fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Scheme.IsGermInjectiveAt (X : Scheme.{u}) (x : X) : Prop where
cond : ∃ (U : X.Opens) (hx : x ∈ U), IsAffineOpen U ∧ Function.Injective (X.presheaf.germ U x hx) | class | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.IsGermInjectiveAt | The germ map at `x` is injective if there exists some affine `U ∋ x`
such that the map `Γ(X, U) ⟶ X_x` is injective |
injective_germ_basicOpen (U : X.Opens) (hU : IsAffineOpen U)
(x : X) (hx : x ∈ U) (f : Γ(X, U))
(hf : x ∈ X.basicOpen f)
(H : Function.Injective (X.presheaf.germ U x hx)) :
Function.Injective (X.presheaf.germ (X.basicOpen f) x hf) := by
rw [RingHom.injective_iff_ker_eq_bot, RingHom.ker_eq_bot_iff_eq_z... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | injective_germ_basicOpen | null |
Scheme.exists_germ_injective (X : Scheme.{u}) (x : X) [X.IsGermInjectiveAt x] :
∃ (U : X.Opens) (hx : x ∈ U),
IsAffineOpen U ∧ Function.Injective (X.presheaf.germ U x hx) :=
Scheme.IsGermInjectiveAt.cond | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.exists_germ_injective | null |
Scheme.exists_le_and_germ_injective (X : Scheme.{u}) (x : X) [X.IsGermInjectiveAt x]
(V : X.Opens) (hxV : x ∈ V) :
∃ (U : X.Opens) (hx : x ∈ U),
IsAffineOpen U ∧ U ≤ V ∧ Function.Injective (X.presheaf.germ U x hx) := by
obtain ⟨U, hx, hU, H⟩ := Scheme.IsGermInjectiveAt.cond (x := x)
obtain ⟨f, hf, hxf... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.exists_le_and_germ_injective | null |
isGermInjectiveAt_iff_of_isOpenImmersion {x : X} [IsOpenImmersion f] :
Y.IsGermInjectiveAt (f.base x) ↔ X.IsGermInjectiveAt x := by
refine ⟨fun H ↦ ?_, fun _ ↦ inferInstance⟩
obtain ⟨U, hxU, hU, hU', H⟩ :=
Y.exists_le_and_germ_injective (f.base x) (V := f.opensRange) ⟨x, rfl⟩
obtain ⟨V, hV⟩ := (IsOpenImme... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | isGermInjectiveAt_iff_of_isOpenImmersion | null |
Scheme.IsGermInjective (X : Scheme.{u}) := ∀ x : X, X.IsGermInjectiveAt x | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.IsGermInjective | The class of schemes such that for each `x : X`,
`Γ(X, U) ⟶ X_x` is injective for some affine `U` containing `x`.
This is typically satisfied when `X` is integral or locally Noetherian. |
Scheme.IsGermInjective.of_openCover
{X : Scheme.{u}} (𝒰 : X.OpenCover) [∀ i, (𝒰.X i).IsGermInjective] : X.IsGermInjective := by
intro x
rw [← (𝒰.covers x).choose_spec]
infer_instance | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.IsGermInjective.of_openCover | null |
protected
Scheme.IsGermInjective.Spec
(H : ∀ I : Ideal R, I.IsPrime →
∃ f : R, f ∉ I ∧ ∀ (x y : R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0) :
(Spec R).IsGermInjective := by
refine fun p ↦ ⟨?_⟩
obtain ⟨f, hf, H⟩ := H p.asIdeal p.2
refine ⟨PrimeSpectrum.basicOpen f, hf, ?_, ?_⟩
· rw [← basicOpen_eq_... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | Scheme.IsGermInjective.Spec | null |
@[stacks 0BX6]
spread_out_unique_of_isGermInjective {x : X} [X.IsGermInjectiveAt x]
(f g : X ⟶ Y) (e : f.base x = g.base x)
(H : f.stalkMap x =
Y.presheaf.stalkSpecializes (Inseparable.of_eq e.symm).specializes ≫ g.stalkMap x) :
∃ (U : X.Opens), x ∈ U ∧ U.ι ≫ f = U.ι ≫ g := by
obtain ⟨_, ⟨V : Y.Open... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | spread_out_unique_of_isGermInjective | Let `x : X` and `f g : X ⟶ Y` be two morphisms such that `f x = g x`.
If `f` and `g` agree on the stalk of `x`, then they agree on an open neighborhood of `x`,
provided `X` is "germ-injective" at `x` (e.g. when it's integral or locally Noetherian).
