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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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algebra_map_pushforward_stalk :
algebra_map R ((Spec.Top_map f _* (structure_sheaf S).1).stalk p) =
f ≫ structure_sheaf.to_pushforward_stalk f p | rfl | lemma | algebraic_geometry.structure_sheaf.algebra_map_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"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pushforward_stalk_alg_hom :
S →ₐ[R] (Spec.Top_map (algebra_map R S) _* (structure_sheaf S).1).stalk p | { commutes' := λ _, rfl, ..(structure_sheaf.to_pushforward_stalk (algebra_map R S) p) } | def | algebraic_geometry.structure_sheaf.to_pushforward_stalk_alg_hom | 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"
] | [
"algebra_map"
] | This is the `alg_hom` version of `to_pushforward_stalk`, which is the map `S ⟶ (f∗ 𝒪ₛ)ₚ` for some
algebra `R ⟶ S` and some `p : Spec R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localized_module_to_pushforward_stalk_alg_hom_aux (y) :
∃ (x : S × p.as_ideal.prime_compl), x.2 • y = to_pushforward_stalk_alg_hom R S p x.1 | begin
obtain ⟨U, hp, s, e⟩ := Top.presheaf.germ_exist _ _ y,
obtain ⟨_, ⟨r, rfl⟩, hpr : p ∈ prime_spectrum.basic_open r,
hrU : prime_spectrum.basic_open r ≤ U⟩ := prime_spectrum.is_topological_basis_basic_opens
.exists_subset_of_mem_open (show p ∈ ↑U, from hp) U.2,
change prime_spectrum.basic_open r ≤ U... | lemma | algebraic_geometry.structure_sheaf.is_localized_module_to_pushforward_stalk_alg_hom_aux | 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_exist",
"algebra.smul_def",
"algebra_map",
"le_top",
"map_mul",
"map_pow",
"mul_comm",
"prime_spectrum.basic_open",
"prime_spectrum.is_topological_basis_basic_opens",
"submonoid.smul_def",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localized_module_to_pushforward_stalk_alg_hom :
is_localized_module p.as_ideal.prime_compl (to_pushforward_stalk_alg_hom R S p).to_linear_map | begin
apply is_localized_module.mk_of_algebra,
{ intros x hx, rw [algebra_map_pushforward_stalk, to_pushforward_stalk_comp, comp_apply],
exact (is_localization.map_units ((structure_sheaf R).presheaf.stalk p) ⟨x, hx⟩).map _ },
{ apply is_localized_module_to_pushforward_stalk_alg_hom_aux },
{ intros x hx,
... | instance | algebraic_geometry.structure_sheaf.is_localized_module_to_pushforward_stalk_alg_hom | 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_eq",
"Top.presheaf.pushforward_obj_map",
"algebra.smul_def",
"algebra_map",
"is_localization.mk'_eq_zero_iff",
"is_localization.mk'_one",
"is_localized_module",
"map_pow",
"prime_spectrum.basic_open",
"prime_spectrum.is_topological_basis_basic_opens",
"ring_hom.to_fun_eq_coe",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk (X : PresheafedSpace C) (x : X) : C | X.presheaf.stalk x | abbreviation | algebraic_geometry.PresheafedSpace.stalk | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | The stalk at `x` of a `PresheafedSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_map {X Y : PresheafedSpace.{v} C} (α : X ⟶ Y) (x : X) : Y.stalk (α.base x) ⟶ X.stalk x | (stalk_functor C (α.base x)).map (α.c) ≫ X.presheaf.stalk_pushforward C α.base x | def | algebraic_geometry.PresheafedSpace.stalk_map | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | A morphism of presheafed spaces induces a morphism of stalks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_map_germ {X Y : PresheafedSpace.{v} C} (α : X ⟶ Y) (U : opens Y.carrier)
(x : (opens.map α.base).obj U) :
Y.presheaf.germ ⟨α.base x, x.2⟩ ≫ stalk_map α ↑x = α.c.app (op U) ≫ X.presheaf.germ x | by rw [stalk_map, stalk_functor_map_germ_assoc, stalk_pushforward_germ] | lemma | algebraic_geometry.PresheafedSpace.stalk_map_germ | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_stalk_iso {U : Top} (X : PresheafedSpace.{v} C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) (x : U) :
(X.restrict h).stalk x ≅ X.stalk (f x) | begin
-- As a left adjoint, the functor `h.is_open_map.functor_nhds x` is initial.
haveI := initial_of_adjunction (h.is_open_map.adjunction_nhds x),
-- Typeclass resolution knows that the opposite of an initial functor is final. The result
-- follows from the general fact that postcomposing with a final functor... | def | algebraic_geometry.PresheafedSpace.restrict_stalk_iso | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top",
"open_embedding"
] | For an open embedding `f : U ⟶ X` and a point `x : U`, we get an isomorphism between the stalk
of `X` at `f x` and the stalk of the restriction of `X` along `f` at t `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_stalk_iso_hom_eq_germ {U : Top} (X : PresheafedSpace.{v} C) {f : U ⟶ (X : Top.{v})}
(h : open_embedding f) (V : opens U) (x : U) (hx : x ∈ V) :
(X.restrict h).presheaf.germ ⟨x, hx⟩ ≫ (restrict_stalk_iso X h x).hom =
X.presheaf.germ ⟨f x, show f x ∈ h.is_open_map.functor.obj V, from ⟨x, hx, rfl⟩⟩ | colimit.ι_pre ((open_nhds.inclusion (f x)).op ⋙ X.presheaf)
(h.is_open_map.functor_nhds x).op (op ⟨V, hx⟩) | lemma | algebraic_geometry.PresheafedSpace.restrict_stalk_iso_hom_eq_germ | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_stalk_iso_inv_eq_germ {U : Top} (X : PresheafedSpace.{v} C) {f : U ⟶ (X : Top.{v})}
(h : open_embedding f) (V : opens U) (x : U) (hx : x ∈ V) :
X.presheaf.germ ⟨f x, show f x ∈ h.is_open_map.functor.obj V, from ⟨x, hx, rfl⟩⟩ ≫
(restrict_stalk_iso X h x).inv = (X.restrict h).presheaf.germ ⟨x, hx⟩ | by rw [← restrict_stalk_iso_hom_eq_germ, category.assoc, iso.hom_inv_id, category.comp_id] | lemma | algebraic_geometry.PresheafedSpace.restrict_stalk_iso_inv_eq_germ | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_stalk_iso_inv_eq_of_restrict {U : Top} (X : PresheafedSpace.{v} C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) (x : U) :
(X.restrict_stalk_iso h x).inv = stalk_map (X.of_restrict h) x | begin
ext V,
induction V using opposite.rec,
let i : (h.is_open_map.functor_nhds x).obj ((open_nhds.map f x).obj V) ⟶ V :=
hom_of_le (set.image_preimage_subset f _),
erw [iso.comp_inv_eq, colimit.ι_map_assoc, colimit.ι_map_assoc, colimit.ι_pre],
simp_rw category.assoc,
erw colimit.ι_pre ((open_nhds.incl... | lemma | algebraic_geometry.PresheafedSpace.restrict_stalk_iso_inv_eq_of_restrict | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top",
"open_embedding",
"opposite.rec",
"set.image_preimage_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_stalk_map_is_iso {U : Top} (X : PresheafedSpace.{v} C)
{f : U ⟶ (X : Top.{v})} (h : open_embedding f) (x : U) :
