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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basic_open_res {U V : (opens X)ᵒᵖ} (i : U ⟶ V) (f : X.presheaf.obj U) :
@basic_open X (unop V) (X.presheaf.map i f) = (unop V) ⊓ @basic_open X (unop U) f | begin
induction U using opposite.rec,
induction V using opposite.rec,
let g := i.unop, have : i = g.op := rfl, clear_value g, subst this,
ext, split,
{ rintro ⟨x, (hx : is_unit _), rfl⟩,
rw germ_res_apply at hx,
exact ⟨x.2, g x, hx, rfl⟩ },
{ rintros ⟨hxV, x, hx, rfl⟩,
refine ⟨⟨x, hxV⟩, (_ : is_... | lemma | algebraic_geometry.RingedSpace.basic_open_res | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit",
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_res_eq {U V : (opens X)ᵒᵖ} (i : U ⟶ V) [is_iso i]
(f : X.presheaf.obj U) :
@basic_open X (unop V) (X.presheaf.map i f) = @RingedSpace.basic_open X (unop U) f | begin
apply le_antisymm,
{ rw X.basic_open_res i f, exact inf_le_right },
{ have := X.basic_open_res (inv i) (X.presheaf.map i f),
rw [← comp_apply, ← X.presheaf.map_comp, is_iso.hom_inv_id, X.presheaf.map_id] at this,
erw this,
exact inf_le_right }
end | lemma | algebraic_geometry.RingedSpace.basic_open_res_eq | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul {U : opens X} (f g : X.presheaf.obj (op U)) :
X.basic_open (f * g) = X.basic_open f ⊓ X.basic_open g | begin
ext1,
dsimp [RingedSpace.basic_open],
rw ←set.image_inter subtype.coe_injective,
congr,
ext,
simp_rw map_mul,
exact is_unit.mul_iff,
end | lemma | algebraic_geometry.RingedSpace.basic_open_mul | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit.mul_iff",
"map_mul",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_of_is_unit {U : opens X} {f : X.presheaf.obj (op U)} (hf : is_unit f) :
X.basic_open f = U | begin
apply le_antisymm,
{ exact X.basic_open_le f },
intros x hx,
erw X.mem_basic_open f (⟨x, hx⟩ : U),
exact ring_hom.is_unit_map _ hf
end | lemma | algebraic_geometry.RingedSpace.basic_open_of_is_unit | algebraic_geometry | src/algebraic_geometry/ringed_space.lean | [
"algebra.category.Ring.filtered_colimits",
"algebraic_geometry.sheafed_space",
"topology.sheaves.stalks",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits"
] | [
"is_unit",
"ring_hom.is_unit_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scheme extends to_LocallyRingedSpace : LocallyRingedSpace | (local_affine : ∀ x : to_LocallyRingedSpace, ∃ (U : open_nhds x) (R : CommRing),
nonempty (to_LocallyRingedSpace.restrict U.open_embedding ≅
Spec.to_LocallyRingedSpace.obj (op R))) | structure | algebraic_geometry.Scheme | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | We define `Scheme` as a `X : LocallyRingedSpace`,
along with a proof that every point has an open neighbourhood `U`
so that that the restriction of `X` to `U` is isomorphic,
as a locally ringed space, to `Spec.to_LocallyRingedSpace.obj (op R)`
for some `R : CommRing`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (X Y : Scheme) : Type* | X.to_LocallyRingedSpace ⟶ Y.to_LocallyRingedSpace | def | algebraic_geometry.Scheme.hom | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | A morphism between schemes is a morphism between the underlying locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sheaf (X : Scheme) | X.to_SheafedSpace.sheaf | abbreviation | algebraic_geometry.Scheme.sheaf | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | The structure sheaf of a Scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_LocallyRingedSpace : Scheme ⥤ LocallyRingedSpace | induced_functor _ | def | algebraic_geometry.Scheme.forget_to_LocallyRingedSpace | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | The forgetful functor from `Scheme` to `LocallyRingedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_LocallyRingedSpace_preimage {X Y : Scheme} (f : X ⟶ Y) :
Scheme.forget_to_LocallyRingedSpace.preimage f = f | rfl | lemma | algebraic_geometry.Scheme.forget_to_LocallyRingedSpace_preimage | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Top : Scheme ⥤ Top | Scheme.forget_to_LocallyRingedSpace ⋙ LocallyRingedSpace.forget_to_Top | def | algebraic_geometry.Scheme.forget_to_Top | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"Top"