TODO: The condition on `X` is unnecessary when `Y` is locally of finit... |
spread_out_unique_of_isGermInjective' {x : X} [X.IsGermInjectiveAt x]
(f g : X ⟶ Y)
(e : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g) :
∃ (U : X.Opens), x ∈ U ∧ U.ι ≫ f = U.ι ≫ g := by
fapply spread_out_unique_of_isGermInjective
· simpa using congr(($e).base (IsLocalRing.closedPoint _))
· apply Spec... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | spread_out_unique_of_isGermInjective' | A variant of `spread_out_unique_of_isGermInjective`
whose condition is an equality of scheme morphisms instead of ring homomorphisms. |
exists_lift_of_germInjective_aux {U : X.Opens} {x : X} (hxU)
(φ : A ⟶ X.presheaf.stalk x) (φRA : R ⟶ A) (φRX : R ⟶ Γ(X, U))
(hφRA : RingHom.FiniteType φRA.hom)
(e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU) :
∃ (V : X.Opens) (hxV : x ∈ V),
V ≤ U ∧ RingHom.range φ.hom ≤ RingHom.range (X.presheaf.ger... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | exists_lift_of_germInjective_aux | null |
exists_lift_of_germInjective {x : X} [X.IsGermInjectiveAt x] {U : X.Opens} (hxU : x ∈ U)
(φ : A ⟶ X.presheaf.stalk x) (φRA : R ⟶ A) (φRX : R ⟶ Γ(X, U))
(hφRA : RingHom.FiniteType φRA.hom)
(e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU) :
∃ (V : X.Opens) (hxV : x ∈ V) (φ' : A ⟶ Γ(X, V)) (i : V ≤ U), IsAffi... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | exists_lift_of_germInjective | Suppose `X` is a scheme, `x : X` such that the germ map at `x` is (locally) injective,
and `U` is a neighborhood of `x`.
Given a commutative diagram of `CommRingCat`
```
R ⟶ Γ(X, U)
↓ ↓
A ⟶ 𝒪_{X, x}
```
such that `R` is of finite type over `A`, we may lift `A ⟶ 𝒪_{X, x}` to some `A ⟶ Γ(X, V)`. |
@[stacks 0BX6]
spread_out_of_isGermInjective [LocallyOfFiniteType sY] {x : X} [X.IsGermInjectiveAt x] {y : Y}
(e : sX.base x = sY.base y) (φ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x)
(h : sY.stalkMap y ≫ φ =
S.presheaf.stalkSpecializes (Inseparable.of_eq e).specializes ≫ sX.stalkMap x) :
∃ (U : X.Ope... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | spread_out_of_isGermInjective | Given `S`-schemes `X Y` and points `x : X` `y : Y` over `s : S`.
Suppose we have the following diagram of `S`-schemes
```
Spec 𝒪_{X, x} ⟶ X
|
Spec(φ)
↓
Spec 𝒪_{Y, y} ⟶ Y
```
Then the map `Spec(φ)` spreads out to an `S`-morphism on an open subscheme `U ⊆ X`,
```
Spec 𝒪_{X, x} ⟶ U ⊆ X
| |
S... |
spread_out_of_isGermInjective' [LocallyOfFiniteType sY] {x : X} [X.IsGermInjectiveAt x]
(φ : Spec (X.presheaf.stalk x) ⟶ Y)
(h : φ ≫ sY = X.fromSpecStalk x ≫ sX) :
∃ (U : X.Opens) (hxU : x ∈ U) (f : U.toScheme ⟶ Y),
φ = U.fromSpecStalkOfMem x hxU ≫ f ∧ f ≫ sY = U.ι ≫ sX := by
have := spread_out_of_i... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.FiniteType",
"Mathlib.AlgebraicGeometry.Noetherian",
"Mathlib.AlgebraicGeometry.Stalk",
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/SpreadingOut.lean | spread_out_of_isGermInjective' | Given `S`-schemes `X Y`, a point `x : X`, and a `S`-morphism `φ : Spec 𝒪_{X, x} ⟶ Y`,
we may spread it out to an `S`-morphism `f : U ⟶ Y`
provided that `Y` is locally of finite type over `S` and
`X` is "germ-injective" at `x` (e.g. when it's integral or locally Noetherian).
TODO: The condition on `X` is unnecessary w... |
noncomputable IsAffineOpen.fromSpecStalk
{X : Scheme} {U : X.Opens} (hU : IsAffineOpen U) {x : X} (hxU : x ∈ U) :
Spec (X.presheaf.stalk x) ⟶ X :=
Spec.map (X.presheaf.germ _ x hxU) ≫ hU.fromSpec | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | IsAffineOpen.fromSpecStalk | A morphism from `Spec(O_x)` to `X`, which is defined with the help of an affine open
neighborhood `U` of `x`. |
IsAffineOpen.fromSpecStalk_eq (x : X) (hxU : x ∈ U) (hxV : x ∈ V) :
hU.fromSpecStalk hxU = hV.fromSpecStalk hxV := by
obtain ⟨U', h₁, h₂, h₃ : U' ≤ U ⊓ V⟩ :=
Opens.isBasis_iff_nbhd.mp (isBasis_affine_open X) (show x ∈ U ⊓ V from ⟨hxU, hxV⟩)
transitivity fromSpecStalk h₁ h₂
· delta fromSpecStalk
rw [← ... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | IsAffineOpen.fromSpecStalk_eq | The morphism from `Spec(O_x)` to `X` given by `IsAffineOpen.fromSpec` does not depend on the affine