is_iso (stalk_map (X.of_restrict h) x) | by { rw ← restrict_stalk_iso_inv_eq_of_restrict, apply_instance } | instance | algebraic_geometry.PresheafedSpace.of_restrict_stalk_map_is_iso | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (X : PresheafedSpace.{v} C) (x : X) : stalk_map (𝟙 X) x = 𝟙 (X.stalk x) | begin
dsimp [stalk_map],
simp only [stalk_pushforward.id],
rw [←map_comp],
convert (stalk_functor C x).map_id X.presheaf,
tidy,
end | lemma | algebraic_geometry.PresheafedSpace.stalk_map.id | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {X Y Z : PresheafedSpace.{v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
stalk_map (α ≫ β) x =
(stalk_map β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
(stalk_map α x : Y.stalk (α.base x) ⟶ X.stalk x) | begin
dsimp [stalk_map, stalk_functor, stalk_pushforward],
ext U,
induction U using opposite.rec,
cases U,
simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc, colimit.ι_pre,
whisker_left_app, whisker_right_app,
assoc, id_comp, map_id, map_comp],
dsimp,
simp only [map_id, assoc, pushforward.comp_i... | lemma | algebraic_geometry.PresheafedSpace.stalk_map.comp | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"map_comp",
"map_id",
"opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr {X Y : PresheafedSpace.{v} C} (α β : X ⟶ Y) (h₁ : α = β) (x x': X) (h₂ : x = x') :
stalk_map α x ≫ eq_to_hom (show X.stalk x = X.stalk x', by rw h₂) =
eq_to_hom (show Y.stalk (α.base x) = Y.stalk (β.base x'), by rw [h₁, h₂]) ≫ stalk_map β x' | stalk_hom_ext _ $ λ U hx, by { subst h₁, subst h₂, simp } | lemma | algebraic_geometry.PresheafedSpace.stalk_map.congr | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_hom {X Y : PresheafedSpace.{v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
stalk_map α x =
eq_to_hom (show Y.stalk (α.base x) = Y.stalk (β.base x), by rw h) ≫ stalk_map β x | by rw [← stalk_map.congr α β h x x rfl, eq_to_hom_refl, category.comp_id] | lemma | algebraic_geometry.PresheafedSpace.stalk_map.congr_hom | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_point {X Y : PresheafedSpace.{v} C} (α : X ⟶ Y) (x x' : X) (h : x = x') :
stalk_map α x ≫ eq_to_hom (show X.stalk x = X.stalk x', by rw h) =
eq_to_hom (show Y.stalk (α.base x) = Y.stalk (α.base x'), by rw h) ≫ stalk_map α x' | by rw stalk_map.congr α α rfl x x' h | lemma | algebraic_geometry.PresheafedSpace.stalk_map.congr_point | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso {X Y : PresheafedSpace.{v} C} (α : X ⟶ Y) [is_iso α] (x : X) :
is_iso (stalk_map α x) | { out := begin
let β : Y ⟶ X := category_theory.inv α,
have h_eq : (α ≫ β).base x = x,
{ rw [is_iso.hom_inv_id α, id_base, Top.id_app] },
-- Intuitively, the inverse of the stalk map of `α` at `x` should just be the stalk map of `β`
-- at `α x`. Unfortunately, we have a problem with dependent type theory here... | instance | algebraic_geometry.PresheafedSpace.stalk_map.is_iso | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [
"Top.id_app",
"category_theory.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_iso {X Y : PresheafedSpace.{v} C} (α : X ≅ Y) (x : X) :
Y.stalk (α.hom.base x) ≅ X.stalk x | as_iso (stalk_map α.hom x) | def | algebraic_geometry.PresheafedSpace.stalk_map.stalk_iso | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | An isomorphism between presheafed spaces induces an isomorphism of stalks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_specializes_stalk_map {X Y : PresheafedSpace.{v} C} (f : X ⟶ Y) {x y : X} (h : x ⤳ y) :
Y.presheaf.stalk_specializes (f.base.map_specializes h) ≫ stalk_map f x =
stalk_map f y ≫ X.presheaf.stalk_specializes h | by { delta PresheafedSpace.stalk_map, simp [stalk_map] } | lemma | algebraic_geometry.PresheafedSpace.stalk_map.stalk_specializes_stalk_map | algebraic_geometry | src/algebraic_geometry/stalks.lean | [
"algebraic_geometry.presheafed_space",
"category_theory.limits.final",
"topology.sheaves.stalks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_spectrum.Top : Top | Top.of (prime_spectrum R) | def | algebraic_geometry.prime_spectrum.Top | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"Top",
"Top.of",
"prime_spectrum"
] | The prime spectrum, just as a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localizations (P : prime_spectrum.Top R) : Type u | localization.at_prime P.as_ideal | def | algebraic_geometry.structure_sheaf.localizations | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.at_prime"
] | The type family over `prime_spectrum R` consisting of the localization over each point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction {U : opens (prime_spectrum.Top R)} (f : Π x : U, localizations R x) : Prop | ∃ (r s : R), ∀ x : U,
¬ (s ∈ x.1.as_ideal) ∧ f x * algebra_map _ _ s = algebra_map _ _ r | def | algebraic_geometry.structure_sheaf.is_fraction | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map"
] | The predicate saying that a dependent function on an open `U` is realised as a fixed fraction
`r / s` in each of the stalks (which are localizations at various prime ideals). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction.eq_mk' {U : opens (prime_spectrum.Top R)} {f : Π x : U, localizations R x}
(hf : is_fraction f) :
∃ (r s : R) , ∀ x : U, ∃ (hs : s ∉ x.1.as_ideal), f x =
is_localization.mk' (localization.at_prime _) r
(⟨s, hs⟩ : (x : prime_spectrum.Top R).as_ideal.prime_compl) | begin
rcases hf with ⟨r, s, h⟩,
refine ⟨r, s, λ x, ⟨(h x).1, (is_localization.mk'_eq_iff_eq_mul.mpr _).symm⟩⟩,
exact (h x).2.symm,
end | lemma | algebraic_geometry.structure_sheaf.is_fraction.eq_mk' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'",
"localization.at_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_prelocal : prelocal_predicate (localizations R) | { pred := λ U f, is_fraction f,
res := by { rintro V U i f ⟨r, s, w⟩, exact ⟨r, s, λ x, w (i x)⟩ } } | def | algebraic_geometry.structure_sheaf.is_fraction_prelocal | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | The predicate `is_fraction` is "prelocal",
in the sense that if it holds on `U` it holds on any open subset `V` of `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_fraction : local_predicate (localizations R) | (is_fraction_prelocal R).sheafify | def | algebraic_geometry.structure_sheaf.is_locally_fraction | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | We will define the structure sheaf as
the subsheaf of all dependent functions in `Π x : U, localizations R x`
consisting of those functions which can locally be expressed as a ratio of
(the images in the localization of) elements of `R`.