] | The forgetful functor from `Scheme` to `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_val_base (X : Scheme) : (𝟙 X : _).1.base = 𝟙 _ | rfl | lemma | algebraic_geometry.Scheme.id_val_base | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_app {X : Scheme} (U : (opens X.carrier)ᵒᵖ) :
(𝟙 X : _).val.c.app U = X.presheaf.map
(eq_to_hom (by { induction U using opposite.rec, cases U, refl })) | PresheafedSpace.id_c_app X.to_PresheafedSpace U | lemma | algebraic_geometry.Scheme.id_app | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val = f.val ≫ g.val | rfl | lemma | algebraic_geometry.Scheme.comp_val | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coe_base {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base | rfl | lemma | algebraic_geometry.Scheme.comp_coe_base | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val_base {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base | rfl | lemma | algebraic_geometry.Scheme.comp_val_base | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val_c_app {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(f ≫ g).val.c.app U = g.val.c.app U ≫ f.val.c.app _ | rfl | lemma | algebraic_geometry.Scheme.comp_val_c_app | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_app {X Y : Scheme} {f g : X ⟶ Y} (e : f = g) (U) :
f.val.c.app U = g.val.c.app U ≫ X.presheaf.map (eq_to_hom (by subst e)) | by { subst e, dsimp, simp } | lemma | algebraic_geometry.Scheme.congr_app | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_eq {X Y : Scheme} (f : X ⟶ Y) {U V : opens Y.carrier} (e : U = V) :
f.val.c.app (op U) = Y.presheaf.map (eq_to_hom e.symm).op ≫
f.val.c.app (op V) ≫ X.presheaf.map (eq_to_hom (congr_arg (opens.map f.val.base).obj e)).op | begin
rw [← is_iso.inv_comp_eq, ← functor.map_inv, f.val.c.naturality, presheaf.pushforward_obj_map],
congr
end | lemma | algebraic_geometry.Scheme.app_eq | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_LocallyRingedSpace_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] :
@is_iso LocallyRingedSpace _ _ _ f | forget_to_LocallyRingedSpace.map_is_iso f | instance | algebraic_geometry.Scheme.is_LocallyRingedSpace_iso | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_val_c_app {X Y : Scheme} (f : X ⟶ Y) [is_iso f] (U : opens X.carrier) :
(inv f).val.c.app (op U) = X.presheaf.map (eq_to_hom $ by { rw is_iso.hom_inv_id, ext1, refl } :
(opens.map (f ≫ inv f).1.base).obj U ⟶ U).op ≫
inv (f.val.c.app (op $ (opens.map _).obj U)) | begin
rw [is_iso.eq_comp_inv],
erw ← Scheme.comp_val_c_app,
rw [Scheme.congr_app (is_iso.hom_inv_id f),
Scheme.id_app, ← functor.map_comp, eq_to_hom_trans, eq_to_hom_op],
refl
end | lemma | algebraic_geometry.Scheme.inv_val_c_app | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.app_le {X Y : Scheme}
(f : X ⟶ Y) {V : opens X.carrier} {U : opens Y.carrier} (e : V ≤ (opens.map f.1.base).obj U) :
Y.presheaf.obj (op U) ⟶ X.presheaf.obj (op V) | f.1.c.app (op U) ≫ X.presheaf.map (hom_of_le e).op | abbreviation | algebraic_geometry.Scheme.hom.app_le | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | Given a morphism of schemes `f : X ⟶ Y`, and open sets `U ⊆ Y`, `V ⊆ f ⁻¹' U`,
this is the induced map `Γ(Y, U) ⟶ Γ(X, V)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec_obj (R : CommRing) : Scheme | { local_affine := λ x,
⟨⟨⊤, trivial⟩, R, ⟨(Spec.to_LocallyRingedSpace.obj (op R)).restrict_top_iso⟩⟩,
to_LocallyRingedSpace := Spec.LocallyRingedSpace_obj R } | def | algebraic_geometry.Scheme.Spec_obj | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | The spectrum of a commutative ring, as a scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec_obj_to_LocallyRingedSpace (R : CommRing) :
(Spec_obj R).to_LocallyRingedSpace = Spec.LocallyRingedSpace_obj R | rfl | lemma | algebraic_geometry.Scheme.Spec_obj_to_LocallyRingedSpace | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_map {R S : CommRing} (f : R ⟶ S) :
Spec_obj S ⟶ Spec_obj R | (Spec.LocallyRingedSpace_map f : Spec.LocallyRingedSpace_obj S ⟶ Spec.LocallyRingedSpace_obj R) | def | algebraic_geometry.Scheme.Spec_map | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | The induced map of a ring homomorphism on the ring spectra, as a morphism of schemes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec_map_id (R : CommRing) :