open neighborhood of `x` we choose. |
noncomputable Scheme.fromSpecStalk (X : Scheme) (x : X) :
Spec (X.presheaf.stalk x) ⟶ X :=
(isAffineOpen_opensRange (X.affineCover.f (X.affineCover.idx x))).fromSpecStalk
(X.affineCover.covers x)
@[simps over] noncomputable | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Scheme.fromSpecStalk | If `x` is a point of `X`, this is the canonical morphism from `Spec(O_x)` to `X`. |
@[simp]
IsAffineOpen.fromSpecStalk_eq_fromSpecStalk {x : X} (hxU : x ∈ U) :
hU.fromSpecStalk hxU = X.fromSpecStalk x := fromSpecStalk_eq .. | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | IsAffineOpen.fromSpecStalk_eq_fromSpecStalk | null |
IsAffineOpen.fromSpecStalk_isPreimmersion {X : Scheme.{u}} {U : Opens X}
(hU : IsAffineOpen U) (x : X) (hx : x ∈ U) : IsPreimmersion (hU.fromSpecStalk hx) := by
dsimp [IsAffineOpen.fromSpecStalk]
haveI : IsPreimmersion (Spec.map (X.presheaf.germ U x hx)) :=
letI : Algebra Γ(X, U) (X.presheaf.stalk x) := (X.... | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | IsAffineOpen.fromSpecStalk_isPreimmersion | null |
IsAffineOpen.fromSpecStalk_closedPoint {U : Opens X} (hU : IsAffineOpen U)
{x : X} (hxU : x ∈ U) :
(hU.fromSpecStalk hxU).base (closedPoint (X.presheaf.stalk x)) = x := by
rw [IsAffineOpen.fromSpecStalk, Scheme.comp_base_apply]
rw [← hU.primeIdealOf_eq_map_closedPoint ⟨x, hxU⟩, hU.fromSpec_primeIdealOf ⟨x, ... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | IsAffineOpen.fromSpecStalk_closedPoint | null |
@[simp]
fromSpecStalk_closedPoint {x : X} :
(X.fromSpecStalk x).base (closedPoint (X.presheaf.stalk x)) = x :=
IsAffineOpen.fromSpecStalk_closedPoint _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | fromSpecStalk_closedPoint | null |
fromSpecStalk_app {x : X} (hxU : x ∈ U) :
(X.fromSpecStalk x).app U =
X.presheaf.germ U x hxU ≫
(ΓSpecIso (X.presheaf.stalk x)).inv ≫
(Spec (X.presheaf.stalk x)).presheaf.map (homOfLE le_top).op := by
obtain ⟨_, ⟨V : X.Opens, hV, rfl⟩, hxV, hVU⟩ := (isBasis_affine_open X).exists_subset_of_... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | fromSpecStalk_app | null |
fromSpecStalk_appTop {x : X} :
(X.fromSpecStalk x).appTop =
X.presheaf.germ ⊤ x trivial ≫
(ΓSpecIso (X.presheaf.stalk x)).inv ≫
(Spec (X.presheaf.stalk x)).presheaf.map (homOfLE le_top).op :=
fromSpecStalk_app ..
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | fromSpecStalk_appTop | null |
Spec_map_stalkSpecializes_fromSpecStalk {x y : X} (h : x ⤳ y) :
Spec.map (X.presheaf.stalkSpecializes h) ≫ X.fromSpecStalk y = X.fromSpecStalk x := by
obtain ⟨_, ⟨U, hU, rfl⟩, hyU, -⟩ :=
(isBasis_affine_open X).exists_subset_of_mem_open (Set.mem_univ y) isOpen_univ
have hxU : x ∈ U := h.mem_open U.2 hyU
r... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_map_stalkSpecializes_fromSpecStalk | null |
@[reassoc (attr := simp)]
Spec_map_stalkMap_fromSpecStalk {x} :
Spec.map (f.stalkMap x) ≫ Y.fromSpecStalk _ = X.fromSpecStalk x ≫ f := by
obtain ⟨_, ⟨U, hU, rfl⟩, hxU, -⟩ := (isBasis_affine_open Y).exists_subset_of_mem_open
(Set.mem_univ (f.base x)) isOpen_univ
obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := (isBasis... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_map_stalkMap_fromSpecStalk | null |
Spec_fromSpecStalk (R : CommRingCat) (x) :
(Spec R).fromSpecStalk x =
Spec.map ((ΓSpecIso R).inv ≫ (Spec R).presheaf.germ ⊤ x trivial) := by
rw [← (isAffineOpen_top (Spec R)).fromSpecStalk_eq_fromSpecStalk (x := x) trivial,
IsAffineOpen.fromSpecStalk, IsAffineOpen.fromSpec_top, isoSpec_Spec_inv,
← S... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_fromSpecStalk | null |
Spec_fromSpecStalk' (R : CommRingCat) (x) :
(Spec R).fromSpecStalk x = Spec.map (StructureSheaf.toStalk R _) :=
Spec_fromSpecStalk _ _
@[stacks 01J7] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_fromSpecStalk' | A variant of `Spec_fromSpecStalk` that breaks abstraction boundaries. |
range_fromSpecStalk {x : X} :
Set.range (X.fromSpecStalk x).base = { y | y ⤳ x } := by
ext y
constructor
· rintro ⟨y, rfl⟩
exact ((IsLocalRing.specializes_closedPoint y).map (X.fromSpecStalk x).base.hom.2).trans