Quoting Hartshorne:
For an open set $U ⊆ Spec A$, we define $𝒪(U)$ to be the se... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_locally_fraction_pred
{U : opens (prime_spectrum.Top R)} (f : Π x : U, localizations R x) :
(is_locally_fraction R).pred f =
∀ x : U, ∃ (V) (m : x.1 ∈ V) (i : V ⟶ U),
∃ (r s : R), ∀ y : V,
¬ (s ∈ y.1.as_ideal) ∧
f (i y : U) * algebra_map _ _ s = algebra_map _ _ r | rfl | lemma | algebraic_geometry.structure_sheaf.is_locally_fraction_pred | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sections_subring (U : (opens (prime_spectrum.Top R))ᵒᵖ) :
subring (Π x : unop U, localizations R x) | { carrier := { f | (is_locally_fraction R).pred f },
zero_mem' :=
begin
refine λ x, ⟨unop U, x.2, 𝟙 _, 0, 1, λ y, ⟨_, _⟩⟩,
{ rw ←ideal.ne_top_iff_one, exact y.1.is_prime.1, },
{ simp, },
end,
one_mem' :=
begin
refine λ x, ⟨unop U, x.2, 𝟙 _, 1, 1, λ y, ⟨_, _⟩⟩,
{ rw ←ideal.ne_top_iff_one,... | def | algebraic_geometry.structure_sheaf.sections_subring | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"mul_assoc",
"mul_comm",
"mul_left_comm",
"neg_mul",
"pi.mul_apply",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_neg",
"subring"
] | The functions satisfying `is_locally_fraction` form a subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structure_sheaf_in_Type : sheaf (Type u) (prime_spectrum.Top R) | subsheaf_to_Types (is_locally_fraction R) | def | algebraic_geometry.structure_sheaf_in_Type | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | The structure sheaf (valued in `Type`, not yet `CommRing`) is the subsheaf consisting of
functions satisfying `is_locally_fraction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring_structure_sheaf_in_Type_obj (U : (opens (prime_spectrum.Top R))ᵒᵖ) :
comm_ring ((structure_sheaf_in_Type R).1.obj U) | (sections_subring R U).to_comm_ring | instance | algebraic_geometry.comm_ring_structure_sheaf_in_Type_obj | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
structure_presheaf_in_CommRing : presheaf CommRing (prime_spectrum.Top R) | { obj := λ U, CommRing.of ((structure_sheaf_in_Type R).1.obj U),
map := λ U V i,
{ to_fun := ((structure_sheaf_in_Type R).1.map i),
map_zero' := rfl,
map_add' := λ x y, rfl,
map_one' := rfl,
map_mul' := λ x y, rfl, }, } | def | algebraic_geometry.structure_presheaf_in_CommRing | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing",
"CommRing.of"
] | The structure presheaf, valued in `CommRing`, constructed by dressing up the `Type` valued
structure presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
structure_presheaf_comp_forget :
structure_presheaf_in_CommRing R ⋙ (forget CommRing) ≅ (structure_sheaf_in_Type R).1 | nat_iso.of_components
(λ U, iso.refl _)
(by tidy) | def | algebraic_geometry.structure_presheaf_comp_forget | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing"
] | Some glue, verifying that that structure presheaf valued in `CommRing` agrees
with the `Type` valued structure presheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Spec.structure_sheaf : sheaf CommRing (prime_spectrum.Top R) | ⟨structure_presheaf_in_CommRing R,
-- We check the sheaf condition under `forget CommRing`.
(is_sheaf_iff_is_sheaf_comp _ _).mpr
(is_sheaf_of_iso (structure_presheaf_comp_forget R).symm
(structure_sheaf_in_Type R).cond)⟩ | def | algebraic_geometry.Spec.structure_sheaf | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing"
] | The structure sheaf on $Spec R$, valued in `CommRing`.
