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 ⌀ |
|---|---|---|---|---|---|---|
spec_le_iff (R : CommRingCat) (p q : Spec R) : p ≤ q ↔ q.asIdeal ≤ p.asIdeal := by
aesop (add simp PrimeSpectrum.le_iff_specializes) | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation",
"Mathlib.RingTheory.Spectrum.Prime.Polynomial",
"Mathlib.AlgebraicGeometry.PullbackCarrier"
] | Mathlib/AlgebraicGeometry/AffineSpace.lean | spec_le_iff | null |
@[stacks 01Z2]
Scheme.nonempty_of_isLimit [IsCofilteredOrEmpty I]
[∀ {i j} (f : i ⟶ j), IsAffineHom (D.map f)] [∀ i, Nonempty (D.obj i)]
[∀ i, CompactSpace (D.obj i)] :
Nonempty c.pt := by
classical
cases isEmpty_or_nonempty I
· have e := (isLimitEquivIsTerminalOfIsEmpty _ _ hc).uniqueUpToIso specULif... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | Scheme.nonempty_of_isLimit | Suppose we have a cofiltered diagram of nonempty quasi-compact schemes,
whose transition maps are affine. Then the limit is also nonempty. |
exists_mem_of_isClosed_of_nonempty
[IsCofilteredOrEmpty I]
[∀ {i j} (f : i ⟶ j), IsAffineHom (D.map f)]
(Z : ∀ (i : I), Set (D.obj i))
(hZc : ∀ (i : I), IsClosed (Z i))
(hZne : ∀ i, (Z i).Nonempty)
(hZcpt : ∀ i, IsCompact (Z i))
(hmapsTo : ∀ {i i' : I} (f : i ⟶ i'), Set.MapsTo (D.map f).base... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | exists_mem_of_isClosed_of_nonempty | Suppose we have a cofiltered diagram of schemes whose transition maps are affine. The limit of
a family of compatible nonempty quasicompact closed sets in the diagram is also nonempty. |
@[stacks 01Z3]
exists_mem_of_isClosed_of_nonempty'
[IsCofilteredOrEmpty I]
[∀ {i j} (f : i ⟶ j), IsAffineHom (D.map f)]
{j : I}
(Z : ∀ (i : I), (i ⟶ j) → Set (D.obj i))
(hZc : ∀ i hij, IsClosed (Z i hij))
(hZne : ∀ i hij, (Z i hij).Nonempty)
(hZcpt : ∀ i hij, IsCompact (Z i hij))
(hstab ... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | exists_mem_of_isClosed_of_nonempty' | A variant of `exists_mem_of_isClosed_of_nonempty` where the closed sets are only defined
for the objects over a given `j : I`. |
ExistsHomHomCompEqCompAux where
/-- (Implementation) The limit cone. See the section docstring. -/
c : Cone D
/-- (Implementation) The limit cone is a limit. See the section docstring. -/
hc : IsLimit c
/-- (Implementation) The index on which `a` and `b` lives. See the section docstring. -/
i : I
/-- (Imp... | structure | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | ExistsHomHomCompEqCompAux | Subsumed by `Scheme.exists_hom_hom_comp_eq_comp_of_locallyOfFiniteType`. -/
private nonrec lemma Scheme.exists_hom_hom_comp_eq_comp_of_isAffine_of_locallyOfFiniteType
[IsAffine S] [IsAffine X] [∀ i, IsAffine (D.obj i)] [IsAffine c.pt]
{i : I} (a : D.obj i ⟶ X) (ha : t.app i = a ≫ f)
{j : I} (b : D.obj j ⟶ X... |
exists_index : ∃ (i' : I) (hii' : i' ⟶ A.i),
((D.map hii' ≫ pullback.lift A.a A.b (A.ha.symm.trans A.hb)).base ⁻¹'
((Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X : Set <|
↑(pullback f f))ᶜ)) = ∅ := by
let W := Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
by_contra! h
let ... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | exists_index | null |
i' : I := A.exists_index.choose | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | i' | (Implementation)
The index `i'` such that `a` and `b` restricted onto `i'` maps into the diagonal components.
See the section docstring. |
hii' : A.i' ⟶ A.i := A.exists_index.choose_spec.choose | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | hii' | (Implementation) The map from `i'` to `i`. See the section docstring. |
g : D.obj A.i' ⟶ pullback f f :=
(D.map A.hii' ≫ pullback.lift A.a A.b (A.ha.symm.trans A.hb))
omit [LocallyOfFiniteType f] in | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | g | (Implementation)
The map sending `x` to `(a x, b x)`, which should fall in the diagonal component.
See the section docstring. |
range_g_subset :
Set.range A.g.base ⊆ Scheme.Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X := by
simpa [ExistsHomHomCompEqCompAux.hii', g] using A.exists_index.choose_spec.choose_spec | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | range_g_subset | null |
noncomputable 𝒰D₀ : Scheme.OpenCover.{u} (D.obj A.i') :=
Scheme.Cover.mkOfCovers (Σ i : A.𝒰S.I₀, (A.𝒰X i).I₀) _
(fun i ↦ ((Scheme.Pullback.diagonalCover f A.𝒰S A.𝒰X).pullback₁ A.g).f ⟨i.1, i.2, i.2⟩)
(fun x ↦ by simpa [← Set.mem_range, Scheme.Pullback.range_fst,
Scheme.Pullback.diagonalCoverDiago... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | 𝒰D₀ | (Implementation)
The covering of `D(i')` by the pullback of the diagonal components of `X ×ₛ X`.
