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stringclasses 11
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stringclasses 32
values | imports
listlengths 1
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| filename
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| symbolic_name
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algHom_ext' {A} [Semiring A] [Algebra S A] ⦃f g : tsze R M →ₐ[S] A⦄
(hinl : f.comp (inlAlgHom S R M) = g.comp (inlAlgHom S R M))
(hinr : f.toLinearMap.comp (inrHom R M |>.restrictScalars S) =
g.toLinearMap.comp (inrHom R M |>.restrictScalars S)) : f = g :=
AlgHom.toLinearMap_injective <|
linearMap_ext (AlgHom.congr_fun hinl) (LinearMap.congr_fun hinr)
variable {A : Type*} [Semiring A] [Algebra S A] [Algebra R' A]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
algHom_ext'
| null |
lift (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) : tsze R M →ₐ[S] A :=
AlgHom.ofLinearMap
((f.comp <| fstHom S R M).toLinearMap + g ∘ₗ (sndHom R M |>.restrictScalars S))
(show f 1 + g (0 : M) = 1 by rw [map_zero, map_one, add_zero])
(TrivSqZeroExt.ind fun r₁ m₁ =>
TrivSqZeroExt.ind fun r₂ m₂ => by
dsimp
simp only [add_zero, zero_add, add_mul, mul_add, hg]
rw [← map_mul, LinearMap.map_add, add_comm (g _), add_assoc, hfg, hgf])
|
def
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift
|
Assemble an algebra morphism `TrivSqZeroExt R M →ₐ[S] A` from separate morphisms on `R` and `M`.
Namely, we require that for an algebra morphism `f : R →ₐ[S] A` and a linear map `g : M →ₗ[S] A`,
we have:
* `g x * g y = 0`: the elements of `M` continue to square to zero.
* `g (r •> x) = f r * g x` and `g (x <• r) = g x * f r`: scalar multiplication on the left and
right is sent to left- and right- multiplication by the image under `f`.
See `TrivSqZeroExt.liftEquiv` for this as an equiv; namely that any such algebra morphism can be
factored in this way.
When `R` is commutative, this can be invoked with `f = Algebra.ofId R A`, which satisfies `hfg` and
`hgf`. This version is captured as an equiv by `TrivSqZeroExt.liftEquivOfComm`.
|
lift_def (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r) (x : tsze R M) :
lift f g hg hfg hgf x = f x.fst + g x.snd :=
rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_def
| null |
lift_apply_inl (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(r : R) :
lift f g hg hfg hgf (inl r) = f r :=
show f r + g 0 = f r by rw [map_zero, add_zero]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_apply_inl
| null |
lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(m : M) :
lift f g hg hfg hgf (inr m) = g m :=
show f 0 + g m = g m by rw [map_zero, zero_add]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_apply_inr
| null |
lift_comp_inlHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) :
(lift f g hg hfg hgf).comp (inlAlgHom S R M) = f :=
AlgHom.ext <| lift_apply_inl f g hg hfg hgf
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_comp_inlHom
| null |
lift_comp_inrHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) :
(lift f g hg hfg hgf).toLinearMap.comp (inrHom R M |>.restrictScalars S) = g :=
LinearMap.ext <| lift_apply_inr f g hg hfg hgf
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_comp_inrHom
| null |
@[simp]
lift_inlAlgHom_inrHom :
lift (inlAlgHom _ _ _) (inrHom R M |>.restrictScalars S)
(inr_mul_inr R) (fun _ _ => (inl_mul_inr _ _).symm) (fun _ _ => (inr_mul_inl _ _).symm) =
AlgHom.id S (tsze R M) :=
algHom_ext' (lift_comp_inlHom _ _ _ _ _) (lift_comp_inrHom _ _ _ _ _)
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
lift_inlAlgHom_inrHom
|
When applied to `inr` and `inl` themselves, `lift` is the identity.