This is provided as a bundled `SheafedSpace` as `Spec.SheafedSpace R` later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
res_apply (U V : opens (prime_spectrum.Top R)) (i : V ⟶ U)
(s : (structure_sheaf R).1.obj (op U)) (x : V) :
((structure_sheaf R).1.map i.op s).1 x = (s.1 (i x) : _) | rfl | lemma | algebraic_geometry.structure_sheaf.res_apply | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (f g : R) (U : opens (prime_spectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : prime_spectrum.Top R).as_ideal.prime_compl) :
(structure_sheaf R).1.obj (op U) | ⟨λ x, is_localization.mk' _ f ⟨g, hu x x.2⟩,
λ x, ⟨U, x.2, 𝟙 _, f, g, λ y, ⟨hu y y.2, is_localization.mk'_spec _ _ _⟩⟩⟩ | def | algebraic_geometry.structure_sheaf.const | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'",
"is_localization.mk'_spec"
] | The section of `structure_sheaf R` on an open `U` sending each `x ∈ U` to the element
`f/g` in the localization of `R` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply (f g : R) (U : opens (prime_spectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : prime_spectrum.Top R).as_ideal.prime_compl) (x : U) :
(const R f g U hu).1 x = is_localization.mk' _ f ⟨g, hu x x.2⟩ | rfl | lemma | algebraic_geometry.structure_sheaf.const_apply | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_apply' (f g : R) (U : opens (prime_spectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : prime_spectrum.Top R).as_ideal.prime_compl) (x : U)
(hx : g ∈ (as_ideal (x : prime_spectrum.Top R)).prime_compl) :
(const R f g U hu).1 x = is_localization.mk' _ f ⟨g, hx⟩ | rfl | lemma | algebraic_geometry.structure_sheaf.const_apply' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_const (U) (s : (structure_sheaf R).1.obj (op U)) (x : prime_spectrum.Top R)
(hx : x ∈ U) :
∃ (V : opens (prime_spectrum.Top R)) (hxV : x ∈ V) (i : V ⟶ U) (f g : R) hg,
const R f g V hg = (structure_sheaf R).1.map i.op s | let ⟨V, hxV, iVU, f, g, hfg⟩ := s.2 ⟨x, hx⟩ in
⟨V, hxV, iVU, f, g, λ y hyV, (hfg ⟨y, hyV⟩).1, subtype.eq $ funext $ λ y,
is_localization.mk'_eq_iff_eq_mul.2 $ eq.symm $ (hfg y).2⟩ | lemma | algebraic_geometry.structure_sheaf.exists_const | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"exists_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
res_const (f g : R) (U hu V hv i) :
(structure_sheaf R).1.map i (const R f g U hu) = const R f g V hv | rfl | lemma | algebraic_geometry.structure_sheaf.res_const | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
res_const' (f g : R) (V hv) :
(structure_sheaf R).1.map (hom_of_le hv).op (const R f g (basic_open g) (λ _, id)) =
const R f g V hv | rfl | lemma | algebraic_geometry.structure_sheaf.res_const' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_zero (f : R) (U hu) : const R 0 f U hu = 0 | subtype.eq $ funext $ λ x, is_localization.mk'_eq_iff_eq_mul.2 $
by erw [ring_hom.map_zero, subtype.val_eq_coe, subring.coe_zero, pi.zero_apply, zero_mul] | lemma | algebraic_geometry.structure_sheaf.const_zero | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"ring_hom.map_zero",
"subring.coe_zero",
"subtype.val_eq_coe",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_self (f : R) (U hu) : const R f f U hu = 1 | subtype.eq $ funext $ λ x, is_localization.mk'_self _ _ | lemma | algebraic_geometry.structure_sheaf.const_self | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_one (U) : const R 1 1 U (λ p _, submonoid.one_mem _) = 1 | const_self R 1 U _ | lemma | algebraic_geometry.structure_sheaf.const_one | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"submonoid.one_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_add (f₁ f₂ g₁ g₂ : R) (U hu₁ hu₂) :
const R f₁ g₁ U hu₁ + const R f₂ g₂ U hu₂ =
const R (f₁ * g₂ + f₂ * g₁) (g₁ * g₂) U (λ x hx, submonoid.mul_mem _ (hu₁ x hx) (hu₂ x hx)) | subtype.eq $ funext $ λ x, eq.symm $
by convert is_localization.mk'_add f₁ f₂ ⟨g₁, hu₁ x x.2⟩ ⟨g₂, hu₂ x x.2⟩ | lemma | algebraic_geometry.structure_sheaf.const_add | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'_add",
"submonoid.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_mul (f₁ f₂ g₁ g₂ : R) (U hu₁ hu₂) :
const R f₁ g₁ U hu₁ * const R f₂ g₂ U hu₂ =
const R (f₁ * f₂) (g₁ * g₂) U (λ x hx, submonoid.mul_mem _ (hu₁ x hx) (hu₂ x hx)) | subtype.eq $ funext $ λ x, eq.symm $
by convert is_localization.mk'_mul _ f₁ f₂ ⟨g₁, hu₁ x x.2⟩ ⟨g₂, hu₂ x x.2⟩ | lemma | algebraic_geometry.structure_sheaf.const_mul | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'_mul",
"submonoid.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_ext {f₁ f₂ g₁ g₂ : R} {U hu₁ hu₂} (h : f₁ * g₂ = f₂ * g₁) :
const R f₁ g₁ U hu₁ = const R f₂ g₂ U hu₂ | subtype.eq $ funext $ λ x, is_localization.mk'_eq_of_eq
(by rw [mul_comm, subtype.coe_mk, ←h, mul_comm, subtype.coe_mk]) | lemma | algebraic_geometry.structure_sheaf.const_ext | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.mk'_eq_of_eq",
"mul_comm",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_congr {f₁ f₂ g₁ g₂ : R} {U hu} (hf : f₁ = f₂) (hg : g₁ = g₂) :
const R f₁ g₁ U hu = const R f₂ g₂ U (hg ▸ hu) | by substs hf hg | lemma | algebraic_geometry.structure_sheaf.const_congr | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_mul_rev (f g : R) (U hu₁ hu₂) :
const R f g U hu₁ * const R g f U hu₂ = 1 | by rw [const_mul, const_congr R rfl (mul_comm g f), const_self] | lemma | algebraic_geometry.structure_sheaf.const_mul_rev | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_mul_cancel (f g₁ g₂ : R) (U hu₁ hu₂) :
const R f g₁ U hu₁ * const R g₁ g₂ U hu₂ = const R f g₂ U hu₂ | by { rw [const_mul, const_ext], rw mul_assoc } | lemma | algebraic_geometry.structure_sheaf.const_mul_cancel | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_mul_cancel' (f g₁ g₂ : R) (U hu₁ hu₂) :
const R g₁ g₂ U hu₂ * const R f g₁ U hu₁ = const R f g₂ U hu₂ | by rw [mul_comm, const_mul_cancel] | lemma | algebraic_geometry.structure_sheaf.const_mul_cancel' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_open (U : opens (prime_spectrum.Top R)) :
CommRing.of R ⟶ (structure_sheaf R).1.obj (op U) | { to_fun := λ f, ⟨λ x, algebra_map R _ f,