See the section docstring. |
noncomputable 𝒰D : Scheme.OpenCover.{u} (D.obj A.i') :=
A.𝒰D₀.bind fun _ ↦ Scheme.affineCover _
attribute [-simp] cast_eq eq_mpr_eq_cast | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | 𝒰D | (Implementation) An affine open cover refining `𝒰D₀`. See the section docstring. |
D' (j : A.𝒰D.I₀) : Over A.i' ⥤ Scheme :=
Over.post D ⋙ Over.pullback (A.𝒰D.f j) ⋙ Over.forget _ | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | D' | (Implementation) The diagram restricted to `Over i'`. See the section docstring. |
c' (j : A.𝒰D.I₀) : Cone (A.D' j) :=
(Over.pullback (A.𝒰D.f j) ⋙ Over.forget _).mapCone ((Over.conePost _ _).obj A.c)
attribute [local instance] IsCofiltered.isConnected | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | c' | (Implementation) The limit cone restricted to `Over i'`. See the section docstring. |
hc' (j : A.𝒰D.I₀) : IsLimit (A.c' j) :=
isLimitOfPreserves (Over.pullback (A.𝒰D.f j) ⋙ Over.forget _) (Over.isLimitConePost _ A.hc)
variable [∀ i, IsAffineHom (A.c.π.app i)] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | hc' | (Implementation)
The limit cone restricted to `Over i'` is still a limit because the diagram is cofiltered.
See the section docstring. |
exists_eq (j : A.𝒰D.I₀) : ∃ (k : I) (hki' : k ⟶ A.i'),
(A.𝒰D.pullback₁ (D.map hki')).f j ≫ D.map (hki' ≫ A.hii') ≫ A.a =
(A.𝒰D.pullback₁ (D.map hki')).f j ≫ D.map (hki' ≫ A.hii') ≫ A.b := by
have : IsAffine (A.𝒰D.X j) := by dsimp [𝒰D]; infer_instance
have (i : _) : IsAffine ((Over.post D ⋙ Over.pullb... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | exists_eq | null |
@[stacks 01ZC "Injective part of (1) => (3)"]
Scheme.exists_hom_hom_comp_eq_comp_of_locallyOfFiniteType
{i : I} (a : D.obj i ⟶ X) (ha : t.app i = a ≫ f)
{j : I} (b : D.obj j ⟶ X) (hb : t.app j = b ≫ f)
(hab : c.π.app i ≫ a = c.π.app j ≫ b) :
∃ (k : I) (hik : k ⟶ i) (hjk : k ⟶ j),
D.map hik ≫ a = D... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.FinitePresentation",
"Mathlib.AlgebraicGeometry.IdealSheaf.Functorial",
"Mathlib.AlgebraicGeometry.Morphisms.Separated",
"Mathlib.CategoryTheory.Filtered.Final",
"Mathlib.CategoryTheory.Monad.Limits"
] | Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean | Scheme.exists_hom_hom_comp_eq_comp_of_locallyOfFiniteType | Given a cofiltered diagram `D` of quasi-compact `S`-schemes with affine transition maps,
and another scheme `X` of finite type over `S`.
Then the canonical map `colim Homₛ(Dᵢ, X) ⟶ Homₛ(lim Dᵢ, X)` is injective.
In other words, for each pair of `a : Homₛ(Dᵢ, X)` and `b : Homₛ(Dⱼ, X)` that give rise to the
same map `Ho... |
Scheme.Hom.fiber (f : X.Hom Y) (y : Y) : Scheme := pullback f (Y.fromSpecResidueField y) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiber | `f.fiber y` is the scheme-theoretic fiber of `f` at `y`. |
Scheme.Hom.fiberι (f : X.Hom Y) (y : Y) : f.fiber y ⟶ X := pullback.fst _ _ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberι | `f.fiberι y : f.fiber y ⟶ X` is the embedding of the scheme-theoretic fiber into `X`. |
Scheme.Hom.fiberToSpecResidueField (f : X.Hom Y) (y : Y) :
f.fiber y ⟶ Spec (Y.residueField y) :=
pullback.snd _ _ | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberToSpecResidueField | The canonical map from the scheme-theoretic fiber to the residue field. |
@[reducible] Scheme.Hom.fiberOverSpecResidueField
(f : X.Hom Y) (y : Y) : (f.fiber y).Over (Spec (Y.residueField y)) where
hom := f.fiberToSpecResidueField y | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberOverSpecResidueField | The fiber of `f` at `y` is naturally a `κ(y)`-scheme. |
Scheme.Hom.fiberToSpecResidueField_apply (f : X.Hom Y) (y : Y) (x : f.fiber y) :
(f.fiberToSpecResidueField y).base x = IsLocalRing.closedPoint (Y.residueField y) :=
Subsingleton.elim (α := PrimeSpectrum _) _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberToSpecResidueField_apply | null |
Scheme.Hom.range_fiberι (f : X.Hom Y) (y : Y) :
Set.range (f.fiberι y).base = f.base ⁻¹' {y} := by
simp [fiber, fiberι, Scheme.Pullback.range_fst, Scheme.range_fromSpecResidueField] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.range_fiberι | null |
Scheme.Hom.fiberHomeo (f : X.Hom Y) (y : Y) : f.fiber y ≃ₜ f.base ⁻¹' {y} :=