|
range_inlAlgHom_sup_adjoin_range_inr :
(inlAlgHom S R M).range ⊔ Algebra.adjoin S (Set.range inr) = (⊤ : Subalgebra S (tsze R M)) := by
refine top_unique fun x hx => ?_; clear hx
rw [← x.inl_fst_add_inr_snd_eq]
refine add_mem ?_ ?_
· exact le_sup_left (α := Subalgebra S _) <| Set.mem_range_self x.fst
· exact le_sup_right (α := Subalgebra S _) <| Algebra.subset_adjoin <| Set.mem_range_self x.snd
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
range_inlAlgHom_sup_adjoin_range_inr
| null |
range_liftAux (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) :
(lift f g hg hfg hgf).range = f.range ⊔ Algebra.adjoin S (Set.range g) := by
simp_rw [← Algebra.map_top, ← range_inlAlgHom_sup_adjoin_range_inr, Algebra.map_sup,
AlgHom.map_adjoin, ← AlgHom.range_comp, lift_comp_inlHom, ← Set.range_comp, Function.comp_def,
lift_apply_inr, Algebra.map_top]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
range_liftAux
| null |
@[simps! apply symm_apply_coe]
liftEquiv :
{fg : (R →ₐ[S] A) × (M →ₗ[S] A) //
(∀ x y, fg.2 x * fg.2 y = 0) ∧
(∀ r x, fg.2 (r •> x) = fg.1 r * fg.2 x) ∧
(∀ r x, fg.2 (x <• r) = fg.2 x * fg.1 r)} ≃ (tsze R M →ₐ[S] A) where
toFun fg := lift fg.val.1 fg.val.2 fg.prop.1 fg.prop.2.1 fg.prop.2.2
invFun F :=
⟨(F.comp (inlAlgHom _ _ _), F.toLinearMap ∘ₗ (inrHom _ _ |>.restrictScalars _)),
(fun _x _y =>
(map_mul F _ _).symm.trans <| (F.congr_arg <| inr_mul_inr _ _ _).trans (map_zero F)),
(fun _r _x => (F.congr_arg (inl_mul_inr _ _).symm).trans (map_mul F _ _)),
(fun _r _x => (F.congr_arg (inr_mul_inl _ _).symm).trans (map_mul F _ _))⟩
left_inv _f := Subtype.ext <| Prod.ext (lift_comp_inlHom _ _ _ _ _) (lift_comp_inrHom _ _ _ _ _)
right_inv _F := algHom_ext' (lift_comp_inlHom _ _ _ _ _) (lift_comp_inrHom _ _ _ _ _)
|
def
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
liftEquiv
|
A universal property of the trivial square-zero extension, providing a unique
`TrivSqZeroExt R M →ₐ[R] A` for every pair of maps `f : R →ₐ[S] A` and `g : M →ₗ[S] A`,
where the range of `g` has no non-zero products, and scaling the input to `g` on the left or right
amounts to a corresponding multiplication by `f` in the output.
This isomorphism is named to match the very similar `Complex.lift`.
|
@[simps! apply symm_apply_coe]
liftEquivOfComm :
{ f : M →ₗ[R'] A // ∀ x y, f x * f y = 0 } ≃ (tsze R' M →ₐ[R'] A) := by
refine Equiv.trans ?_ liftEquiv
exact {
toFun := fun f => ⟨(Algebra.ofId _ _, f.val), f.prop,
fun r x => by simp [Algebra.smul_def, Algebra.ofId_apply],
fun r x => by simp [Algebra.smul_def, Algebra.ofId_apply, Algebra.commutes]⟩
invFun := fun fg => ⟨fg.val.2, fg.prop.1⟩ }
|
def
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
liftEquivOfComm
|
A simplified version of `TrivSqZeroExt.liftEquiv` for the commutative case.
|
map (f : M →ₗ[R'] N) : TrivSqZeroExt R' M →ₐ[R'] TrivSqZeroExt R' N :=
liftEquivOfComm ⟨inrHom R' N ∘ₗ f, fun _ _ => inr_mul_inr _ _ _⟩
@[simp]
|
def
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map
|
Functoriality of `TrivSqZeroExt` when the ring is commutative: a linear map
`f : M →ₗ[R'] N` induces a morphism of `R'`-algebras from `TrivSqZeroExt R' M` to
`TrivSqZeroExt R' N`.