(specializes_of_eq fromSpecStalk_closedPoint)
· rintro (hy : y ⤳ x)
have := fromSp... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | range_fromSpecStalk | null |
noncomputable
Opens.fromSpecStalkOfMem {X : Scheme.{u}} (U : X.Opens) (x : X) (hxU : x ∈ U) :
Spec (X.presheaf.stalk x) ⟶ U :=
Spec.map (inv (U.ι.stalkMap ⟨x, hxU⟩)) ≫ U.toScheme.fromSpecStalk ⟨x, hxU⟩
@[reassoc (attr := simp)] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Opens.fromSpecStalkOfMem | The canonical map `Spec 𝒪_{X, x} ⟶ U` given `x ∈ U ⊆ X`. |
Opens.fromSpecStalkOfMem_ι {X : Scheme.{u}} (U : X.Opens) (x : X) (hxU : x ∈ U) :
U.fromSpecStalkOfMem x hxU ≫ U.ι = X.fromSpecStalk x := by
simp only [Opens.fromSpecStalkOfMem, Spec.map_inv, Category.assoc, IsIso.inv_comp_eq]
exact (Scheme.Spec_map_stalkMap_fromSpecStalk U.ι (x := ⟨x, hxU⟩)).symm | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Opens.fromSpecStalkOfMem_ι | null |
@[reassoc]
fromSpecStalk_toSpecΓ (X : Scheme.{u}) (x : X) :
X.fromSpecStalk x ≫ X.toSpecΓ = Spec.map (X.presheaf.germ ⊤ x trivial) := by
rw [Scheme.toSpecΓ_naturality, ← SpecMap_ΓSpecIso_hom, ← Spec.map_comp,
Scheme.fromSpecStalk_appTop]
simp
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | fromSpecStalk_toSpecΓ | null |
Opens.fromSpecStalkOfMem_toSpecΓ {X : Scheme.{u}} (U : X.Opens) (x : X) (hxU : x ∈ U) :
U.fromSpecStalkOfMem x hxU ≫ U.toSpecΓ = Spec.map (X.presheaf.germ U x hxU) := by
rw [fromSpecStalkOfMem, Opens.toSpecΓ, Category.assoc, fromSpecStalk_toSpecΓ_assoc,
← Spec.map_comp, ← Spec.map_comp]
congr 1
rw [IsIso.... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Opens.fromSpecStalkOfMem_toSpecΓ | null |
noncomputable
stalkClosedPointIso :
(Spec R).presheaf.stalk (closedPoint R) ≅ R :=
StructureSheaf.stalkIso _ _ ≪≫ (IsLocalization.atUnits R
(closedPoint R).asIdeal.primeCompl fun _ ↦ not_not.mp).toRingEquiv.toCommRingCatIso.symm | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | stalkClosedPointIso | For a local ring `(R, 𝔪)`,
this is the isomorphism between the stalk of `Spec R` at `𝔪` and `R`. |
stalkClosedPointIso_inv :
(stalkClosedPointIso R).inv = StructureSheaf.toStalk R _ := by
ext x
exact StructureSheaf.localizationToStalk_of _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | stalkClosedPointIso_inv | null |
ΓSpecIso_hom_stalkClosedPointIso_inv :
(Scheme.ΓSpecIso R).hom ≫ (stalkClosedPointIso R).inv =
(Spec R).presheaf.germ ⊤ (closedPoint _) trivial := by
rw [stalkClosedPointIso_inv, ← Iso.eq_inv_comp]
rfl
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | ΓSpecIso_hom_stalkClosedPointIso_inv | null |
germ_stalkClosedPointIso_hom :
(Spec R).presheaf.germ ⊤ (closedPoint _) trivial ≫ (stalkClosedPointIso R).hom =
(Scheme.ΓSpecIso R).hom := by
rw [← ΓSpecIso_hom_stalkClosedPointIso_inv, Category.assoc, Iso.inv_hom_id, Category.comp_id] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | germ_stalkClosedPointIso_hom | null |
Spec_stalkClosedPointIso :
Spec.map (stalkClosedPointIso R).inv = (Spec R).fromSpecStalk (closedPoint R) := by
rw [stalkClosedPointIso_inv, Scheme.Spec_fromSpecStalk'] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_stalkClosedPointIso | null |
noncomputable
stalkClosedPointTo :
X.presheaf.stalk (f.base (closedPoint R)) ⟶ R :=
f.stalkMap (closedPoint R) ≫ (stalkClosedPointIso R).hom | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | stalkClosedPointTo | Given a local ring `(R, 𝔪)` and a morphism `f : Spec R ⟶ X`,
they induce a (local) ring homomorphism `φ : 𝒪_{X, f 𝔪} ⟶ R`.
This is inverse to `φ ↦ Spec.map φ ≫ X.fromSpecStalk (f 𝔪)`. See `SpecToEquivOfLocalRing`. |
isLocalHom_stalkClosedPointTo :
IsLocalHom (stalkClosedPointTo f).hom :=
inferInstanceAs <| IsLocalHom (f.stalkMap (closedPoint R) ≫ (stalkClosedPointIso R).hom).hom | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | isLocalHom_stalkClosedPointTo | null |
isLocalHom_stalkClosedPointTo' {R : Type u} [CommRing R] [IsLocalRing R]
(f : Spec(R) ⟶ X) :
IsLocalHom (stalkClosedPointTo f).hom :=
isLocalHom_stalkClosedPointTo f | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | isLocalHom_stalkClosedPointTo' | Copy of `isLocalHom_stalkClosedPointTo` which unbundles the comm ring.