.trans (f.fiberι y).isEmbedding.toHomeomorph (.setCongr (f.range_fiberι y))
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberHomeo | The scheme-theoretic fiber of `f` at `y` is homeomorphic to `f ⁻¹' {y}`. |
Scheme.Hom.fiberHomeo_apply (f : X.Hom Y) (y : Y) (x : f.fiber y) :
(f.fiberHomeo y x).1 = (f.fiberι y).base x := rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberHomeo_apply | null |
Scheme.Hom.fiberι_fiberHomeo_symm (f : X.Hom Y) (y : Y) (x : f.base ⁻¹' {y}) :
(f.fiberι y).base ((f.fiberHomeo y).symm x) = x :=
congr($((f.fiberHomeo y).apply_symm_apply x).1) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberι_fiberHomeo_symm | null |
Scheme.Hom.asFiber (f : X.Hom Y) (x : X) : f.fiber (f.base x) :=
(f.fiberHomeo (f.base x)).symm ⟨x, rfl⟩
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.asFiber | A point `x` as a point in the fiber of `f` at `f x`. |
Scheme.Hom.fiberι_asFiber (f : X.Hom Y) (x : X) : (f.fiberι _).base (f.asFiber x) = x :=
f.fiberι_fiberHomeo_symm _ _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.fiberι_asFiber | null |
QuasiCompact.isCompact_preimage_singleton (f : X ⟶ Y) [QuasiCompact f] (y : Y) :
IsCompact (f.base ⁻¹' {y}) :=
f.range_fiberι y ▸ isCompact_range (f.fiberι y).continuous | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | QuasiCompact.isCompact_preimage_singleton | null |
IsFinite.finite_preimage_singleton (f : X ⟶ Y) [IsFinite f] (y : Y) :
(f.base ⁻¹' {y}).Finite :=
f.range_fiberι y ▸ Set.finite_range (f.fiberι y).base | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | IsFinite.finite_preimage_singleton | null |
Scheme.Hom.finite_preimage (f : X.Hom Y) [IsFinite f] {s : Set Y} (hs : s.Finite) :
(f.base ⁻¹' s).Finite := by
rw [← Set.biUnion_of_singleton s, Set.preimage_iUnion₂]
exact hs.biUnion fun _ _ ↦ IsFinite.finite_preimage_singleton f _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.finite_preimage | null |
Scheme.Hom.discrete_fiber (f : X ⟶ Y) (y : Y) [IsFinite f] :
DiscreteTopology (f.fiber y) := inferInstance | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.PullbackCarrier",
"Mathlib.AlgebraicGeometry.Morphisms.Finite",
"Mathlib.RingTheory.Spectrum.Prime.Jacobson"
] | Mathlib/AlgebraicGeometry/Fiber.lean | Scheme.Hom.discrete_fiber | null |
noncomputable Scheme.functionField [IrreducibleSpace X] : CommRingCat :=
X.presheaf.stalk (genericPoint X) | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | Scheme.functionField | The function field of an irreducible scheme is the local ring at its generic point.
Despite the name, this is a field only when the scheme is integral. |
noncomputable Scheme.germToFunctionField [IrreducibleSpace X] (U : X.Opens)
[h : Nonempty U] : Γ(X, U) ⟶ X.functionField :=
X.presheaf.germ U
(genericPoint X)
(((genericPoint_spec X).mem_open_set_iff U.isOpen).mpr (by simpa using h)) | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | Scheme.germToFunctionField | The restriction map from a component to the function field. |
germ_injective_of_isIntegral [IsIntegral X] {U : X.Opens} (x : X) (hx : x ∈ U) :
Function.Injective (X.presheaf.germ U x hx) := by
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ U x hx).hom.map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ hx hx _ _ hy
cases Subsingleton.... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | germ_injective_of_isIntegral | null |
Scheme.germToFunctionField_injective [IsIntegral X] (U : X.Opens) [Nonempty U] :
Function.Injective (X.germToFunctionField U) :=
germ_injective_of_isIntegral _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | Scheme.germToFunctionField_injective | null |
genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[hX : IrreducibleSpace X] [IrreducibleSpace Y] :
f.base (genericPoint X) = genericPoint Y := by
apply ((genericPoint_spec Y).eq _).symm
convert (genericPoint_spec X).image (show Continuous f.base by fun_prop)
symm
rw [... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | genericPoint_eq_of_isOpenImmersion | null |
noncomputable stalkFunctionFieldAlgebra [IrreducibleSpace X] (x : X) :
Algebra (X.presheaf.stalk x) X.functionField := by
apply RingHom.toAlgebra