Spec_map (𝟙 R) = 𝟙 (Spec_obj R) | Spec.LocallyRingedSpace_map_id R | lemma | algebraic_geometry.Scheme.Spec_map_id | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
Spec_map (f ≫ g) = Spec_map g ≫ Spec_map f | Spec.LocallyRingedSpace_map_comp f g | lemma | algebraic_geometry.Scheme.Spec_map_comp | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec : CommRingᵒᵖ ⥤ Scheme | { obj := λ R, Spec_obj (unop R),
map := λ R S f, Spec_map f.unop,
map_id' := λ R, by rw [unop_id, Spec_map_id],
map_comp' := λ R S T f g, by rw [unop_comp, Spec_map_comp] } | def | algebraic_geometry.Scheme.Spec | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u} empty : Scheme.{u} | { carrier := Top.of pempty,
presheaf := (category_theory.functor.const _).obj (CommRing.of punit),
is_sheaf := presheaf.is_sheaf_of_is_terminal _ CommRing.punit_is_terminal,
local_ring := λ x, pempty.elim x,
local_affine := λ x, pempty.elim x } | def | algebraic_geometry.Scheme.empty | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing.of",
"CommRing.punit_is_terminal",
"Top.of",
"category_theory.functor.const",
"local_ring",
"pempty",
"pempty.elim"
] | The empty scheme. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : Schemeᵒᵖ ⥤ CommRing | (induced_functor Scheme.to_LocallyRingedSpace).op ⋙ LocallyRingedSpace.Γ | def | algebraic_geometry.Scheme.Γ | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing"
] | The global sections, notated Gamma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ_def : Γ = (induced_functor Scheme.to_LocallyRingedSpace).op ⋙ LocallyRingedSpace.Γ | rfl | lemma | algebraic_geometry.Scheme.Γ_def | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj (X : Schemeᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.Scheme.Γ_obj | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj_op (X : Scheme) : Γ.obj (op X) = X.presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.Scheme.Γ_obj_op | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map {X Y : Schemeᵒᵖ} (f : X ⟶ Y) :
Γ.map f = f.unop.1.c.app (op ⊤) | rfl | lemma | algebraic_geometry.Scheme.Γ_map | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map_op {X Y : Scheme} (f : X ⟶ Y) :
Γ.map f.op = f.1.c.app (op ⊤) | rfl | lemma | algebraic_geometry.Scheme.Γ_map_op | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open : opens X.carrier | X.to_LocallyRingedSpace.to_RingedSpace.basic_open f | def | algebraic_geometry.Scheme.basic_open | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | The subset of the underlying space where the given section does not vanish. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_basic_open (x : U) : ↑x ∈ X.basic_open f ↔ is_unit (X.presheaf.germ x f) | RingedSpace.mem_basic_open _ _ _ | lemma | algebraic_geometry.Scheme.mem_basic_open | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_basic_open_top (f : X.presheaf.obj (op ⊤)) (x : X.carrier) :
x ∈ X.basic_open f ↔ is_unit (X.presheaf.germ (⟨x, trivial⟩ : (⊤ : opens _)) f) | RingedSpace.mem_basic_open _ f ⟨x, trivial⟩ | lemma | algebraic_geometry.Scheme.mem_basic_open_top | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_res (i : op U ⟶ op V) :
X.basic_open (X.presheaf.map i f) = V ⊓ X.basic_open f | RingedSpace.basic_open_res _ i f | lemma | algebraic_geometry.Scheme.basic_open_res | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_res_eq (i : op U ⟶ op V) [is_iso i] :
X.basic_open (X.presheaf.map i f) = X.basic_open f | RingedSpace.basic_open_res_eq _ i f | lemma | algebraic_geometry.Scheme.basic_open_res_eq | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_le : X.basic_open f ≤ U | RingedSpace.basic_open_le _ _ | lemma | algebraic_geometry.Scheme.basic_open_le | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_basic_open {X Y : Scheme} (f : X ⟶ Y) {U : opens Y.carrier}
(r : Y.presheaf.obj $ op U) :
(opens.map f.1.base).obj (Y.basic_open r) =