Useful for use in combination with `CommRingCat.of K` for a field `K`. |
preimage_eq_top_of_closedPoint_mem
{U : Opens X} (hU : f.base (closedPoint R) ∈ U) : f ⁻¹ᵁ U = ⊤ :=
IsLocalRing.closed_point_mem_iff.mp hU | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | preimage_eq_top_of_closedPoint_mem | null |
stalkClosedPointTo_comp (g : X ⟶ Y) :
stalkClosedPointTo (f ≫ g) = g.stalkMap _ ≫ stalkClosedPointTo f := by
rw [stalkClosedPointTo, Scheme.stalkMap_comp]
exact Category.assoc _ _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | stalkClosedPointTo_comp | null |
germ_stalkClosedPointTo_Spec {R S : CommRingCat} [IsLocalRing S] (φ : R ⟶ S) :
(Spec R).presheaf.germ ⊤ _ trivial ≫ stalkClosedPointTo (Spec.map φ) =
(ΓSpecIso R).hom ≫ φ := by
rw [stalkClosedPointTo, Scheme.stalkMap_germ_assoc, ← Iso.inv_comp_eq,
← ΓSpecIso_inv_naturality_assoc]
simp_rw [Opens.map_to... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | germ_stalkClosedPointTo_Spec | null |
germ_stalkClosedPointTo (U : Opens X) (hU : f.base (closedPoint R) ∈ U) :
X.presheaf.germ U _ hU ≫ stalkClosedPointTo f = f.app U ≫
((Spec R).presheaf.mapIso (eqToIso (preimage_eq_top_of_closedPoint_mem f hU).symm).op ≪≫
ΓSpecIso R).hom := by
rw [stalkClosedPointTo, Scheme.stalkMap_germ_assoc, Iso.t... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | germ_stalkClosedPointTo | null |
germ_stalkClosedPointTo_Spec_fromSpecStalk
{x : X} (f : X.presheaf.stalk x ⟶ R) [IsLocalHom f.hom] (U : Opens X) (hU) :
X.presheaf.germ U _ hU ≫ stalkClosedPointTo (Spec.map f ≫ X.fromSpecStalk x) =
X.presheaf.germ U x (by simpa using hU) ≫ f := by
have : (Spec.map f ≫ X.fromSpecStalk x).base (closedPoi... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | germ_stalkClosedPointTo_Spec_fromSpecStalk | null |
stalkClosedPointTo_fromSpecStalk (x : X) :
stalkClosedPointTo (X.fromSpecStalk x) =
(X.presheaf.stalkCongr (by rw [fromSpecStalk_closedPoint]; rfl)).hom := by
refine TopCat.Presheaf.stalk_hom_ext _ fun U hxU ↦ ?_
simp only [TopCat.Presheaf.stalkCongr_hom, TopCat.Presheaf.germ_stalkSpecializes]
have : X.... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | stalkClosedPointTo_fromSpecStalk | null |
Spec_stalkClosedPointTo_fromSpecStalk :
Spec.map (stalkClosedPointTo f) ≫ X.fromSpecStalk _ = f := by
obtain ⟨_, ⟨U, hU, rfl⟩, hxU, -⟩ := (isBasis_affine_open X).exists_subset_of_mem_open
(Set.mem_univ (f.base (closedPoint R))) isOpen_univ
have := IsAffineOpen.Spec_map_appLE_fromSpec f hU (isAffineOpen_top ... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | Spec_stalkClosedPointTo_fromSpecStalk | null |
SpecToEquivOfLocalRing_eq_iff
{f₁ f₂ : Σ x, { f : X.presheaf.stalk x ⟶ R // IsLocalHom f.hom }} :
f₁ = f₂ ↔ ∃ h₁ : f₁.1 = f₂.1, f₁.2.1 =
(X.presheaf.stalkCongr (by rw [h₁]; rfl)).hom ≫ f₂.2.1 := by
constructor
· rintro rfl; simp
· obtain ⟨x₁, ⟨f₁, h₁⟩⟩ := f₁
obtain ⟨x₂, ⟨f₂, h₂⟩⟩ := f₂
rintr... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | SpecToEquivOfLocalRing_eq_iff | useful lemma for applications of `SpecToEquivOfLocalRing` |
@[simps]
noncomputable
SpecToEquivOfLocalRing :
(Spec R ⟶ X) ≃ Σ x, { f : X.presheaf.stalk x ⟶ R // IsLocalHom f.hom } where
toFun f := ⟨f.base (closedPoint R), Scheme.stalkClosedPointTo f, inferInstance⟩
invFun xf := Spec.map xf.2.1 ≫ X.fromSpecStalk xf.1
left_inv := Scheme.Spec_stalkClosedPointTo_fromSpecSt... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.AffineScheme",
"Mathlib.AlgebraicGeometry.Morphisms.Preimmersion"
] | Mathlib/AlgebraicGeometry/Stalk.lean | SpecToEquivOfLocalRing | Given a local ring `R` and scheme `X`, morphisms `Spec R ⟶ X` corresponds to pairs
`(x, f)` where `x : X` and `f : 𝒪_{X, x} ⟶ R` is a local ring homomorphism. |
PrimeSpectrum.Top : TopCat :=
TopCat.of (PrimeSpectrum R) | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | PrimeSpectrum.Top | The prime spectrum, just as a topological space. |
Localizations (P : PrimeSpectrum.Top R) : Type u :=
Localization.AtPrime P.asIdeal | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | Localizations | The type family over `PrimeSpectrum R` consisting of the localization over each point. |
commRingLocalizations (P : PrimeSpectrum.Top R) : CommRing <| Localizations R P :=