exact (X.presheaf.stalkSpecializes ((genericPoint_spec X).specializes trivial)).hom | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | stalkFunctionFieldAlgebra | null |
functionField_isScalarTower [IrreducibleSpace X] (U : X.Opens) (x : U)
[Nonempty U] : IsScalarTower Γ(X, U) (X.presheaf.stalk x) X.functionField := by
apply IsScalarTower.of_algebraMap_eq'
simp_rw [RingHom.algebraMap_toAlgebra]
change _ = (X.presheaf.germ U x x.2 ≫ _).hom
rw [X.presheaf.germ_stalkSpecialize... | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | functionField_isScalarTower | null |
@[simp]
genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
genericPoint (Spec R) = (⊥ : PrimeSpectrum R) := by
apply (genericPoint_spec (Spec R)).eq
rw [isGenericPoint_def]
rw [← PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure, PrimeSpectrum.vanishingIdeal_singleton]
rw [← PrimeSpectrum.zeroLo... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | genericPoint_eq_bot_of_affine | null |
functionField_isFractionRing_of_affine (R : CommRingCat.{u}) [IsDomain R] :
IsFractionRing R (Spec R).functionField := by
convert StructureSheaf.IsLocalization.to_stalk R (genericPoint (Spec R))
delta IsFractionRing IsLocalization.AtPrime
apply Eq.to_iff
congr 1
rw [genericPoint_eq_bot_of_affine]
ext
... | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | functionField_isFractionRing_of_affine | null |
IsAffineOpen.primeIdealOf_genericPoint {X : Scheme} [IsIntegral X] {U : X.Opens}
(hU : IsAffineOpen U) [h : Nonempty U] :
hU.primeIdealOf
⟨genericPoint X,
((genericPoint_spec X).mem_open_set_iff U.isOpen).mpr (by simpa using h)⟩ =
genericPoint (Spec Γ(X, U)) := by
haveI : IsAffine _ :=... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | IsAffineOpen.primeIdealOf_genericPoint | null |
functionField_isFractionRing_of_isAffineOpen [IsIntegral X] (U : X.Opens)
(hU : IsAffineOpen U) [Nonempty U] :
IsFractionRing Γ(X, U) X.functionField := by
haveI : IsAffine _ := hU
haveI : IsIntegral U :=
@isIntegral_of_isAffine_of_isDomain _ _ _
(by rw [Scheme.Opens.toScheme_presheaf_obj, Opens.i... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Properties"
] | Mathlib/AlgebraicGeometry/FunctionField.lean | functionField_isFractionRing_of_isAffineOpen | null |
toΓSpecFun : X → PrimeSpectrum (Γ.obj (op X)) := fun x =>
comap (X.presheaf.Γgerm x).hom (IsLocalRing.closedPoint (X.presheaf.stalk x)) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecFun | The canonical map from the underlying set to the prime spectrum of `Γ(X)`. |
notMem_prime_iff_unit_in_stalk (r : Γ.obj (op X)) (x : X) :
r ∉ (X.toΓSpecFun x).asIdeal ↔ IsUnit (X.presheaf.Γgerm x r) := by
simp [toΓSpecFun, IsLocalRing.closedPoint]
@[deprecated (since := "2025-05-23")]
alias not_mem_prime_iff_unit_in_stalk := notMem_prime_iff_unit_in_stalk | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | notMem_prime_iff_unit_in_stalk | null |
toΓSpec_preimage_basicOpen_eq (r : Γ.obj (op X)) :
X.toΓSpecFun ⁻¹' basicOpen r = SetLike.coe (X.toRingedSpace.basicOpen r) := by
ext
dsimp
simp only [Set.mem_preimage, SetLike.mem_coe]
rw [X.toRingedSpace.mem_top_basicOpen]
exact notMem_prime_iff_unit_in_stalk .. | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpec_preimage_basicOpen_eq | The preimage of a basic open in `Spec Γ(X)` under the unit is the basic
open in `X` defined by the same element (they are equal as sets). |
toΓSpec_continuous : Continuous X.toΓSpecFun := by
rw [isTopologicalBasis_basic_opens.continuous_iff]
rintro _ ⟨r, rfl⟩
rw [X.toΓSpec_preimage_basicOpen_eq r]
exact (X.toRingedSpace.basicOpen r).2 | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpec_continuous | `toΓSpecFun` is continuous. |
toΓSpecBase : X.toTopCat ⟶ Spec.topObj (Γ.obj (op X)) :=
TopCat.ofHom
{ toFun := X.toΓSpecFun
continuous_toFun := X.toΓSpec_continuous }
variable (r : Γ.obj (op X)) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecBase | The canonical (bundled) continuous map from the underlying topological
space of `X` to the prime spectrum of its global sections. |
toΓSpecMapBasicOpen : Opens X :=
(Opens.map X.toΓSpecBase).obj (basicOpen r) | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecMapBasicOpen | The preimage in `X` of a basic open in `Spec Γ(X)` (as an open set). |