@Scheme.basic_open X ((opens.map f.1.base).obj U) (f.1.c.app _ r) | LocallyRingedSpace.preimage_basic_open f r | lemma | algebraic_geometry.Scheme.preimage_basic_open | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_zero (U : opens X.carrier) : X.basic_open (0 : X.presheaf.obj $ op U) = ⊥ | LocallyRingedSpace.basic_open_zero _ U | lemma | algebraic_geometry.Scheme.basic_open_zero | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_mul : X.basic_open (f * g) = X.basic_open f ⊓ X.basic_open g | RingedSpace.basic_open_mul _ _ _ | lemma | algebraic_geometry.Scheme.basic_open_mul | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_of_is_unit {f : X.presheaf.obj (op U)} (hf : is_unit f) : X.basic_open f = U | RingedSpace.basic_open_of_is_unit _ hf | lemma | algebraic_geometry.Scheme.basic_open_of_is_unit | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_of_affine {R : CommRing} (f : R) :
(Scheme.Spec.obj $ op R).basic_open ((Spec_Γ_identity.app R).inv f) =
prime_spectrum.basic_open f | begin
ext,
erw Scheme.mem_basic_open_top,
suffices : is_unit (structure_sheaf.to_stalk R x f) ↔ f ∉ prime_spectrum.as_ideal x,
{ exact this },
erw [← is_unit_map_iff (structure_sheaf.stalk_to_fiber_ring_hom R x),
structure_sheaf.stalk_to_fiber_ring_hom_to_stalk],
exact (is_localization.at_prime.is_unit_... | lemma | algebraic_geometry.basic_open_eq_of_affine | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"is_localization.at_prime.is_unit_to_map_iff",
"is_unit",
"is_unit_map_iff",
"localization.at_prime",
"prime_spectrum.basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_eq_of_affine' {R : CommRing}
(f : (Spec.to_SheafedSpace.obj (op R)).presheaf.obj (op ⊤)) :
(Scheme.Spec.obj $ op R).basic_open f =
prime_spectrum.basic_open ((Spec_Γ_identity.app R).hom f) | begin
convert basic_open_eq_of_affine ((Spec_Γ_identity.app R).hom f),
exact (iso.hom_inv_id_apply _ _).symm
end | lemma | algebraic_geometry.basic_open_eq_of_affine' | algebraic_geometry | src/algebraic_geometry/Scheme.lean | [
"algebraic_geometry.Spec",
"algebra.category.Ring.constructions"
] | [
"CommRing",
"prime_spectrum.basic_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
SheafedSpace extends PresheafedSpace.{v} C | (is_sheaf : presheaf.is_sheaf) | structure | algebraic_geometry.SheafedSpace | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | A `SheafedSpace C` is a topological space equipped with a sheaf of `C`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_carrier : has_coe (SheafedSpace C) Top | { coe := λ X, X.carrier } | instance | algebraic_geometry.SheafedSpace.coe_carrier | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sheaf (X : SheafedSpace C) : sheaf C (X : Top.{v}) | ⟨X.presheaf, X.is_sheaf⟩ | def | algebraic_geometry.SheafedSpace.sheaf | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | Extract the `sheaf C (X : Top)` from a `SheafedSpace C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_coe (X : SheafedSpace.{v} C) : X.carrier = (X : Top.{v}) | rfl | lemma | algebraic_geometry.SheafedSpace.as_coe | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (carrier) (presheaf) (h) :
(({ carrier := carrier, presheaf := presheaf, is_sheaf := h } : SheafedSpace.{v} C) :
Top.{v}) = carrier | rfl | lemma | algebraic_geometry.SheafedSpace.mk_coe | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit (X : Top) : SheafedSpace (discrete unit) | { is_sheaf := presheaf.is_sheaf_unit _,
..@PresheafedSpace.const (discrete unit) _ X ⟨⟨⟩⟩ } | def | algebraic_geometry.SheafedSpace.unit | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [
"Top"
] | The trivial `unit` valued sheaf on any topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_PresheafedSpace : (SheafedSpace.{v} C) ⥤ (PresheafedSpace.{v} C) | induced_functor _ | def | algebraic_geometry.SheafedSpace.forget_to_PresheafedSpace | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | Forgetting the sheaf condition is a functor from `SheafedSpace C` to `PresheafedSpace C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_PresheafedSpace_iso {X Y : SheafedSpace.{v} C} (f : X ⟶ Y) [is_iso f] :
@is_iso (PresheafedSpace C) _ _ _ f | SheafedSpace.forget_to_PresheafedSpace.map_is_iso f | instance | algebraic_geometry.SheafedSpace.is_PresheafedSpace_iso | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_base (X : SheafedSpace C) :
((𝟙 X) : X ⟶ X).base = (𝟙 (X : Top.{v})) | rfl | lemma | algebraic_geometry.SheafedSpace.id_base | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_c (X : SheafedSpace C) :