λ x, ⟨U, x.2, 𝟙 _, f, 1, λ y, ⟨(ideal.ne_top_iff_one _).1 y.1.2.1,
by { rw [ring_hom.map_one, mul_one], refl } ⟩⟩⟩,
map_one' := subtype.eq $ funext $ λ x, ring_hom.map_one _,
map_mul' := λ f g, subtype.eq $ funext $ λ x, ring_hom.map_mul _ _ _,
map_zero' := sub... | def | algebraic_geometry.structure_sheaf.to_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"algebra_map",
"ideal.ne_top_iff_one",
"mul_one",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_zero"
] | The canonical ring homomorphism interpreting an element of `R` as
a section of the structure sheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_open_res (U V : opens (prime_spectrum.Top R)) (i : V ⟶ U) :
to_open R U ≫ (structure_sheaf R).1.map i.op = to_open R V | rfl | lemma | algebraic_geometry.structure_sheaf.to_open_res | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_open_apply (U : opens (prime_spectrum.Top R)) (f : R) (x : U) :
(to_open R U f).1 x = algebra_map _ _ f | rfl | lemma | algebraic_geometry.structure_sheaf.to_open_apply | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_open_eq_const (U : opens (prime_spectrum.Top R)) (f : R) : to_open R U f =
const R f 1 U (λ x _, (ideal.ne_top_iff_one _).1 x.2.1) | subtype.eq $ funext $ λ x, eq.symm $ is_localization.mk'_one _ f | lemma | algebraic_geometry.structure_sheaf.to_open_eq_const | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"ideal.ne_top_iff_one",
"is_localization.mk'_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_stalk (x : prime_spectrum.Top R) : CommRing.of R ⟶ (structure_sheaf R).presheaf.stalk x | (to_open R ⊤ ≫ (structure_sheaf R).presheaf.germ ⟨x, ⟨⟩⟩ : _) | def | algebraic_geometry.structure_sheaf.to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of"
] | The canonical ring homomorphism interpreting an element of `R` as an element of
the stalk of `structure_sheaf R` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_open_germ (U : opens (prime_spectrum.Top R)) (x : U) :
to_open R U ≫ (structure_sheaf R).presheaf.germ x =
to_stalk R x | by { rw [← to_open_res R ⊤ U (hom_of_le le_top : U ⟶ ⊤), category.assoc, presheaf.germ_res], refl } | lemma | algebraic_geometry.structure_sheaf.to_open_germ | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
germ_to_open (U : opens (prime_spectrum.Top R)) (x : U) (f : R) :
(structure_sheaf R).presheaf.germ x (to_open R U f) = to_stalk R x f | by { rw ← to_open_germ, refl } | lemma | algebraic_geometry.structure_sheaf.germ_to_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
germ_to_top (x : prime_spectrum.Top R) (f : R) :
(structure_sheaf R).presheaf.germ (⟨x, trivial⟩ : (⊤ : opens (prime_spectrum.Top R)))
(to_open R ⊤ f) =
to_stalk R x f | rfl | lemma | algebraic_geometry.structure_sheaf.germ_to_top | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_to_basic_open_self (f : R) : is_unit (to_open R (basic_open f) f) | is_unit_of_mul_eq_one _ (const R 1 f (basic_open f) (λ _, id)) $
by rw [to_open_eq_const, const_mul_rev] | lemma | algebraic_geometry.structure_sheaf.is_unit_to_basic_open_self | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_unit",
"is_unit_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_to_stalk (x : prime_spectrum.Top R) (f : x.as_ideal.prime_compl) :
is_unit (to_stalk R x (f : R)) | by { erw ← germ_to_open R (basic_open (f : R)) ⟨x, f.2⟩ (f : R),
exact ring_hom.is_unit_map _ (is_unit_to_basic_open_self R f) } | lemma | algebraic_geometry.structure_sheaf.is_unit_to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_unit",
"ring_hom.is_unit_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_to_stalk (x : prime_spectrum.Top R) :
CommRing.of (localization.at_prime x.as_ideal) ⟶ (structure_sheaf R).presheaf.stalk x | show localization.at_prime x.as_ideal →+* _, from
is_localization.lift (is_unit_to_stalk R x) | def | algebraic_geometry.structure_sheaf.localization_to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"is_localization.lift",
"localization.at_prime"
] | The canonical ring homomorphism from the localization of `R` at `p` to the stalk
of the structure sheaf at the point `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_to_stalk_of (x : prime_spectrum.Top R) (f : R) :
localization_to_stalk R x (algebra_map _ (localization _) f) = to_stalk R x f | is_localization.lift_eq _ f | lemma | algebraic_geometry.structure_sheaf.localization_to_stalk_of | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"is_localization.lift_eq",
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_to_stalk_mk' (x : prime_spectrum.Top R) (f : R)
(s : (as_ideal x).prime_compl) :
localization_to_stalk R x (is_localization.mk' _ f s : localization _) =
(structure_sheaf R).presheaf.germ (⟨x, s.2⟩ : basic_open (s : R))
(const R f s (basic_open s) (λ _, id)) | (is_localization.lift_mk'_spec _ _ _ _).2 $
by erw [← germ_to_open R (basic_open s) ⟨x, s.2⟩, ← germ_to_open R (basic_open s) ⟨x, s.2⟩,
← ring_hom.map_mul, to_open_eq_const, to_open_eq_const, const_mul_cancel'] | lemma | algebraic_geometry.structure_sheaf.localization_to_stalk_mk' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.lift_mk'_spec",
"is_localization.mk'",
"localization",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_to_localization (U : opens (prime_spectrum.Top R)) (x : prime_spectrum.Top R)
(hx : x ∈ U) :
(structure_sheaf R).1.obj (op U) ⟶ CommRing.of (localization.at_prime x.as_ideal) | { to_fun := λ s, (s.1 ⟨x, hx⟩ : _),
map_one' := rfl,
map_mul' := λ _ _, rfl,
map_zero' := rfl,
map_add' := λ _ _, rfl } | def | algebraic_geometry.structure_sheaf.open_to_localization | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"localization.at_prime"
] | The ring homomorphism that takes a section of the structure sheaf of `R` on the open set `U`,
implemented as a subtype of dependent functions to localizations at prime ideals, and evaluates
the section on the point corresponding to a given prime ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_open_to_localization (U : opens (prime_spectrum.Top R)) (x : prime_spectrum.Top R)
(hx : x ∈ U) :
(open_to_localization R U x hx :