toΓSpecMapBasicOpen_eq : X.toΓSpecMapBasicOpen r = X.toRingedSpace.basicOpen r :=
Opens.ext (X.toΓSpec_preimage_basicOpen_eq r) | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecMapBasicOpen_eq | The preimage is the basic open in `X` defined by the same element `r`. |
toToΓSpecMapBasicOpen :
X.presheaf.obj (op ⊤) ⟶ X.presheaf.obj (op <| X.toΓSpecMapBasicOpen r) :=
X.presheaf.map (X.toΓSpecMapBasicOpen r).leTop.op | abbrev | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toToΓSpecMapBasicOpen | The map from the global sections `Γ(X)` to the sections on the (preimage of) a basic open. |
isUnit_res_toΓSpecMapBasicOpen : IsUnit (X.toToΓSpecMapBasicOpen r r) := by
convert
(X.presheaf.map <| (eqToHom <| X.toΓSpecMapBasicOpen_eq r).op).hom.isUnit_map
(X.toRingedSpace.isUnit_res_basicOpen r)
rw [← CommRingCat.comp_apply, ← Functor.map_comp]
congr | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | isUnit_res_toΓSpecMapBasicOpen | `r` is a unit as a section on the basic open defined by `r`. |
toΓSpecCApp :
(structureSheaf <| Γ.obj <| op X).val.obj (op <| basicOpen r) ⟶
X.presheaf.obj (op <| X.toΓSpecMapBasicOpen r) :=
CommRingCat.ofHom <|
IsLocalization.Away.lift
(R := Γ.obj (op X))
(S := (structureSheaf ↑(Γ.obj (op X))).val.obj (op (basicOpen r)))
r
(isUnit_res_toΓS... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecCApp | Define the sheaf hom on individual basic opens for the unit. |
toΓSpecCApp_iff
(f :
(structureSheaf <| Γ.obj <| op X).val.obj (op <| basicOpen r) ⟶
X.presheaf.obj (op <| X.toΓSpecMapBasicOpen r)) :
toOpen _ (basicOpen r) ≫ f = X.toToΓSpecMapBasicOpen r ↔ f = X.toΓSpecCApp r := by
have loc_inst := IsLocalization.to_basicOpen (Γ.obj (op X)) r
refine Concret... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecCApp_iff | Characterization of the sheaf hom on basic opens,
direction ← (next lemma) is used at various places, but → is not used in this file. |
toΓSpecCApp_spec : toOpen _ (basicOpen r) ≫ X.toΓSpecCApp r = X.toToΓSpecMapBasicOpen r :=
(X.toΓSpecCApp_iff r _).2 rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecCApp_spec | null |
@[simps app]
toΓSpecCBasicOpens :
(inducedFunctor basicOpen).op ⋙ (structureSheaf (Γ.obj (op X))).1 ⟶
(inducedFunctor basicOpen).op ⋙ ((TopCat.Sheaf.pushforward _ X.toΓSpecBase).obj X.𝒪).1 where
app r := X.toΓSpecCApp r.unop
naturality r s f := by
apply (StructureSheaf.to_basicOpen_epi (Γ.obj (op X))... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecCBasicOpens | The sheaf hom on all basic opens, commuting with restrictions. |
@[simps]
toΓSpecSheafedSpace : X.toSheafedSpace ⟶ Spec.toSheafedSpace.obj (op (Γ.obj (op X))) where
base := X.toΓSpecBase
c :=
TopCat.Sheaf.restrictHomEquivHom (structureSheaf (Γ.obj (op X))).1 _ isBasis_basic_opens
X.toΓSpecCBasicOpens | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecSheafedSpace | The canonical morphism of sheafed spaces from `X` to the spectrum of its global sections. |
toΓSpecSheafedSpace_app_eq :
X.toΓSpecSheafedSpace.c.app (op (basicOpen r)) = X.toΓSpecCApp r := by
apply TopCat.Sheaf.extend_hom_app _ _ _
@[reassoc] theorem toΓSpecSheafedSpace_app_spec (r : Γ.obj (op X)) :
toOpen (Γ.obj (op X)) (basicOpen r) ≫ X.toΓSpecSheafedSpace.c.app (op (basicOpen r)) =
X.toToΓS... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpecSheafedSpace_app_eq | null |
toStalk_stalkMap_toΓSpec (x : X) :
toStalk _ _ ≫ X.toΓSpecSheafedSpace.stalkMap x = X.presheaf.Γgerm x := by
rw [PresheafedSpace.Hom.stalkMap,
← toOpen_germ _ (basicOpen (1 : Γ.obj (op X))) _ (by rw [basicOpen_one]; trivial),
← Category.assoc, Category.assoc (toOpen _ _), stalkFunctor_map_germ, ← Category... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toStalk_stalkMap_toΓSpec | The map on stalks induced by the unit commutes with maps from `Γ(X)` to
stalks (in `Spec Γ(X)` and in `X`). |
@[simps! base]
toΓSpec : X ⟶ Spec.locallyRingedSpaceObj (Γ.obj (op X)) where
__ := X.toΓSpecSheafedSpace
prop := by
intro x
let p : PrimeSpectrum (Γ.obj (op X)) := X.toΓSpecFun x
constructor
let S := (structureSheaf _).presheaf.stalk p
rintro (t : S) ht
obtain ⟨⟨r, s⟩, he⟩ := IsLocalization.... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpec | The canonical morphism from `X` to the spectrum of its global sections. |