Note that we cannot neatly state the non-commutative case, as we do not have morphisms of bimodules.
|
map_inl (f : M →ₗ[R'] N) (r : R') : map f (inl r) = inl r := by
rw [map, liftEquivOfComm_apply, lift_apply_inl, Algebra.ofId_apply, algebraMap_eq_inl]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_inl
| null |
map_inr (f : M →ₗ[R'] N) (x : M) : map f (inr x) = inr (f x) := by
rw [map, liftEquivOfComm_apply, lift_apply_inr, LinearMap.comp_apply, inrHom_apply]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_inr
| null |
fst_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : fst (map f x) = fst x := by
simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
fst_map
| null |
snd_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : snd (map f x) = f (snd x) := by
simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
snd_map
| null |
map_comp_inlAlgHom (f : M →ₗ[R'] N) :
(map f).comp (inlAlgHom R' R' M) = inlAlgHom R' R' N :=
AlgHom.ext <| map_inl _
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_comp_inlAlgHom
| null |
map_comp_inrHom (f : M →ₗ[R'] N) :
(map f).toLinearMap ∘ₗ inrHom R' M = inrHom R' N ∘ₗ f :=
LinearMap.ext <| map_inr _
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_comp_inrHom
| null |
fstHom_comp_map (f : M →ₗ[R'] N) :
(fstHom R' R' N).comp (map f) = fstHom R' R' M :=
AlgHom.ext <| fst_map _
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
fstHom_comp_map
| null |
sndHom_comp_map (f : M →ₗ[R'] N) :
sndHom R' N ∘ₗ (map f).toLinearMap = f ∘ₗ sndHom R' M :=
LinearMap.ext <| snd_map _
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
sndHom_comp_map
| null |
map_id : map (LinearMap.id : M →ₗ[R'] M) = AlgHom.id R' _ := by
apply algHom_ext
simp only [map_inr, LinearMap.id_coe, id_eq, AlgHom.coe_id, forall_const]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_id
| null |
map_comp_map (f : M →ₗ[R'] N) (g : N →ₗ[R'] P) :
map (g.comp f) = (map g).comp (map f) := by
apply algHom_ext
simp only [map_inr, LinearMap.coe_comp, Function.comp_apply, AlgHom.coe_comp, forall_const]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.Algebra.GroupWithZero.Invertible",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice"
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
map_comp_map
| null |
AffineScheme :=
Scheme.Spec.EssImageSubcategory
deriving Category
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
AffineScheme
|
The category of affine schemes
|
IsAffine (X : Scheme) : Prop where
affine : IsIso X.toSpecΓ
attribute [instance] IsAffine.affine
|
class
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
IsAffine
|
A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism.
|
@[simps! -isSimp hom]
Scheme.isoSpec (X : Scheme) [IsAffine X] : X ≅ Spec Γ(X, ⊤) :=
asIso X.toSpecΓ
@[reassoc]
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isoSpec
|
The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme.
|
Scheme.isoSpec_hom_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
X.isoSpec.hom ≫ Spec.map (f.appTop) = f ≫ Y.isoSpec.hom := by
simp only [isoSpec, asIso_hom, Scheme.toSpecΓ_naturality]
@[reassoc]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isoSpec_hom_naturality
| null |
Scheme.isoSpec_inv_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
Spec.map (f.appTop) ≫ Y.isoSpec.inv = X.isoSpec.inv ≫ f := by
rw [Iso.eq_inv_comp, isoSpec, asIso_hom, ← Scheme.toSpecΓ_naturality_assoc, isoSpec,
asIso_inv, IsIso.hom_inv_id, Category.comp_id]
@[reassoc (attr := simp)]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isoSpec_inv_naturality
| null |
Scheme.toSpecΓ_isoSpec_inv (X : Scheme.{u}) [IsAffine X] :
X.toSpecΓ ≫ X.isoSpec.inv = 𝟙 _ :=
X.isoSpec.hom_inv_id
@[reassoc (attr := simp)]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.toSpecΓ_isoSpec_inv
| null |
Scheme.isoSpec_inv_toSpecΓ (X : Scheme.{u}) [IsAffine X] :
X.isoSpec.inv ≫ X.toSpecΓ = 𝟙 _ :=
X.isoSpec.inv_hom_id
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isoSpec_inv_toSpecΓ
| null |
@[simps]
AffineScheme.mk (X : Scheme) (_ : IsAffine X) : AffineScheme :=
⟨X, ΓSpec.adjunction.mem_essImage_of_unit_isIso _⟩
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
AffineScheme.mk
|
Construct an affine scheme from a scheme and the information that it is affine.