((𝟙 X) : X ⟶ X).c = eq_to_hom (presheaf.pushforward.id_eq X.presheaf).symm | rfl | lemma | algebraic_geometry.SheafedSpace.id_c | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_c_app (X : SheafedSpace C) (U) :
((𝟙 X) : X ⟶ X).c.app U = eq_to_hom (by { induction U using opposite.rec, cases U, refl }) | by { induction U using opposite.rec, cases U, simp only [id_c], dsimp, simp, } | lemma | algebraic_geometry.SheafedSpace.id_c_app | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_base {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).base = f.base ≫ g.base | rfl | lemma | algebraic_geometry.SheafedSpace.comp_base | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_c_app {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :
(α ≫ β).c.app U = (β.c).app U ≫ (α.c).app (op ((opens.map (β.base)).obj (unop U))) | rfl | lemma | algebraic_geometry.SheafedSpace.comp_c_app | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_c_app' {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :
(α ≫ β).c.app (op U) = (β.c).app (op U) ≫ (α.c).app (op ((opens.map (β.base)).obj U)) | rfl | lemma | algebraic_geometry.SheafedSpace.comp_c_app' | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_app {X Y : SheafedSpace C} {α β : X ⟶ Y} (h : α = β) (U) :
α.c.app U = β.c.app U ≫ X.presheaf.map (eq_to_hom (by subst h)) | PresheafedSpace.congr_app h U | lemma | algebraic_geometry.SheafedSpace.congr_app | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : SheafedSpace C ⥤ Top | { obj := λ X, (X : Top.{v}),
map := λ X Y f, f.base } | def | algebraic_geometry.SheafedSpace.forget | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [
"Top"
] | The forgetful functor from `SheafedSpace` to `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict {U : Top} (X : SheafedSpace C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) : SheafedSpace C | { is_sheaf := is_sheaf_of_open_embedding h X.is_sheaf,
..X.to_PresheafedSpace.restrict h } | def | algebraic_geometry.SheafedSpace.restrict | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [
"Top",
"open_embedding"
] | The restriction of a sheafed space along an open embedding into the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_top_iso (X : SheafedSpace C) :
X.restrict (opens.open_embedding ⊤) ≅ X | forget_to_PresheafedSpace.preimage_iso X.to_PresheafedSpace.restrict_top_iso | def | algebraic_geometry.SheafedSpace.restrict_top_iso | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | The restriction of a sheafed space `X` to the top subspace is isomorphic to `X` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ : (SheafedSpace C)ᵒᵖ ⥤ C | forget_to_PresheafedSpace.op ⋙ PresheafedSpace.Γ | def | algebraic_geometry.SheafedSpace.Γ | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | The global sections, notated Gamma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Γ_def : (Γ : _ ⥤ C) = forget_to_PresheafedSpace.op ⋙ PresheafedSpace.Γ | rfl | lemma | algebraic_geometry.SheafedSpace.Γ_def | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj (X : (SheafedSpace C)ᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.SheafedSpace.Γ_obj | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_obj_op (X : SheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) | rfl | lemma | algebraic_geometry.SheafedSpace.Γ_obj_op | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map {X Y : (SheafedSpace C)ᵒᵖ} (f : X ⟶ Y) :
Γ.map f = f.unop.c.app (op ⊤) | rfl | lemma | algebraic_geometry.SheafedSpace.Γ_map | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Γ_map_op {X Y : SheafedSpace C} (f : X ⟶ Y) :
Γ.map f.op = f.c.app (op ⊤) | rfl | lemma | algebraic_geometry.SheafedSpace.Γ_map_op | algebraic_geometry | src/algebraic_geometry/sheafed_space.lean | [
"algebraic_geometry.presheafed_space.has_colimits",
"topology.sheaves.functors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.Top_obj (R : CommRing) : Top | Top.of (prime_spectrum R) | def | algebraic_geometry.Spec.Top_obj | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"Top",
"Top.of",
"prime_spectrum"
] | The spectrum of a commutative ring, as a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.Top_map {R S : CommRing} (f : R ⟶ S) :