toΓSpec_preimage_zeroLocus_eq {X : LocallyRingedSpace.{u}}
(s : Set (X.presheaf.obj (op ⊤))) :
X.toΓSpec.base ⁻¹' PrimeSpectrum.zeroLocus s = X.toRingedSpace.zeroLocus s := by
simp only [RingedSpace.zeroLocus]
have (i : LocallyRingedSpace.Γ.obj (op X)) (_ : i ∈ s) :
(SetLike.coe (X.toRingedSpace.basic... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toΓSpec_preimage_zeroLocus_eq | On a locally ringed space `X`, the preimage of the zero locus of the prime spectrum
of `Γ(X, ⊤)` under `toΓSpec` agrees with the associated zero locus on `X`. |
comp_ring_hom_ext {X : LocallyRingedSpace.{u}} {R : CommRingCat.{u}} {f : R ⟶ Γ.obj (op X)}
{β : X ⟶ Spec.locallyRingedSpaceObj R}
(w : X.toΓSpec.base ≫ (Spec.locallyRingedSpaceMap f).base = β.base)
(h :
∀ r : R,
f ≫ X.presheaf.map (homOfLE le_top : (Opens.map β.base).obj (basicOpen r) ⟶ _).op... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | comp_ring_hom_ext | null |
Γ_Spec_left_triangle : toSpecΓ (Γ.obj (op X)) ≫ X.toΓSpec.c.app (op ⊤) = 𝟙 _ := by
unfold toSpecΓ
rw [← toOpen_res _ (basicOpen (1 : Γ.obj (op X))) ⊤ (eqToHom basicOpen_one.symm),
Category.assoc, NatTrans.naturality, ← Category.assoc]
erw [X.toΓSpecSheafedSpace_app_spec 1, ← Functor.map_comp]
convert eqToH... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Γ_Spec_left_triangle | `toSpecΓ _` is an isomorphism so these are mutually two-sided inverses. |
identityToΓSpec : 𝟭 LocallyRingedSpace.{u} ⟶ Γ.rightOp ⋙ Spec.toLocallyRingedSpace where
app := LocallyRingedSpace.toΓSpec
naturality X Y f := by
symm
apply LocallyRingedSpace.comp_ring_hom_ext
· ext1 x
dsimp
change PrimeSpectrum.comap (f.c.app (op ⊤)).hom (X.toΓSpecFun x) = Y.toΓSpecFun (f... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | identityToΓSpec | The unit as a natural transformation. |
left_triangle (X : LocallyRingedSpace) :
SpecΓIdentity.inv.app (Γ.obj (op X)) ≫ (identityToΓSpec.app X).c.app (op ⊤) = 𝟙 _ :=
X.Γ_Spec_left_triangle | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | left_triangle | null |
right_triangle (R : CommRingCat) :
identityToΓSpec.app (Spec.toLocallyRingedSpace.obj <| op R) ≫
Spec.toLocallyRingedSpace.map (SpecΓIdentity.inv.app R).op =
𝟙 _ := by
apply LocallyRingedSpace.comp_ring_hom_ext
· ext (p : PrimeSpectrum R)
dsimp
refine PrimeSpectrum.ext (Ideal.ext fun x =>... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | right_triangle | `SpecΓIdentity` is iso so these are mutually two-sided inverses. |
@[simps]
locallyRingedSpaceAdjunction : Γ.rightOp ⊣ Spec.toLocallyRingedSpace.{u} where
unit := identityToΓSpec
counit := (NatIso.op SpecΓIdentity).inv
left_triangle_components X := by
simp only [Functor.id_obj, Functor.rightOp_obj, Γ_obj, Functor.comp_obj,
Spec.toLocallyRingedSpace_obj, Spec.locallyRin... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | locallyRingedSpaceAdjunction | The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `LocallyRingedSpace`. |
@[simp]
toSpecΓ_unop (R : CommRingCatᵒᵖ) :
AlgebraicGeometry.toSpecΓ (Opposite.unop R) = toOpen R.unop ⊤ := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toSpecΓ_unop | `@[simp]`-normal form of `locallyRingedSpaceAdjunction_counit_app`. |
@[simp]
toSpecΓ_of (R : Type u) [CommRing R] :
AlgebraicGeometry.toSpecΓ (CommRingCat.of R) = toOpen R ⊤ := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toSpecΓ_of | `@[simp]`-normal form of `locallyRingedSpaceAdjunction_counit_app'`. |
locallyRingedSpaceAdjunction_counit_app (R : CommRingCatᵒᵖ) :
locallyRingedSpaceAdjunction.counit.app R =
(toOpen R.unop ⊤).op := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | locallyRingedSpaceAdjunction_counit_app | null |
locallyRingedSpaceAdjunction_counit_app' (R : Type u) [CommRing R] :
locallyRingedSpaceAdjunction.counit.app (op <| CommRingCat.of R) =
(toOpen R ⊤).op := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | locallyRingedSpaceAdjunction_counit_app' | null |
locallyRingedSpaceAdjunction_homEquiv_apply
{X : LocallyRingedSpace} {R : CommRingCatᵒᵖ}
(f : Γ.rightOp.obj X ⟶ R) :
locallyRingedSpaceAdjunction.homEquiv X R f =
identityToΓSpec.app X ≫ Spec.locallyRingedSpaceMap f.unop := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | locallyRingedSpaceAdjunction_homEquiv_apply | null |