Also see `AffineScheme.of` for a typeclass version.
|
AffineScheme.of (X : Scheme) [h : IsAffine X] : AffineScheme :=
AffineScheme.mk X h
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
AffineScheme.of
|
Construct an affine scheme from a scheme. Also see `AffineScheme.mk` for a non-typeclass
version.
|
AffineScheme.ofHom {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
AffineScheme.of X ⟶ AffineScheme.of Y :=
f
@[simp]
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
AffineScheme.ofHom
|
Type check a morphism of schemes as a morphism in `AffineScheme`.
|
essImage_Spec {X : Scheme} : Scheme.Spec.essImage X ↔ IsAffine X :=
⟨fun h => ⟨Functor.essImage.unit_isIso h⟩,
fun _ => ΓSpec.adjunction.mem_essImage_of_unit_isIso _⟩
@[deprecated (since := "2025-04-08")] alias mem_Spec_essImage := essImage_Spec
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
essImage_Spec
| null |
isAffine_affineScheme (X : AffineScheme.{u}) : IsAffine X.obj :=
⟨Functor.essImage.unit_isIso X.property⟩
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isAffine_affineScheme
| null |
isAffine_Spec (R : CommRingCat) : IsAffine (Spec R) :=
AlgebraicGeometry.isAffine_affineScheme ⟨_, Scheme.Spec.obj_mem_essImage (op R)⟩
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isAffine_Spec
| null |
IsAffine.of_isIso {X Y : Scheme} (f : X ⟶ Y) [IsIso f] [h : IsAffine Y] : IsAffine X := by
rw [← essImage_Spec] at h ⊢; exact Functor.essImage.ofIso (asIso f).symm h
@[deprecated (since := "2025-03-31")] alias isAffine_of_isIso := IsAffine.of_isIso
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
IsAffine.of_isIso
| null |
noncomputable
arrowIsoSpecΓOfIsAffine {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
Arrow.mk f ≅ Arrow.mk (Spec.map f.appTop) :=
Arrow.isoMk X.isoSpec Y.isoSpec (ΓSpec.adjunction.unit_naturality _)
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
arrowIsoSpecΓOfIsAffine
|
If `f : X ⟶ Y` is a morphism between affine schemes, the corresponding arrow is isomorphic
to the arrow of the morphism on prime spectra induced by the map on global sections.
|
arrowIsoΓSpecOfIsAffine {A B : CommRingCat} (f : A ⟶ B) :
Arrow.mk f ≅ Arrow.mk ((Spec.map f).appTop) :=
Arrow.isoMk (Scheme.ΓSpecIso _).symm (Scheme.ΓSpecIso _).symm
(Scheme.ΓSpecIso_inv_naturality f).symm
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
arrowIsoΓSpecOfIsAffine
|
If `f : A ⟶ B` is a ring homomorphism, the corresponding arrow is isomorphic
to the arrow of the morphism induced on global sections by the map on prime spectra.
|
Scheme.isoSpec_Spec (R : CommRingCat.{u}) :
(Spec R).isoSpec = Scheme.Spec.mapIso (Scheme.ΓSpecIso R).op :=
Iso.ext (SpecMap_ΓSpecIso_hom R).symm
@[simp] theorem Scheme.isoSpec_Spec_hom (R : CommRingCat.{u}) :
(Spec R).isoSpec.hom = Spec.map (Scheme.ΓSpecIso R).hom :=
(SpecMap_ΓSpecIso_hom R).symm
@[simp] theorem Scheme.isoSpec_Spec_inv (R : CommRingCat.{u}) :
(Spec R).isoSpec.inv = Spec.map (Scheme.ΓSpecIso R).inv :=
congr($(isoSpec_Spec R).inv)
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isoSpec_Spec
| null |
ext_of_isAffine {X Y : Scheme} [IsAffine Y] {f g : X ⟶ Y} (e : f.appTop = g.appTop) :
f = g := by
rw [← cancel_mono Y.toSpecΓ, Scheme.toSpecΓ_naturality, Scheme.toSpecΓ_naturality, e]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
ext_of_isAffine
| null |
Spec : CommRingCatᵒᵖ ⥤ AffineScheme :=
Scheme.Spec.toEssImage
/-! We copy over instances from `Scheme.Spec.toEssImage`. -/
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Spec
|
The `Spec` functor into the category of affine schemes.