inferInstanceAs <| CommRing <| Localization.AtPrime P.asIdeal | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | commRingLocalizations | null |
localRingLocalizations (P : PrimeSpectrum.Top R) : IsLocalRing <| Localizations R P :=
inferInstanceAs <| IsLocalRing <| Localization.AtPrime P.asIdeal | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localRingLocalizations | null |
IsFraction {U : Opens (PrimeSpectrum.Top R)} (f : ∀ x : U, Localizations R x) : Prop :=
∃ r s : R, ∀ x : U, s ∉ x.1.asIdeal ∧ f x * algebraMap _ _ s = algebraMap _ _ r | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | IsFraction | 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). |
IsFraction.eq_mk' {U : Opens (PrimeSpectrum.Top R)} {f : ∀ x : U, Localizations R x}
(hf : IsFraction f) :
∃ r s : R,
∀ x : U,
∃ hs : s ∉ x.1.asIdeal,
f x =
IsLocalization.mk' (Localization.AtPrime _) r
(⟨s, hs⟩ : (x : PrimeSpectrum.Top R).asIdeal.primeCompl) :=... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | IsFraction.eq_mk' | null |
isFractionPrelocal : PrelocalPredicate (Localizations R) where
pred {_} f := IsFraction f
res := by rintro V U i f ⟨r, s, w⟩; exact ⟨r, s, fun x => w (i x)⟩ | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isFractionPrelocal | The predicate `IsFraction` is "prelocal",
in the sense that if it holds on `U` it holds on any open subset `V` of `U`. |
isLocallyFraction : LocalPredicate (Localizations R) :=
(isFractionPrelocal R).sheafify
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isLocallyFraction | 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... |
isLocallyFraction_pred {U : Opens (PrimeSpectrum.Top R)} (f : ∀ x : U, Localizations R x) :
(isLocallyFraction R).pred f =
∀ x : U,
∃ (V : _) (_ : x.1 ∈ V) (i : V ⟶ U),
∃ r s : R,
∀ y : V, s ∉ y.1.asIdeal ∧ f (i y : U) * algebraMap _ _ s = algebraMap _ _ r :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isLocallyFraction_pred | null |
sectionsSubring (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
Subring (∀ x : U.unop, Localizations R x) where
carrier := { f | (isLocallyFraction R).pred f }
zero_mem' := by
refine fun x => ⟨unop U, x.2, 𝟙 _, 0, 1, fun y => ⟨?_, ?_⟩⟩
· rw [← Ideal.ne_top_iff_one]; exact y.1.isPrime.1
· simp
one_mem' :=... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | sectionsSubring | The functions satisfying `isLocallyFraction` form a subring. |
structureSheafInType : Sheaf (Type u) (PrimeSpectrum.Top R) :=
subsheafToTypes (isLocallyFraction R) | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | structureSheafInType | The structure sheaf (valued in `Type`, not yet `CommRingCat`) is the subsheaf consisting of
functions satisfying `isLocallyFraction`. |
commRingStructureSheafInTypeObj (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
CommRing ((structureSheafInType R).1.obj U) :=
(sectionsSubring R U).toCommRing
open PrimeSpectrum | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | commRingStructureSheafInTypeObj | null |
@[simps obj_carrier]
structurePresheafInCommRing : Presheaf CommRingCat (PrimeSpectrum.Top R) where
obj U := CommRingCat.of ((structureSheafInType R).1.obj U)
map {_ _} i := CommRingCat.ofHom
{ toFun := (structureSheafInType R).1.map i
map_zero' := rfl
map_add' := fun _ _ => rfl
map_one' := rf... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | structurePresheafInCommRing | The structure presheaf, valued in `CommRingCat`, constructed by dressing up the `Type`-valued
structure presheaf. |
structurePresheafCompForget :
structurePresheafInCommRing R ⋙ forget CommRingCat ≅ (structureSheafInType R).1 :=
NatIso.ofComponents fun _ => Iso.refl _
open TopCat.Presheaf | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | structurePresheafCompForget | Some glue, verifying that the structure presheaf valued in `CommRingCat` agrees
with the `Type`-valued structure presheaf. |
Spec.structureSheaf : Sheaf CommRingCat (PrimeSpectrum.Top R) :=
⟨structurePresheafInCommRing R,
(-- We check the sheaf condition under `forget CommRingCat`.
isSheaf_iff_isSheaf_comp
_ _).mpr
(isSheaf_of_iso (structurePresheafCompForget R).symm (structureSheafInType R).cond)⟩
open Spec (... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | Spec.structureSheaf | The structure sheaf on $Spec R$, valued in `CommRingCat`.