(structure_sheaf R).1.obj (op U) → localization.at_prime x.as_ideal) =
(λ s, (s.1 ⟨x, hx⟩ : _)) | rfl | lemma | algebraic_geometry.structure_sheaf.coe_open_to_localization | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.at_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_to_localization_apply (U : opens (prime_spectrum.Top R)) (x : prime_spectrum.Top R)
(hx : x ∈ U)
(s : (structure_sheaf R).1.obj (op U)) :
open_to_localization R U x hx s = (s.1 ⟨x, hx⟩ : _) | rfl | lemma | algebraic_geometry.structure_sheaf.open_to_localization_apply | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom (x : prime_spectrum.Top R) :
(structure_sheaf R).presheaf.stalk x ⟶ CommRing.of (localization.at_prime x.as_ideal) | limits.colimit.desc (((open_nhds.inclusion x).op) ⋙ (structure_sheaf R).1)
{ X := _,
ι :=
{ app := λ U, open_to_localization R ((open_nhds.inclusion _).obj (unop U)) x (unop U).2, } } | def | algebraic_geometry.structure_sheaf.stalk_to_fiber_ring_hom | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"localization.at_prime"
] | The ring homomorphism from the stalk of the structure sheaf of `R` at a point corresponding to
a prime ideal `p` to the localization of `R` at `p`,
formed by gluing the `open_to_localization` maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
germ_comp_stalk_to_fiber_ring_hom (U : opens (prime_spectrum.Top R)) (x : U) :
(structure_sheaf R).presheaf.germ x ≫ stalk_to_fiber_ring_hom R x =
open_to_localization R U x x.2 | limits.colimit.ι_desc _ _ | lemma | algebraic_geometry.structure_sheaf.germ_comp_stalk_to_fiber_ring_hom | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom_germ' (U : opens (prime_spectrum.Top R))
(x : prime_spectrum.Top R) (hx : x ∈ U) (s : (structure_sheaf R).1.obj (op U)) :
stalk_to_fiber_ring_hom R x ((structure_sheaf R).presheaf.germ ⟨x, hx⟩ s) = (s.1 ⟨x, hx⟩ : _) | ring_hom.ext_iff.1 (germ_comp_stalk_to_fiber_ring_hom R U ⟨x, hx⟩ : _) s | lemma | algebraic_geometry.structure_sheaf.stalk_to_fiber_ring_hom_germ' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom_germ (U : opens (prime_spectrum.Top R)) (x : U)
(s : (structure_sheaf R).1.obj (op U)) :
stalk_to_fiber_ring_hom R x ((structure_sheaf R).presheaf.germ x s) = s.1 x | by { cases x, exact stalk_to_fiber_ring_hom_germ' R U _ _ _ } | lemma | algebraic_geometry.structure_sheaf.stalk_to_fiber_ring_hom_germ | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_stalk_comp_stalk_to_fiber_ring_hom (x : prime_spectrum.Top R) :
to_stalk R x ≫ stalk_to_fiber_ring_hom R x = (algebra_map _ _ : R →+* localization _) | by { erw [to_stalk, category.assoc, germ_comp_stalk_to_fiber_ring_hom], refl } | lemma | algebraic_geometry.structure_sheaf.to_stalk_comp_stalk_to_fiber_ring_hom | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_to_fiber_ring_hom_to_stalk (x : prime_spectrum.Top R) (f : R) :
stalk_to_fiber_ring_hom R x (to_stalk R x f) = algebra_map _ (localization _) f | ring_hom.ext_iff.1 (to_stalk_comp_stalk_to_fiber_ring_hom R x) _ | lemma | algebraic_geometry.structure_sheaf.stalk_to_fiber_ring_hom_to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_iso (x : prime_spectrum.Top R) :
(structure_sheaf R).presheaf.stalk x ≅ CommRing.of (localization.at_prime x.as_ideal) | { hom := stalk_to_fiber_ring_hom R x,
inv := localization_to_stalk R x,
hom_inv_id' := (structure_sheaf R).presheaf.stalk_hom_ext $ λ U hxU,
begin
ext s, simp only [comp_apply], rw [id_apply, stalk_to_fiber_ring_hom_germ'],
obtain ⟨V, hxV, iVU, f, g, hg, hs⟩ := exists_const _ _ s x hxU,
erw [← res_app... | def | algebraic_geometry.structure_sheaf.stalk_iso | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"exists_const",
"is_localization.ring_hom_ext",
"localization.at_prime",
"ring_hom.comp",
"ring_hom.comp_apply",
"ring_hom.id",
"ring_hom.id_apply"
] | The ring isomorphism between the stalk of the structure sheaf of `R` at a point `p`
corresponding to a prime ideal in `R` and the localization of `R` at `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_to_fiber_ring_hom_localization_to_stalk (x : prime_spectrum.Top R) :
stalk_to_fiber_ring_hom R x ≫ localization_to_stalk R x = 𝟙 _ | (stalk_iso R x).hom_inv_id | lemma | algebraic_geometry.structure_sheaf.stalk_to_fiber_ring_hom_localization_to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_to_stalk_stalk_to_fiber_ring_hom (x : prime_spectrum.Top R) :
localization_to_stalk R x ≫ stalk_to_fiber_ring_hom R x = 𝟙 _ | (stalk_iso R x).inv_hom_id | lemma | algebraic_geometry.structure_sheaf.localization_to_stalk_stalk_to_fiber_ring_hom | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basic_open (f : R) : localization.away f →+*
(structure_sheaf R).1.obj (op $ basic_open f) | is_localization.away.lift f (is_unit_to_basic_open_self R f) | def | algebraic_geometry.structure_sheaf.to_basic_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.away.lift",
"localization.away"
] | The canonical ring homomorphism interpreting `s ∈ R_f` as a section of the structure sheaf
on the basic open defined by `f ∈ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_basic_open_mk' (s f : R) (g : submonoid.powers s) :
to_basic_open R s (is_localization.mk' (localization.away s) f g) =
const R f g (basic_open s) (λ x hx, submonoid.powers_subset hx g.2) | (is_localization.lift_mk'_spec _ _ _ _).2 $
by rw [to_open_eq_const, to_open_eq_const, const_mul_cancel'] | lemma | algebraic_geometry.structure_sheaf.to_basic_open_mk' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.lift_mk'_spec",
"is_localization.mk'",
"localization.away",
"submonoid.powers",
"submonoid.powers_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_to_basic_open (f : R) :
ring_hom.comp (to_basic_open R f) (algebra_map R (localization.away f)) =
to_open R (basic_open f) | ring_hom.ext $ λ g,
by rw [to_basic_open, is_localization.away.lift, ring_hom.comp_apply, is_localization.lift_eq] | lemma | algebraic_geometry.structure_sheaf.localization_to_basic_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"is_localization.away.lift",
"is_localization.lift_eq",
"localization.away",
"ring_hom.comp",