|
Spec_full : Spec.Full := Functor.Full.toEssImage _
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Spec_full
| null |
Spec_faithful : Spec.Faithful := Functor.Faithful.toEssImage _
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Spec_faithful
| null |
Spec_essSurj : Spec.EssSurj := Functor.EssSurj.toEssImage (F := _)
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Spec_essSurj
| null |
@[simps!]
forgetToScheme : AffineScheme ⥤ Scheme :=
Scheme.Spec.essImage.ι
/-! We copy over instances from `Scheme.Spec.essImageInclusion`. -/
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
forgetToScheme
|
The forgetful functor `AffineScheme ⥤ Scheme`.
|
forgetToScheme_full : forgetToScheme.Full :=
inferInstanceAs Scheme.Spec.essImage.ι.Full
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
forgetToScheme_full
| null |
forgetToScheme_faithful : forgetToScheme.Faithful :=
inferInstanceAs Scheme.Spec.essImage.ι.Faithful
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
forgetToScheme_faithful
| null |
Γ : AffineSchemeᵒᵖ ⥤ CommRingCat :=
forgetToScheme.op ⋙ Scheme.Γ
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Γ
|
The global section functor of an affine scheme.
|
equivCommRingCat : AffineScheme ≌ CommRingCatᵒᵖ :=
equivEssImageOfReflective.symm
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
equivCommRingCat
|
The category of affine schemes is equivalent to the category of commutative rings.
|
ΓIsEquiv : Γ.{u}.IsEquivalence :=
inferInstanceAs (Γ.{u}.rightOp.op ⋙ (opOpEquivalence _).functor).IsEquivalence
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
ΓIsEquiv
| null |
hasColimits : HasColimits AffineScheme.{u} :=
haveI := Adjunction.has_limits_of_equivalence.{u} Γ.{u}
Adjunction.has_colimits_of_equivalence.{u} (opOpEquivalence AffineScheme.{u}).inverse
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
hasColimits
| null |
hasLimits : HasLimits AffineScheme.{u} := by
haveI := Adjunction.has_colimits_of_equivalence Γ.{u}
haveI : HasLimits AffineScheme.{u}ᵒᵖᵒᵖ := Limits.hasLimits_op_of_hasColimits
exact Adjunction.has_limits_of_equivalence (opOpEquivalence AffineScheme.{u}).inverse
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
hasLimits
| null |
noncomputable Γ_preservesLimits : PreservesLimits Γ.{u}.rightOp := inferInstance
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Γ_preservesLimits
| null |
noncomputable forgetToScheme_preservesLimits : PreservesLimits forgetToScheme := by
apply (config := { allowSynthFailures := true })
@preservesLimits_of_natIso _ _ _ _ _ _
(Functor.isoWhiskerRight equivCommRingCat.unitIso forgetToScheme).symm
change PreservesLimits (equivCommRingCat.functor ⋙ Scheme.Spec)
infer_instance
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
forgetToScheme_preservesLimits
| null |
IsAffineOpen {X : Scheme} (U : X.Opens) : Prop :=
IsAffine U
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
IsAffineOpen
|
An open subset of a scheme is affine if the open subscheme is affine.
|
Scheme.affineOpens (X : Scheme) : Set X.Opens :=
{U : X.Opens | IsAffineOpen U}
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.affineOpens
|
The set of affine opens as a subset of `opens X`.