This is provided as a bundled `SheafedSpace` as `Spec.SheafedSpace R` later. |
@[simp]
res_apply (U V : Opens (PrimeSpectrum.Top R)) (i : V ⟶ U)
(s : (structureSheaf R).1.obj (op U)) (x : V) :
((structureSheaf R).1.map i.op s).1 x = (s.1 (i x) :) :=
rfl
/-
Notation in this comment
X = Spec R
OX = structure sheaf
In the following we construct an isomorphism between OX_p and R_p given any... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | res_apply | null |
const (f g : R) (U : Opens (PrimeSpectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : PrimeSpectrum.Top R).asIdeal.primeCompl) :
(structureSheaf R).1.obj (op U) :=
⟨fun x => IsLocalization.mk' _ f ⟨g, hu x x.2⟩, fun x =>
⟨U, x.2, 𝟙 _, f, g, fun y => ⟨hu y y.2, IsLocalization.mk'_spec _ _ _⟩⟩⟩
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const | The section of `structureSheaf R` on an open `U` sending each `x ∈ U` to the element
`f/g` in the localization of `R` at `x`. |
const_apply (f g : R) (U : Opens (PrimeSpectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : PrimeSpectrum.Top R).asIdeal.primeCompl) (x : U) :
(const R f g U hu).1 x =
IsLocalization.mk' (Localization.AtPrime x.1.asIdeal) f ⟨g, hu x x.2⟩ :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_apply | null |
const_apply' (f g : R) (U : Opens (PrimeSpectrum.Top R))
(hu : ∀ x ∈ U, g ∈ (x : PrimeSpectrum.Top R).asIdeal.primeCompl) (x : U)
(hx : g ∈ (x : PrimeSpectrum.Top R).asIdeal.primeCompl) :
(const R f g U hu).1 x = IsLocalization.mk' _ f ⟨g, hx⟩ :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_apply' | null |
exists_const (U) (s : (structureSheaf R).1.obj (op U)) (x : PrimeSpectrum.Top R)
(hx : x ∈ U) :
∃ (V : Opens (PrimeSpectrum.Top R)) (_ : x ∈ V) (i : V ⟶ U) (f g : R) (hg : _),
const R f g V hg = (structureSheaf R).1.map i.op s :=
let ⟨V, hxV, iVU, f, g, hfg⟩ := s.2 ⟨x, hx⟩
⟨V, hxV, iVU, f, g, fun y hy... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | exists_const | null |
res_const (f g : R) (U hu V hv i) :
(structureSheaf R).1.map i (const R f g U hu) = const R f g V hv :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | res_const | null |
res_const' (f g : R) (V hv) :
(structureSheaf R).1.map (homOfLE hv).op (const R f g (PrimeSpectrum.basicOpen g) fun _ => id) =
const R f g V hv :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | res_const' | null |
const_zero (f : R) (U hu) : const R 0 f U hu = 0 :=
Subtype.eq <| funext fun x => IsLocalization.mk'_eq_iff_eq_mul.2 <| by
rw [RingHom.map_zero]
exact (mul_eq_zero_of_left rfl ((algebraMap R (Localizations R x)) _)).symm | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_zero | null |
const_self (f : R) (U hu) : const R f f U hu = 1 :=
Subtype.eq <| funext fun _ => IsLocalization.mk'_self _ _ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_self | null |
const_one (U) : (const R 1 1 U fun _ _ => Submonoid.one_mem _) = 1 :=
const_self R 1 U _ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_one | null |
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 fun x hx =>
Submonoid.mul_mem _ (hu₁ x hx) (hu₂ x hx) :=
Subtype.eq <| funext fun x => Eq.symm <| IsLocalization.mk'_add _ _
⟨g₁, hu₁ x x.2⟩ ⟨g₂, hu₂ x x.2⟩ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_add | null |
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 fun x hx => Submonoid.mul_mem _ (hu₁ x hx) (hu₂ x hx) :=
Subtype.eq <|
funext fun x =>
Eq.symm <| IsLocalization.mk'_mul _ f₁ f₂ ⟨g₁, hu₁ x x.2⟩ ⟨g₂, hu₂ x x.2⟩ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_mul | null |
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 fun x =>
IsLocalization.mk'_eq_of_eq (by rw [mul_comm, Subtype.coe_mk, ← h, mul_comm, Subtype.coe_mk]) | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_ext | null |
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; rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_congr | null |
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] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_mul_rev | null |
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] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_mul_cancel | null |
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] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | const_mul_cancel' | null |
toOpen (U : Opens (PrimeSpectrum.Top R)) :
CommRingCat.of R ⟶ (structureSheaf R).1.obj (op U) := CommRingCat.ofHom
{ toFun f :=
⟨fun _ => algebraMap R _ f, fun x =>
⟨U, x.2, 𝟙 _, f, 1, fun y =>
⟨(Ideal.ne_top_iff_one _).1 y.1.2.1, by simp [RingHom.map_one, mul_one]⟩⟩⟩
map_one' := Subt... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen | The canonical ring homomorphism interpreting an element of `R` as
a section of the structure sheaf. |
toOpen_res (U V : Opens (PrimeSpectrum.Top R)) (i : V ⟶ U) :
toOpen R U ≫ (structureSheaf R).1.map i.op = toOpen R V :=
rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_res | null |
toOpen_apply (U : Opens (PrimeSpectrum.Top R)) (f : R) (x : U) :
(toOpen R U f).1 x = algebraMap _ _ f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_apply | null |
toOpen_eq_const (U : Opens (PrimeSpectrum.Top R)) (f : R) :
toOpen R U f = const R f 1 U fun x _ => (Ideal.ne_top_iff_one _).1 x.2.1 :=