locallyRingedSpaceAdjunction_homEquiv_apply'
{X : LocallyRingedSpace} {R : Type u} [CommRing R]
(f : CommRingCat.of R ⟶ Γ.obj <| op X) :
locallyRingedSpaceAdjunction.homEquiv X (op <| CommRingCat.of R) (op f) =
identityToΓSpec.app X ≫ Spec.locallyRingedSpaceMap f := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | locallyRingedSpaceAdjunction_homEquiv_apply' | null |
toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app
{X : LocallyRingedSpace} {R : Type u} [CommRing R]
(f : Γ.rightOp.obj X ⟶ op (CommRingCat.of R)) (U) :
StructureSheaf.toOpen R U.unop ≫
(locallyRingedSpaceAdjunction.homEquiv X (op <| CommRingCat.of R) f).c.app U =
f.unop ≫ X.presheaf.map (homO... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app | null |
adjunction : Scheme.Γ.rightOp ⊣ Scheme.Spec.{u} where
unit :=
{ app := fun X ↦ ⟨locallyRingedSpaceAdjunction.{u}.unit.app X.toLocallyRingedSpace⟩
naturality := fun _ _ f ↦
Scheme.Hom.ext' (locallyRingedSpaceAdjunction.{u}.unit.naturality f.toLRSHom) }
counit := (NatIso.op Scheme.SpecΓIdentity.{u}).inv
... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction | The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `Scheme`. |
adjunction_homEquiv_apply {X : Scheme} {R : CommRingCatᵒᵖ}
(f : (op <| Scheme.Γ.obj <| op X) ⟶ R) :
ΓSpec.adjunction.homEquiv X R f = ⟨locallyRingedSpaceAdjunction.homEquiv X.1 R f⟩ := rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction_homEquiv_apply | null |
adjunction_homEquiv_symm_apply {X : Scheme} {R : CommRingCatᵒᵖ}
(f : X ⟶ Scheme.Spec.obj R) :
(ΓSpec.adjunction.homEquiv X R).symm f =
(locallyRingedSpaceAdjunction.homEquiv X.1 R).symm f.toLRSHom := rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction_homEquiv_symm_apply | null |
adjunction_counit_app' {R : CommRingCatᵒᵖ} :
ΓSpec.adjunction.counit.app R = locallyRingedSpaceAdjunction.counit.app R := rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction_counit_app' | null |
adjunction_counit_app {R : CommRingCatᵒᵖ} :
ΓSpec.adjunction.counit.app R = (Scheme.ΓSpecIso (unop R)).inv.op := rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction_counit_app | null |
_root_.AlgebraicGeometry.Scheme.toSpecΓ (X : Scheme.{u}) : X ⟶ Spec Γ(X, ⊤) :=
ΓSpec.adjunction.unit.app X
@[simp] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | _root_.AlgebraicGeometry.Scheme.toSpecΓ | The canonical map `X ⟶ Spec Γ(X, ⊤)`. This is the unit of the `Γ-Spec` adjunction. |
adjunction_unit_app {X : Scheme} :
ΓSpec.adjunction.unit.app X = X.toSpecΓ := rfl | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | adjunction_unit_app | null |
isIso_locallyRingedSpaceAdjunction_counit :
IsIso.{u + 1, u + 1} locallyRingedSpaceAdjunction.counit :=
(NatIso.op SpecΓIdentity).isIso_inv | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | isIso_locallyRingedSpaceAdjunction_counit | null |
isIso_adjunction_counit : IsIso ΓSpec.adjunction.counit := by
apply (config := { allowSynthFailures := true }) NatIso.isIso_of_isIso_app
intro R
rw [adjunction_counit_app]
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | isIso_adjunction_counit | null |
Scheme.toSpecΓ_base (X : Scheme.{u}) (x) :
(Scheme.toSpecΓ X).base x =
(Spec.map (X.presheaf.germ ⊤ x trivial)).base (IsLocalRing.closedPoint _) := rfl
@[reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Scheme.toSpecΓ_base | null |
Scheme.toSpecΓ_naturality {X Y : Scheme.{u}} (f : X ⟶ Y) :
f ≫ Y.toSpecΓ = X.toSpecΓ ≫ Spec.map (f.appTop) :=
ΓSpec.adjunction.unit.naturality f
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Scheme.toSpecΓ_naturality | null |
Scheme.toSpecΓ_appTop (X : Scheme.{u}) :
X.toSpecΓ.appTop = (Scheme.ΓSpecIso Γ(X, ⊤)).hom := by
have := ΓSpec.adjunction.left_triangle_components X
dsimp at this
rw [← IsIso.eq_comp_inv] at this
simp only [Category.id_comp] at this
rw [← Quiver.Hom.op_inj.eq_iff, this, ← op_inv, IsIso.Iso.inv_inv]
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Scheme.toSpecΓ_appTop | null |
SpecMap_ΓSpecIso_hom (R : CommRingCat.{u}) :
Spec.map ((Scheme.ΓSpecIso R).hom) = (Spec R).toSpecΓ := by
have := ΓSpec.adjunction.right_triangle_components (op R)
dsimp at this
