map_mem_maximal_iff
((structure_sheaf R).presheaf.stalk p) p.as_ideal x,
refl },
{ intro r, apply to_open_res },
end | lemma | algebraic_geometry.Γ_Spec.right_triangle | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"CommRing",
"is_localization.at_prime.to_map_mem_maximal_iff",
"prime_spectrum"
] | `Spec_Γ_identity` is iso so these are mutually two-sided inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LocallyRingedSpace_adjunction : Γ.right_op ⊣ Spec.to_LocallyRingedSpace | adjunction.mk_of_unit_counit
{ unit := identity_to_Γ_Spec,
counit := (nat_iso.op Spec_Γ_identity).inv,
left_triangle' := by { ext X, erw category.id_comp,
exact congr_arg quiver.hom.op (left_triangle X) },
right_triangle' := by { ext1, ext1 R, erw category.id_comp,
exact right_triangle R.unop } } | def | algebraic_geometry.Γ_Spec.LocallyRingedSpace_adjunction | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"quiver.hom.op"
] | The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `LocallyRingedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction : Scheme.Γ.right_op ⊣ Scheme.Spec | LocallyRingedSpace_adjunction.restrict_fully_faithful
Scheme.forget_to_LocallyRingedSpace (𝟭 _)
(nat_iso.of_components (λ X, iso.refl _) (λ _ _ f, by simpa))
(nat_iso.of_components (λ X, iso.refl _) (λ _ _ f, by simpa)) | def | algebraic_geometry.Γ_Spec.adjunction | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `Scheme`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_hom_equiv_apply {X : Scheme} {R : CommRingᵒᵖ}
(f : (op $ Scheme.Γ.obj $ op X) ⟶ R) :
Γ_Spec.adjunction.hom_equiv X R f =
LocallyRingedSpace_adjunction.hom_equiv X.1 R f | by { dsimp [adjunction, adjunction.restrict_fully_faithful], simp } | lemma | algebraic_geometry.Γ_Spec.adjunction_hom_equiv_apply | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_hom_equiv (X : Scheme) (R : CommRingᵒᵖ) :
Γ_Spec.adjunction.hom_equiv X R = LocallyRingedSpace_adjunction.hom_equiv X.1 R | equiv.ext $ λ f, adjunction_hom_equiv_apply f | lemma | algebraic_geometry.Γ_Spec.adjunction_hom_equiv | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_hom_equiv_symm_apply {X : Scheme} {R : CommRingᵒᵖ}
(f : X ⟶ Scheme.Spec.obj R) :
(Γ_Spec.adjunction.hom_equiv X R).symm f =
(LocallyRingedSpace_adjunction.hom_equiv X.1 R).symm f | by { congr' 2, exact adjunction_hom_equiv _ _ } | lemma | algebraic_geometry.Γ_Spec.adjunction_hom_equiv_symm_apply | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_counit_app {R : CommRingᵒᵖ} :
Γ_Spec.adjunction.counit.app R = LocallyRingedSpace_adjunction.counit.app R | by { rw [← adjunction.hom_equiv_symm_id, ← adjunction.hom_equiv_symm_id,
adjunction_hom_equiv_symm_apply], refl } | lemma | algebraic_geometry.Γ_Spec.adjunction_counit_app | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_unit_app {X : Scheme} :
Γ_Spec.adjunction.unit.app X = LocallyRingedSpace_adjunction.unit.app X.1 | by { rw [← adjunction.hom_equiv_id, ← adjunction.hom_equiv_id, adjunction_hom_equiv_apply], refl } | lemma | algebraic_geometry.Γ_Spec.adjunction_unit_app | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_LocallyRingedSpace_adjunction_counit :
is_iso LocallyRingedSpace_adjunction.counit | is_iso.of_iso_inv _ | instance | algebraic_geometry.Γ_Spec.is_iso_LocallyRingedSpace_adjunction_counit | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_adjunction_counit : is_iso Γ_Spec.adjunction.counit | begin
apply_with nat_iso.is_iso_of_is_iso_app { instances := ff },
intro R,
rw adjunction_counit_app,
apply_instance,
end | instance | algebraic_geometry.Γ_Spec.is_iso_adjunction_counit | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_unit_app_app_top (X : Scheme) :
@eq ((Scheme.Spec.obj (op $ X.presheaf.obj (op ⊤))).presheaf.obj (op ⊤) ⟶
((Γ_Spec.adjunction.unit.app X).1.base _* X.presheaf).obj (op ⊤))
((Γ_Spec.adjunction.unit.app X).val.c.app (op ⊤))
(Spec_Γ_identity.hom.app (X.presheaf.obj (op ⊤))) | begin
have := congr_app Γ_Spec.adjunction.left_triangle X,
dsimp at this,
rw ← is_iso.eq_comp_inv at this,
simp only [Γ_Spec.LocallyRingedSpace_adjunction_counit, nat_trans.op_app, category.id_comp,
Γ_Spec.adjunction_counit_app] at this,
rw [← op_inv, nat_iso.inv_inv_app, quiver.hom.op_inj.eq_iff] at this... | lemma | algebraic_geometry.Γ_Spec.adjunction_unit_app_app_top | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.preserves_limits : limits.preserves_limits Scheme.Spec | Γ_Spec.adjunction.right_adjoint_preserves_limits | instance | algebraic_geometry.Spec.preserves_limits | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.full : full Scheme.Spec | R_full_of_counit_is_iso Γ_Spec.adjunction | instance | algebraic_geometry.Spec.full | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.faithful : faithful Scheme.Spec | R_faithful_of_counit_is_iso Γ_Spec.adjunction | instance | algebraic_geometry.Spec.faithful | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.reflective : reflective Scheme.Spec | ⟨⟩ | instance | algebraic_geometry.Spec.reflective | algebraic_geometry | src/algebraic_geometry/Gamma_Spec_adjunction.lean | [
"algebraic_geometry.Scheme",
"category_theory.adjunction.limits",
"category_theory.adjunction.reflective"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
glue_data extends category_theory.glue_data Scheme | (f_open : ∀ i j, is_open_immersion (f i j)) | structure | algebraic_geometry.Scheme.glue_data | algebraic_geometry | src/algebraic_geometry/gluing.lean | [
"algebraic_geometry.presheafed_space.gluing",
"algebraic_geometry.open_immersion.Scheme"
] | [
"category_theory.glue_data"
] | A family of gluing data consists of
1. An index type `J`
2. An scheme `U i` for each `i : J`.
3. An scheme `V i j` for each `i j : J`.
(Note that this is `J × J → Scheme` rather than `J → J → Scheme` to connect to the
limits library easier.)
4. An open immersion `f i j : V i j ⟶ U i` for each `i j : ι`.
5. A transi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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