Subtype.eq <| funext fun _ => Eq.symm <| IsLocalization.mk'_one _ f | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_eq_const | null |
toStalk (x : PrimeSpectrum.Top R) : CommRingCat.of R ⟶ (structureSheaf R).presheaf.stalk x :=
(toOpen R ⊤ ≫ (structureSheaf R).presheaf.germ _ x (by trivial))
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toStalk | The canonical ring homomorphism interpreting an element of `R` as an element of
the stalk of `structureSheaf R` at `x`. |
toOpen_germ (U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R) (hx : x ∈ U) :
toOpen R U ≫ (structureSheaf R).presheaf.germ U x hx = toStalk R x := by
rw [← toOpen_res R ⊤ U (homOfLE le_top : U ⟶ ⊤), Category.assoc, Presheaf.germ_res]; rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_germ | null |
germ_toOpen
(U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R) (hx : x ∈ U) (f : R) :
(structureSheaf R).presheaf.germ U x hx (toOpen R U f) = toStalk R x f := by
rw [← toOpen_germ]; rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | germ_toOpen | null |
toOpen_Γgerm_apply (x : PrimeSpectrum.Top R) (f : R) :
(structureSheaf R).presheaf.Γgerm x (toOpen R ⊤ f) = toStalk R x f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_Γgerm_apply | null |
isUnit_to_basicOpen_self (f : R) : IsUnit (toOpen R (PrimeSpectrum.basicOpen f) f) :=
isUnit_of_mul_eq_one _ (const R 1 f (PrimeSpectrum.basicOpen f) fun _ => id) <| by
rw [toOpen_eq_const, const_mul_rev] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isUnit_to_basicOpen_self | null |
isUnit_toStalk (x : PrimeSpectrum.Top R) (f : x.asIdeal.primeCompl) :
IsUnit (toStalk R x (f : R)) := by
rw [← germ_toOpen R (PrimeSpectrum.basicOpen (f : R)) x f.2 (f : R)]
exact RingHom.isUnit_map _ (isUnit_to_basicOpen_self R f) | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isUnit_toStalk | null |
localizationToStalk (x : PrimeSpectrum.Top R) :
CommRingCat.of (Localization.AtPrime x.asIdeal) ⟶ (structureSheaf R).presheaf.stalk x :=
CommRingCat.ofHom <|
show Localization.AtPrime x.asIdeal →+* _ from IsLocalization.lift (isUnit_toStalk R x)
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localizationToStalk | The canonical ring homomorphism from the localization of `R` at `p` to the stalk
of the structure sheaf at the point `p`. |
localizationToStalk_of (x : PrimeSpectrum.Top R) (f : R) :
localizationToStalk R x (algebraMap _ (Localization _) f) = toStalk R x f :=
IsLocalization.lift_eq (S := Localization x.asIdeal.primeCompl) _ f
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localizationToStalk_of | null |
localizationToStalk_mk' (x : PrimeSpectrum.Top R) (f : R) (s : x.asIdeal.primeCompl) :
localizationToStalk R x (IsLocalization.mk' (Localization.AtPrime x.asIdeal) f s) =
(structureSheaf R).presheaf.germ (PrimeSpectrum.basicOpen (s : R)) x s.2
(const R f s (PrimeSpectrum.basicOpen s) fun _ => id) :=
... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localizationToStalk_mk' | null |
openToLocalization (U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R) (hx : x ∈ U) :
(structureSheaf R).1.obj (op U) ⟶ CommRingCat.of (Localization.AtPrime x.asIdeal) :=
CommRingCat.ofHom
{ toFun s := (s.1 ⟨x, hx⟩ :)
map_one' := rfl
map_mul' _ _ := rfl
map_zero' := rfl
map_add' _ _ := r... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | openToLocalization | 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. |
coe_openToLocalization (U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R)
(hx : x ∈ U) :
(openToLocalization R U x hx :
(structureSheaf R).1.obj (op U) → Localization.AtPrime x.asIdeal) =
fun s => s.1 ⟨x, hx⟩ :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | coe_openToLocalization | null |
openToLocalization_apply (U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R)
(hx : x ∈ U) (s : (structureSheaf R).1.obj (op U)) :
openToLocalization R U x hx s = s.1 ⟨x, hx⟩ :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | openToLocalization_apply | null |
stalkToFiberRingHom (x : PrimeSpectrum.Top R) :
(structureSheaf R).presheaf.stalk x ⟶ CommRingCat.of (Localization.AtPrime x.asIdeal) :=
Limits.colimit.desc ((OpenNhds.inclusion x).op ⋙ (structureSheaf R).1)
{ pt := _
ι := { app := fun U =>
openToLocalization R ((OpenNhds.inclusion _).obj (unop ... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkToFiberRingHom | 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 `openToLocalization` maps. |
germ_comp_stalkToFiberRingHom
(U : Opens (PrimeSpectrum.Top R)) (x : PrimeSpectrum.Top R) (hx : x ∈ U) :
(structureSheaf R).presheaf.germ U x hx ≫ stalkToFiberRingHom R x =
openToLocalization R U x hx :=
Limits.colimit.ι_desc _ _
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | germ_comp_stalkToFiberRingHom | null |
stalkToFiberRingHom_germ (U : Opens (PrimeSpectrum.Top R))
(x : PrimeSpectrum.Top R) (hx : x ∈ U) (s : (structureSheaf R).1.obj (op U)) :
stalkToFiberRingHom R x ((structureSheaf R).presheaf.germ U x hx s) = s.1 ⟨x, hx⟩ :=
RingHom.ext_iff.mp (CommRingCat.hom_ext_iff.mp (germ_comp_stalkToFiberRingHom R U x hx)... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkToFiberRingHom_germ | null |
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