|
isAffineOpen_opensRange {X Y : Scheme} [IsAffine X] (f : X ⟶ Y)
[H : IsOpenImmersion f] : IsAffineOpen (Scheme.Hom.opensRange f) := by
refine .of_isIso (IsOpenImmersion.isoOfRangeEq f (Y.ofRestrict _) ?_).inv
exact Subtype.range_val.symm
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isAffineOpen_opensRange
| null |
isAffineOpen_top (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : X.Opens) := by
convert isAffineOpen_opensRange (𝟙 X)
ext1
exact Set.range_id.symm
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isAffineOpen_top
| null |
exists_isAffineOpen_mem_and_subset {X : Scheme.{u}} {x : X}
{U : X.Opens} (hxU : x ∈ U) : ∃ W : X.Opens, IsAffineOpen W ∧ x ∈ W ∧ W.1 ⊆ U := by
obtain ⟨R, f, hf⟩ := AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset hxU
exact ⟨Scheme.Hom.opensRange f (H := hf.1),
⟨AlgebraicGeometry.isAffineOpen_opensRange f (H := hf.1), hf.2.1, hf.2.2⟩⟩
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
exists_isAffineOpen_mem_and_subset
| null |
Scheme.isAffine_affineCover (X : Scheme) (i : X.affineCover.I₀) :
IsAffine (X.affineCover.X i) :=
isAffine_Spec _
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isAffine_affineCover
| null |
Scheme.isAffine_affineBasisCover (X : Scheme) (i : X.affineBasisCover.I₀) :
IsAffine (X.affineBasisCover.X i) :=
isAffine_Spec _
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isAffine_affineBasisCover
| null |
Scheme.isAffine_affineOpenCover (X : Scheme) (𝒰 : X.AffineOpenCover) (i : 𝒰.I₀) :
IsAffine (𝒰.openCover.X i) :=
inferInstanceAs (IsAffine (Spec (𝒰.X i)))
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.isAffine_affineOpenCover
| null |
isBasis_affine_open (X : Scheme) : Opens.IsBasis X.affineOpens := by
rw [Opens.isBasis_iff_nbhd]
rintro U x (hU : x ∈ (U : Set X))
obtain ⟨S, hS, hxS, hSU⟩ := X.affineBasisCover_is_basis.exists_subset_of_mem_open hU U.isOpen
refine ⟨⟨S, X.affineBasisCover_is_basis.isOpen hS⟩, ?_, hxS, hSU⟩
rcases hS with ⟨i, rfl⟩
exact isAffineOpen_opensRange _
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isBasis_affine_open
| null |
iSup_affineOpens_eq_top (X : Scheme) : ⨆ i : X.affineOpens, (i : X.Opens) = ⊤ := by
apply Opens.ext
rw [Opens.coe_iSup]
apply IsTopologicalBasis.sUnion_eq
rw [← Set.image_eq_range]
exact isBasis_affine_open X
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
iSup_affineOpens_eq_top
| null |
Scheme.map_PrimeSpectrum_basicOpen_of_affine
(X : Scheme) [IsAffine X] (f : Γ(X, ⊤)) :
X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f :=
Scheme.toSpecΓ_preimage_basicOpen _ _
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.map_PrimeSpectrum_basicOpen_of_affine
| null |
isBasis_basicOpen (X : Scheme) [IsAffine X] :
Opens.IsBasis (Set.range (X.basicOpen : Γ(X, ⊤) → X.Opens)) := by
convert PrimeSpectrum.isBasis_basic_opens.of_isInducing
(TopCat.homeoOfIso (Scheme.forgetToTop.mapIso X.isoSpec)).isInducing using 1
ext V
simp only [Set.mem_range, exists_exists_eq_and, Set.mem_setOf,
← Opens.coe_inj (V := V), ← Scheme.toSpecΓ_preimage_basicOpen]
rfl
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isBasis_basicOpen
| null |
noncomputable
Scheme.Opens.toSpecΓ {X : Scheme.{u}} (U : X.Opens) :
U.toScheme ⟶ Spec Γ(X, U) :=
U.toScheme.toSpecΓ ≫ Spec.map U.topIso.inv
@[reassoc (attr := simp)]
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.Opens.toSpecΓ
|
The canonical map `U ⟶ Spec Γ(X, U)` for an open `U ⊆ X`.