rwa [← IsIso.eq_comp_inv, Category.id_comp, ← Spec.map_inv, IsIso.Iso.inv_inv, eq_comm] at this | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | SpecMap_ΓSpecIso_hom | null |
Scheme.toSpecΓ_preimage_basicOpen (X : Scheme.{u}) (r : Γ(X, ⊤)) :
X.toSpecΓ ⁻¹ᵁ (PrimeSpectrum.basicOpen r) = X.basicOpen r := by
rw [← basicOpen_eq_of_affine, Scheme.preimage_basicOpen, ← Scheme.Hom.appTop]
congr
rw [Scheme.toSpecΓ_appTop]
exact Iso.inv_hom_id_apply (C := CommRingCat) _ _
@[reassoc (attr ... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Scheme.toSpecΓ_preimage_basicOpen | null |
toOpen_toSpecΓ_app {X : Scheme.{u}} (U) :
StructureSheaf.toOpen _ _ ≫ X.toSpecΓ.app U =
X.presheaf.map (homOfLE (by exact le_top)).op := by
rw [← StructureSheaf.toOpen_res _ _ _ (homOfLE le_top), Category.assoc,
NatTrans.naturality _ (homOfLE (le_top (a := U))).op]
change (ΓSpec.adjunction.counit.app ... | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | toOpen_toSpecΓ_app | null |
ΓSpecIso_inv_ΓSpec_adjunction_homEquiv {X : Scheme.{u}} {B : CommRingCat} (φ : B ⟶ Γ(X, ⊤)) :
(Scheme.ΓSpecIso B).inv ≫ ((ΓSpec.adjunction.homEquiv X (op B)) φ.op).appTop = φ := by
simp only [Adjunction.homEquiv_apply, Scheme.Spec_map, Opens.map_top, Scheme.comp_app]
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | ΓSpecIso_inv_ΓSpec_adjunction_homEquiv | null |
ΓSpec_adjunction_homEquiv_eq {X : Scheme.{u}} {B : CommRingCat} (φ : B ⟶ Γ(X, ⊤)) :
((ΓSpec.adjunction.homEquiv X (op B)) φ.op).appTop = (Scheme.ΓSpecIso B).hom ≫ φ := by
rw [← Iso.inv_comp_eq, ΓSpecIso_inv_ΓSpec_adjunction_homEquiv] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | ΓSpec_adjunction_homEquiv_eq | null |
ΓSpecIso_obj_hom {X : Scheme.{u}} (U : X.Opens) :
(Scheme.ΓSpecIso Γ(X, U)).hom = (Spec.map U.topIso.inv).appTop ≫
U.toScheme.toSpecΓ.appTop ≫ U.topIso.hom := by simp
/-! Immediate consequences of the adjunction. -/ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | ΓSpecIso_obj_hom | null |
Spec.fullyFaithfulToLocallyRingedSpace : Spec.toLocallyRingedSpace.FullyFaithful :=
ΓSpec.locallyRingedSpaceAdjunction.fullyFaithfulROfIsIsoCounit | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.fullyFaithfulToLocallyRingedSpace | The functor `Spec.toLocallyRingedSpace : CommRingCatᵒᵖ ⥤ LocallyRingedSpace`
is fully faithful. |
Spec.fullyFaithful : Scheme.Spec.FullyFaithful :=
ΓSpec.adjunction.fullyFaithfulROfIsIsoCounit | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.fullyFaithful | Spec is a full functor. -/
instance : Spec.toLocallyRingedSpace.Full :=
Spec.fullyFaithfulToLocallyRingedSpace.full
/-- Spec is a faithful functor. -/
instance : Spec.toLocallyRingedSpace.Faithful :=
Spec.fullyFaithfulToLocallyRingedSpace.faithful
/-- The functor `Spec : CommRingCatᵒᵖ ⥤ Scheme` is fully faithful. |
Spec.full : Scheme.Spec.Full :=
Spec.fullyFaithful.full | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.full | Spec is a full functor. |
Spec.faithful : Scheme.Spec.Faithful :=
Spec.fullyFaithful.faithful | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.faithful | Spec is a faithful functor. |
Spec.map_inj : Spec.map φ = Spec.map ψ ↔ φ = ψ := by
rw [iff_comm, ← Quiver.Hom.op_inj.eq_iff, ← Scheme.Spec.map_injective.eq_iff]
rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.map_inj | null |
Spec.map_injective {R S : CommRingCat} : Function.Injective (Spec.map : (R ⟶ S) → _) :=
fun _ _ ↦ Spec.map_inj.mp
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.map_injective | null |
Spec.map_eq_id {R : CommRingCat} {ϕ : R ⟶ R} : Spec.map ϕ = 𝟙 (Spec R) ↔ ϕ = 𝟙 R := by
simp [← map_inj] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.map_eq_id | null |
Spec.preimage : R ⟶ S := (Scheme.Spec.preimage f).unop
@[simp] lemma Spec.map_preimage : Spec.map (Spec.preimage f) = f := Scheme.Spec.map_preimage f
@[simp] lemma Spec.map_preimage_unop (f : Spec R ⟶ Spec S) :
Spec.map (Spec.fullyFaithful.preimage f).unop = f := Spec.fullyFaithful.map_preimage _
variable (φ) in
@[... | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Adjunction.Limits",
"Mathlib.CategoryTheory.Adjunction.Opposites",
"Mathlib.CategoryTheory.Adjunction.Reflective"
] | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | Spec.preimage | The preimage under Spec. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.