"ring_hom.comp_apply",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basic_open_to_map (s f : R) :
to_basic_open R s (algebra_map R (localization.away s) f) =
const R f 1 (basic_open s) (λ _ _, submonoid.one_mem _) | (is_localization.lift_eq _ _).trans $ to_open_eq_const _ _ _ | lemma | algebraic_geometry.structure_sheaf.to_basic_open_to_map | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"is_localization.lift_eq",
"localization.away",
"submonoid.one_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basic_open_injective (f : R) : function.injective (to_basic_open R f) | begin
intros s t h_eq,
obtain ⟨a, ⟨b, hb⟩, rfl⟩ := is_localization.mk'_surjective (submonoid.powers f) s,
obtain ⟨c, ⟨d, hd⟩, rfl⟩ := is_localization.mk'_surjective (submonoid.powers f) t,
simp only [to_basic_open_mk'] at h_eq,
rw is_localization.eq,
-- We know that the fractions `a/b` and `c/d` are equal a... | lemma | algebraic_geometry.structure_sheaf.to_basic_open_injective | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"ideal",
"is_localization.eq",
"is_localization.mk'_surjective",
"mul_assoc",
"set.not_subset",
"submonoid.powers",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_const_basic_open (U : opens (prime_spectrum.Top R))
(s : (structure_sheaf R).1.obj (op U)) (x : U) :
∃ (f g : R) (i : basic_open g ⟶ U), x.1 ∈ basic_open g ∧
const R f g (basic_open g) (λ y hy, hy) = (structure_sheaf R).1.map i.op s | begin
-- First, any section `s` can be represented as a fraction `f/g` on some open neighborhood of `x`
-- and we may pass to a `basic_open h`, since these form a basis
obtain ⟨V, (hxV : x.1 ∈ V.1), iVU, f, g, (hVDg : V ≤ basic_open g), s_eq⟩ :=
exists_const R U s x.1 x.2,
obtain ⟨_, ⟨h, rfl⟩, hxDh, (hDhV :... | lemma | algebraic_geometry.structure_sheaf.locally_const_basic_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"exists_const",
"ideal.mem_span_singleton'",
"ideal.mul_mem_left",
"ideal.span",
"mul_assoc",
"pow_succ",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_finite_fraction_representation (U : opens (prime_spectrum.Top R))
(s : (structure_sheaf R).1.obj (op U)) {ι : Type*} (t : finset ι) (a h : ι → R)
(iDh : Π i : ι, basic_open (h i) ⟶ U) (h_cover : U ≤ ⨆ i ∈ t, basic_open (h i))
(hs : ∀ i : ι, const R (a i) (h i) (basic_open (h i)) (λ y hy, hy) =
(stru... | begin
-- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
-- coincide in the localization of `R` at `h i * h j`
have fractions_eq : ∀ (i j : ι),
is_localization.mk' (localization.away _) (a i * h j) ⟨h i * h j, submonoid.mem_powers _⟩ =
is_localization.mk' _ (h ... | lemma | algebraic_geometry.structure_sheaf.normalize_finite_fraction_representation | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"finset",
"finset.le_sup",
"is_localization.mk'",
"localization.away",
"mul_pow",
"pow_add",
"pow_one",
"pow_succ",
"ring",
"set_like.coe_mk",
"submonoid.mem_powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_basic_open_surjective (f : R) : function.surjective (to_basic_open R f) | begin
intro s,
-- In this proof, `basic_open f` will play two distinct roles: Firstly, it is an open set in the
-- prime spectrum. Secondly, it is used as an indexing type for various families of objects
-- (open sets, ring elements, ...). In order to make the distinction clear, we introduce a type
-- alias `... | lemma | algebraic_geometry.structure_sheaf.to_basic_open_surjective | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"compl_compl",
"finset.mul_sum",
"finset.set_bUnion_coe",
"finset.sum_mul",
"finsupp.mem_span_image_iff_total",
"finsupp.total_apply_of_mem_supported",
"ideal.mul_mem_left",
"ideal.span",
"is_localization.mk'",
"localization.away",
"mul_assoc",
"ring",
"set.compl_Union",
"set.compl_subset_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_to_basic_open (f : R) : is_iso (show CommRing.of _ ⟶ _, from to_basic_open R f) | begin
haveI : is_iso ((forget CommRing).map (show CommRing.of _ ⟶ _, from to_basic_open R f)) :=
(is_iso_iff_bijective _).mpr ⟨to_basic_open_injective R f, to_basic_open_surjective R f⟩,
exact is_iso_of_reflects_iso _ (forget CommRing),
end | instance | algebraic_geometry.structure_sheaf.is_iso_to_basic_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing",
"CommRing.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic_open_iso (f : R) : (structure_sheaf R).1.obj (op (basic_open f)) ≅
CommRing.of (localization.away f) | (as_iso (show CommRing.of _ ⟶ _, from to_basic_open R f)).symm | def | algebraic_geometry.structure_sheaf.basic_open_iso | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of",
"localization.away"
] | The ring isomorphism between the structure sheaf on `basic_open f` and the localization of `R`
at the submonoid of powers of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stalk_algebra (p : prime_spectrum R) : algebra R ((structure_sheaf R).presheaf.stalk p) | (to_stalk R p).to_algebra | instance | algebraic_geometry.structure_sheaf.stalk_algebra | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_algebra_map (p : prime_spectrum R) (r : R) :
algebra_map R ((structure_sheaf R).presheaf.stalk p) r = to_stalk R p r | rfl | lemma | algebraic_geometry.structure_sheaf.stalk_algebra_map | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.to_stalk (p : prime_spectrum R) :
is_localization.at_prime ((structure_sheaf R).presheaf.stalk p) p.as_ideal | begin
convert (is_localization.is_localization_iff_of_ring_equiv _ (stalk_iso R p).symm
.CommRing_iso_to_ring_equiv).mp localization.is_localization,
apply algebra.algebra_ext,
intro _,
rw stalk_algebra_map,
congr' 1,
erw iso.eq_comp_inv,
exact to_stalk_comp_stalk_to_fiber_ring_hom R p,
end | instance | algebraic_geometry.structure_sheaf.is_localization.to_stalk | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra.algebra_ext",
"is_localization.at_prime",
"is_localization.is_localization_iff_of_ring_equiv",
"prime_spectrum"
] | Stalk of the structure sheaf at a prime p as localization of R | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_algebra (U : (opens (prime_spectrum R))ᵒᵖ) :