Spec.Top_obj S ⟶ Spec.Top_obj R | prime_spectrum.comap f | def | algebraic_geometry.Spec.Top_map | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"prime_spectrum.comap"
] | The induced map of a ring homomorphism on the ring spectra, as a morphism of topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.Top_map_id (R : CommRing) :
Spec.Top_map (𝟙 R) = 𝟙 (Spec.Top_obj R) | prime_spectrum.comap_id | lemma | algebraic_geometry.Spec.Top_map_id | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"prime_spectrum.comap_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.Top_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
Spec.Top_map (f ≫ g) = Spec.Top_map g ≫ Spec.Top_map f | prime_spectrum.comap_comp _ _ | lemma | algebraic_geometry.Spec.Top_map_comp | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"prime_spectrum.comap_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_Top : CommRingᵒᵖ ⥤ Top | { obj := λ R, Spec.Top_obj (unop R),
map := λ R S f, Spec.Top_map f.unop,
map_id' := λ R, by rw [unop_id, Spec.Top_map_id],
map_comp' := λ R S T f g, by rw [unop_comp, Spec.Top_map_comp] } | def | algebraic_geometry.Spec.to_Top | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"Top"
] | The spectrum, as a contravariant functor from commutative rings to topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.SheafedSpace_obj (R : CommRing) : SheafedSpace CommRing | { carrier := Spec.Top_obj R,
presheaf := (structure_sheaf R).1,
is_sheaf := (structure_sheaf R).2 } | def | algebraic_geometry.Spec.SheafedSpace_obj | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | The spectrum of a commutative ring, as a `SheafedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.SheafedSpace_map {R S : CommRing.{u}} (f : R ⟶ S) :
Spec.SheafedSpace_obj S ⟶ Spec.SheafedSpace_obj R | { base := Spec.Top_map f,
c :=
{ app := λ U, comap f (unop U) ((topological_space.opens.map (Spec.Top_map f)).obj (unop U))
(λ p, id),
naturality' := λ U V i, ring_hom.ext $ λ s, subtype.eq $ funext $ λ p, rfl } } | def | algebraic_geometry.Spec.SheafedSpace_map | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"ring_hom.ext",
"topological_space.opens.map"
] | The induced map of a ring homomorphism on the ring spectra, as a morphism of sheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.SheafedSpace_map_id {R : CommRing} :
Spec.SheafedSpace_map (𝟙 R) = 𝟙 (Spec.SheafedSpace_obj R) | PresheafedSpace.ext _ _ (Spec.Top_map_id R) $ nat_trans.ext _ _ $ funext $ λ U,
begin
dsimp,
erw [PresheafedSpace.id_c_app, comap_id], swap,
{ rw [Spec.Top_map_id, topological_space.opens.map_id_obj_unop] },
simpa [eq_to_hom_map],
end | lemma | algebraic_geometry.Spec.SheafedSpace_map_id | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"topological_space.opens.map_id_obj_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.SheafedSpace_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
Spec.SheafedSpace_map (f ≫ g) = Spec.SheafedSpace_map g ≫ Spec.SheafedSpace_map f | PresheafedSpace.ext _ _ (Spec.Top_map_comp f g) $ nat_trans.ext _ _ $ funext $ λ U,
by { dsimp, rw category_theory.functor.map_id, rw category.comp_id, erw comap_comp f g, refl } | lemma | algebraic_geometry.Spec.SheafedSpace_map_comp | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_SheafedSpace : CommRingᵒᵖ ⥤ SheafedSpace CommRing | { obj := λ R, Spec.SheafedSpace_obj (unop R),
map := λ R S f, Spec.SheafedSpace_map f.unop,
map_id' := λ R, by rw [unop_id, Spec.SheafedSpace_map_id],
map_comp' := λ R S T f g, by rw [unop_comp, Spec.SheafedSpace_map_comp] } | def | algebraic_geometry.Spec.to_SheafedSpace | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | Spec, as a contravariant functor from commutative rings to sheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.to_PresheafedSpace : CommRingᵒᵖ ⥤ PresheafedSpace.{u} CommRing.{u} | Spec.to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace | def | algebraic_geometry.Spec.to_PresheafedSpace | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [] | Spec, as a contravariant functor from commutative rings to presheafed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.to_PresheafedSpace_obj (R : CommRingᵒᵖ) :