algebra R ((structure_sheaf R).val.obj U) | (to_open R (unop U)).to_algebra | instance | algebraic_geometry.structure_sheaf.open_algebra | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_algebra_map (U : (opens (prime_spectrum R))ᵒᵖ) (r : R) :
algebra_map R ((structure_sheaf R).val.obj U) r = to_open R (unop U) r | rfl | lemma | algebraic_geometry.structure_sheaf.open_algebra_map | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra_map",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.to_basic_open (r : R) :
is_localization.away r ((structure_sheaf R).val.obj (op $ basic_open r)) | begin
convert (is_localization.is_localization_iff_of_ring_equiv _ (basic_open_iso R r).symm
.CommRing_iso_to_ring_equiv).mp localization.is_localization,
apply algebra.algebra_ext,
intro x,
congr' 1,
exact (localization_to_basic_open R r).symm
end | instance | algebraic_geometry.structure_sheaf.is_localization.to_basic_open | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"algebra.algebra_ext",
"is_localization.away",
"is_localization.is_localization_iff_of_ring_equiv"
] | Sections of the structure sheaf of Spec R on a basic open as localization of R | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_basic_open_epi (r : R) : epi (to_open R (basic_open r)) | ⟨λ S f g h, by { refine is_localization.ring_hom_ext _ _,
swap 5, exact is_localization.to_basic_open R r, exact h }⟩ | instance | algebraic_geometry.structure_sheaf.to_basic_open_epi | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"is_localization.ring_hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_global_factors : to_open R ⊤ =
CommRing.of_hom (algebra_map R (localization.away (1 : R))) ≫ to_basic_open R (1 : R) ≫
(structure_sheaf R).1.map (eq_to_hom (basic_open_one.symm)).op | begin
rw ← category.assoc,
change to_open R ⊤ = (to_basic_open R 1).comp _ ≫ _,
unfold CommRing.of_hom,
rw [localization_to_basic_open R, to_open_res],
end | lemma | algebraic_geometry.structure_sheaf.to_global_factors | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"algebra_map",
"localization.away"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_to_global : is_iso (to_open R ⊤) | begin
let hom := CommRing.of_hom (algebra_map R (localization.away (1 : R))),
haveI : is_iso hom := is_iso.of_iso
((is_localization.at_one R (localization.away (1 : R))).to_ring_equiv.to_CommRing_iso),
rw to_global_factors R,
apply_instance
end | instance | algebraic_geometry.structure_sheaf.is_iso_to_global | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"algebra_map",
"is_localization.at_one",
"localization.away"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
global_sections_iso : CommRing.of R ≅ (structure_sheaf R).1.obj (op ⊤) | as_iso (to_open R ⊤) | def | algebraic_geometry.structure_sheaf.global_sections_iso | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of"
] | The ring isomorphism between the ring `R` and the global sections `Γ(X, 𝒪ₓ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
global_sections_iso_hom (R : CommRing) :
(global_sections_iso R).hom = to_open R ⊤ | rfl | lemma | algebraic_geometry.structure_sheaf.global_sections_iso_hom | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_stalk_stalk_specializes {R : Type*} [comm_ring R]
{x y : prime_spectrum R} (h : x ⤳ y) :
to_stalk R y ≫ (structure_sheaf R).presheaf.stalk_specializes h = to_stalk R x | by { dsimp[to_stalk], simpa [-to_open_germ], } | lemma | algebraic_geometry.structure_sheaf.to_stalk_stalk_specializes | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"comm_ring",
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_to_stalk_stalk_specializes {R : Type*} [comm_ring R]
{x y : prime_spectrum R} (h : x ⤳ y) :
structure_sheaf.localization_to_stalk R y ≫ (structure_sheaf R).presheaf.stalk_specializes h =
CommRing.of_hom (prime_spectrum.localization_map_of_specializes h) ≫
structure_sheaf.localization_to_stalk... | begin
apply is_localization.ring_hom_ext y.as_ideal.prime_compl,
any_goals { dsimp, apply_instance },
erw ring_hom.comp_assoc,
conv_rhs { erw ring_hom.comp_assoc },
dsimp [CommRing.of_hom, localization_to_stalk, prime_spectrum.localization_map_of_specializes],
rw [is_localization.lift_comp, is_localization.... | lemma | algebraic_geometry.structure_sheaf.localization_to_stalk_stalk_specializes | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"comm_ring",
"is_localization.lift_comp",
"is_localization.ring_hom_ext",
"prime_spectrum",
"prime_spectrum.localization_map_of_specializes",
"ring_hom.comp_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stalk_specializes_stalk_to_fiber {R : Type*} [comm_ring R]
{x y : prime_spectrum R} (h : x ⤳ y) :
(structure_sheaf R).presheaf.stalk_specializes h ≫ structure_sheaf.stalk_to_fiber_ring_hom R x =
structure_sheaf.stalk_to_fiber_ring_hom R y ≫
prime_spectrum.localization_map_of_specializes h | begin
change _ ≫ (structure_sheaf.stalk_iso R x).hom = (structure_sheaf.stalk_iso R y).hom ≫ _,
rw [← iso.eq_comp_inv, category.assoc, ← iso.inv_comp_eq],
exact localization_to_stalk_stalk_specializes h,
end | lemma | algebraic_geometry.structure_sheaf.stalk_specializes_stalk_to_fiber | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"comm_ring",
"prime_spectrum",
"prime_spectrum.localization_map_of_specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_fun (f : R →+* S) (U : opens (prime_spectrum.Top R))
(V : opens (prime_spectrum.Top S)) (hUV : V.1 ⊆ (prime_spectrum.comap f) ⁻¹' U.1)
(s : Π x : U, localizations R x) (y : V) : localizations S y | localization.local_ring_hom (prime_spectrum.comap f y.1).as_ideal _ f rfl
(s ⟨(prime_spectrum.comap f y.1), hUV y.2⟩ : _) | def | algebraic_geometry.structure_sheaf.comap_fun | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom",
"prime_spectrum.comap"
] | Given a ring homomorphism `f : R →+* S`, an open set `U` of the prime spectrum of `R` and an open
set `V` of the prime spectrum of `S`, such that `V ⊆ (comap f) ⁻¹' U`, we can push a section `s`
on `U` to a section on `V`, by composing with `localization.local_ring_hom _ _ f` from the left and
`comap f` from the right.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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