Spec.to_PresheafedSpace.obj R = (Spec.SheafedSpace_obj (unop R)).to_PresheafedSpace | rfl | lemma | algebraic_geometry.Spec.to_PresheafedSpace_obj | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_PresheafedSpace_obj_op (R : CommRing) :
Spec.to_PresheafedSpace.obj (op R) = (Spec.SheafedSpace_obj R).to_PresheafedSpace | rfl | lemma | algebraic_geometry.Spec.to_PresheafedSpace_obj_op | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_PresheafedSpace_map (R S : CommRingᵒᵖ) (f : R ⟶ S) :
Spec.to_PresheafedSpace.map f = Spec.SheafedSpace_map f.unop | rfl | lemma | algebraic_geometry.Spec.to_PresheafedSpace_map | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_PresheafedSpace_map_op (R S : CommRing) (f : R ⟶ S) :
Spec.to_PresheafedSpace.map f.op = Spec.SheafedSpace_map f | rfl | lemma | algebraic_geometry.Spec.to_PresheafedSpace_map_op | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.basic_open_hom_ext {X : RingedSpace} {R : CommRing} {α β : X ⟶ Spec.SheafedSpace_obj R}
(w : α.base = β.base) (h : ∀ r : R, let U := prime_spectrum.basic_open r in
(to_open R U ≫ α.c.app (op U)) ≫ X.presheaf.map (eq_to_hom (by rw w)) =
to_open R U ≫ β.c.app (op U)) : α = β | begin
ext1,
{ apply ((Top.sheaf.pushforward β.base).obj X.sheaf).hom_ext _
prime_spectrum.is_basis_basic_opens,
intro r,
apply (structure_sheaf.to_basic_open_epi R r).1,
simpa using h r },
exact w,
end | lemma | algebraic_geometry.Spec.basic_open_hom_ext | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"Top.sheaf.pushforward",
"hom_ext",
"prime_spectrum.basic_open",
"prime_spectrum.is_basis_basic_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.LocallyRingedSpace_obj (R : CommRing) : LocallyRingedSpace | { local_ring := λ x, @@ring_equiv.local_ring _
(show local_ring (localization.at_prime _), by apply_instance) _
(iso.CommRing_iso_to_ring_equiv $ stalk_iso R x).symm,
.. Spec.SheafedSpace_obj R } | def | algebraic_geometry.Spec.LocallyRingedSpace_obj | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"local_ring",
"localization.at_prime",
"ring_equiv.local_ring"
] | The spectrum of a commutative ring, as a `LocallyRingedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_map_to_stalk {R S : CommRing} (f : R ⟶ S) (p : prime_spectrum S) :
to_stalk R (prime_spectrum.comap f p) ≫
PresheafedSpace.stalk_map (Spec.SheafedSpace_map f) p =
f ≫ to_stalk S p | begin
erw [← to_open_germ S ⊤ ⟨p, trivial⟩, ← to_open_germ R ⊤ ⟨prime_spectrum.comap f p, trivial⟩,
category.assoc, PresheafedSpace.stalk_map_germ (Spec.SheafedSpace_map f) ⊤ ⟨p, trivial⟩,
Spec.SheafedSpace_map_c_app, to_open_comp_comap_assoc],
refl
end | lemma | algebraic_geometry.stalk_map_to_stalk | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"prime_spectrum",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom_comp_stalk_iso {R S : CommRing} (f : R ⟶ S) (p : prime_spectrum S) :
(stalk_iso R (prime_spectrum.comap f p)).hom ≫
@category_struct.comp _ _
(CommRing.of (localization.at_prime (prime_spectrum.comap f p).as_ideal))
(CommRing.of (localization.at_prime p.as_ideal)) _
(localization.... | (stalk_iso R (prime_spectrum.comap f p)).eq_inv_comp.mp $ (stalk_iso S p).comp_inv_eq.mpr $
localization.local_ring_hom_unique _ _ _ _ $ λ x, by
rw [stalk_iso_hom, stalk_iso_inv, comp_apply, comp_apply, localization_to_stalk_of,
stalk_map_to_stalk_apply, stalk_to_fiber_ring_hom_to_stalk] | lemma | algebraic_geometry.local_ring_hom_comp_stalk_iso | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"CommRing.of",
"localization.at_prime",
"localization.local_ring_hom",
"localization.local_ring_hom_unique",
"prime_spectrum",
"prime_spectrum.comap"
] | Under the isomorphisms `stalk_iso`, the map `stalk_map (Spec.SheafedSpace_map f) p` corresponds
to the induced local ring homomorphism `localization.local_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.LocallyRingedSpace_map {R S : CommRing} (f : R ⟶ S) :
Spec.LocallyRingedSpace_obj S ⟶ Spec.LocallyRingedSpace_obj R | LocallyRingedSpace.hom.mk (Spec.SheafedSpace_map f) $ λ p, is_local_ring_hom.mk $ λ a ha,
begin
-- Here, we are showing that the map on prime spectra induced by `f` is really a morphism of
-- *locally* ringed spaces, i.e. that the induced map on the stalks is a local ring homomorphism.
rw ← local_ring_hom_comp_st... | def | algebraic_geometry.Spec.LocallyRingedSpace_map | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"prime_spectrum.comap",
"ring_hom.is_unit_map"
] | The induced map of a ring homomorphism on the prime spectra, as a morphism of locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.LocallyRingedSpace_map_id (R : CommRing) :
Spec.LocallyRingedSpace_map (𝟙 R) = 𝟙 (Spec.LocallyRingedSpace_obj R) | LocallyRingedSpace.hom.ext _ _ $
by { rw [Spec.LocallyRingedSpace_map_val, Spec.SheafedSpace_map_id], refl } | lemma | algebraic_geometry.Spec.LocallyRingedSpace_map_id | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.LocallyRingedSpace_map_comp {R S T : CommRing} (f : R ⟶ S) (g : S ⟶ T) :
Spec.LocallyRingedSpace_map (f ≫ g) =
Spec.LocallyRingedSpace_map g ≫ Spec.LocallyRingedSpace_map f | LocallyRingedSpace.hom.ext _ _ $
by { rw [Spec.LocallyRingedSpace_map_val, Spec.SheafedSpace_map_comp], refl } | lemma | algebraic_geometry.Spec.LocallyRingedSpace_map_comp | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec.to_LocallyRingedSpace : CommRingᵒᵖ ⥤ LocallyRingedSpace | { obj := λ R, Spec.LocallyRingedSpace_obj (unop R),
map := λ R S f, Spec.LocallyRingedSpace_map f.unop,
map_id' := λ R, by rw [unop_id, Spec.LocallyRingedSpace_map_id],
map_comp' := λ R S T f g, by rw [unop_comp, Spec.LocallyRingedSpace_map_comp] } | def | algebraic_geometry.Spec.to_LocallyRingedSpace | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [] | Spec, as a contravariant functor from commutative rings to locally ringed spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Spec_Γ (R : CommRing) : R ⟶ Γ.obj (op (Spec.to_LocallyRingedSpace.obj (op R))) | structure_sheaf.to_open R ⊤ | def | algebraic_geometry.to_Spec_Γ | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | The counit morphism `R ⟶ Γ(Spec R)` given by `algebraic_geometry.structure_sheaf.to_open`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_to_Spec_Γ (R : CommRing) : is_iso (to_Spec_Γ R) | by { cases R, apply structure_sheaf.is_iso_to_global } | instance | algebraic_geometry.is_iso_to_Spec_Γ | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_Γ_naturality {R S : CommRing} (f : R ⟶ S) :
f ≫ to_Spec_Γ S = to_Spec_Γ R ≫ Γ.map (Spec.to_LocallyRingedSpace.map f.op).op | by { ext, symmetry, apply localization.local_ring_hom_to_map } | lemma | algebraic_geometry.Spec_Γ_naturality | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"localization.local_ring_hom_to_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Spec_Γ_identity : Spec.to_LocallyRingedSpace.right_op ⋙ Γ ≅ 𝟭 _ | iso.symm $ nat_iso.of_components (λ R, as_iso (to_Spec_Γ R) : _) (λ _ _, Spec_Γ_naturality) | def | algebraic_geometry.Spec_Γ_identity | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [] | The counit (`Spec_Γ_identity.inv.op`) of the adjunction `Γ ⊣ Spec` is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec_map_localization_is_iso (R : CommRing) (M : submonoid R)
(x : prime_spectrum (localization M)) :
is_iso (PresheafedSpace.stalk_map (Spec.to_PresheafedSpace.map
(CommRing.of_hom (algebra_map R (localization M))).op) x) | begin
erw ← local_ring_hom_comp_stalk_iso,
apply_with is_iso.comp_is_iso { instances := ff },
apply_instance,
apply_with is_iso.comp_is_iso { instances := ff },
/- I do not know why this is defeq to the goal, but I'm happy to accept that it is. -/
exact (show is_iso (is_localization.localization_localizatio... | lemma | algebraic_geometry.Spec_map_localization_is_iso | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"CommRing",
"CommRing.of_hom",
"algebra_map",
"is_localization.localization_localization_at_prime_iso_localization",
"localization",
"prime_spectrum",
"submonoid"
] | The stalk map of `Spec M⁻¹R ⟶ Spec R` is an iso for each `p : Spec M⁻¹R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pushforward_stalk :
S ⟶ (Spec.Top_map f _* (structure_sheaf S).1).stalk p | structure_sheaf.to_open S ⊤ ≫
@Top.presheaf.germ _ _ _ _ (Spec.Top_map f _* (structure_sheaf S).1) ⊤ ⟨p, trivial⟩ | def | algebraic_geometry.structure_sheaf.to_pushforward_stalk | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"Top.presheaf.germ"
] | For an algebra `f : R →+* S`, this is the ring homomorphism `S →+* (f∗ 𝒪ₛ)ₚ` for a `p : Spec R`.
This is shown to be the localization at `p` in `is_localized_module_to_pushforward_stalk_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pushforward_stalk_comp :
f ≫ structure_sheaf.to_pushforward_stalk f p =
structure_sheaf.to_stalk R p ≫
(Top.presheaf.stalk_functor _ _).map (Spec.SheafedSpace_map f).c | begin
rw structure_sheaf.to_stalk,
erw category.assoc,
rw Top.presheaf.stalk_functor_map_germ,
exact Spec_Γ_naturality_assoc f _,
end | lemma | algebraic_geometry.structure_sheaf.to_pushforward_stalk_comp | algebraic_geometry | src/algebraic_geometry/Spec.lean | [
"algebraic_geometry.locally_ringed_space",
"algebraic_geometry.structure_sheaf",
"ring_theory.localization.localization_localization",
"topology.sheaves.sheaf_condition.sites",
"topology.sheaves.functors",
"algebra.module.localized_module"
] | [
"Top.presheaf.stalk_functor",
"Top.presheaf.stalk_functor_map_germ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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