|
Scheme.Opens.toSpecΓ_SpecMap_map {X : Scheme} (U V : X.Opens) (h : U ≤ V) :
U.toSpecΓ ≫ Spec.map (X.presheaf.map (homOfLE h).op) = X.homOfLE h ≫ V.toSpecΓ := by
delta Scheme.Opens.toSpecΓ
simp [← Spec.map_comp, ← X.presheaf.map_comp, toSpecΓ_naturality_assoc]
@[simp]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.Opens.toSpecΓ_SpecMap_map
| null |
Scheme.Opens.toSpecΓ_top {X : Scheme} :
(⊤ : X.Opens).toSpecΓ = (⊤ : X.Opens).ι ≫ X.toSpecΓ := by
simp [Scheme.Opens.toSpecΓ, toSpecΓ_naturality]; rfl
@[reassoc]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.Opens.toSpecΓ_top
| null |
Scheme.Opens.toSpecΓ_appTop {X : Scheme.{u}} (U : X.Opens) :
U.toSpecΓ.appTop = (Scheme.ΓSpecIso Γ(X, U)).hom ≫ U.topIso.inv := by
simp [Scheme.Opens.toSpecΓ]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Scheme.Opens.toSpecΓ_appTop
| null |
@[simps! -isSimp inv]
isoSpec :
↑U ≅ Spec Γ(X, U) :=
haveI : IsAffine U := hU
U.toScheme.isoSpec ≪≫ Scheme.Spec.mapIso U.topIso.symm.op
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec
|
The isomorphism `U ≅ Spec Γ(X, U)` for an affine `U`.
|
isoSpec_hom : hU.isoSpec.hom = U.toSpecΓ := rfl
@[reassoc (attr := simp)]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_hom
| null |
toSpecΓ_isoSpec_inv : U.toSpecΓ ≫ hU.isoSpec.inv = 𝟙 _ := hU.isoSpec.hom_inv_id
@[reassoc (attr := simp)]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
toSpecΓ_isoSpec_inv
| null |
isoSpec_inv_toSpecΓ : hU.isoSpec.inv ≫ U.toSpecΓ = 𝟙 _ := hU.isoSpec.inv_hom_id
open IsLocalRing in
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_inv_toSpecΓ
| null |
isoSpec_hom_base_apply (x : U) :
hU.isoSpec.hom.base x = (Spec.map (X.presheaf.germ U x x.2)).base (closedPoint _) := by
dsimp [IsAffineOpen.isoSpec_hom, Scheme.isoSpec_hom, Scheme.toSpecΓ_base, Scheme.Opens.toSpecΓ]
rw [← Scheme.comp_base_apply, ← Spec.map_comp,
(Iso.eq_comp_inv _).mpr (Scheme.Opens.germ_stalkIso_hom U (V := ⊤) x trivial),
X.presheaf.germ_res_assoc, Spec.map_comp, Scheme.comp_base_apply]
congr 1
exact IsLocalRing.comap_closedPoint (U.stalkIso x).inv.hom
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_hom_base_apply
| null |
isoSpec_inv_appTop :
hU.isoSpec.inv.appTop = U.topIso.hom ≫ (Scheme.ΓSpecIso Γ(X, U)).inv := by
simp_rw [Scheme.Opens.toScheme_presheaf_obj, isoSpec_inv, Scheme.isoSpec, asIso_inv,
Scheme.comp_app, Scheme.Opens.topIso_hom, Scheme.ΓSpecIso_inv_naturality,
Scheme.inv_appTop, -- `check_compositions` reports defeq problems starting after this step.
IsIso.inv_comp_eq]
rw [Scheme.toSpecΓ_appTop]
erw [Iso.hom_inv_id_assoc]
simp only [Opens.map_top]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_inv_appTop
| null |
isoSpec_hom_appTop :
hU.isoSpec.hom.appTop = (Scheme.ΓSpecIso Γ(X, U)).hom ≫ U.topIso.inv := by
have := congr(inv $hU.isoSpec_inv_appTop)
rw [IsIso.inv_comp, IsIso.Iso.inv_inv, IsIso.Iso.inv_hom] at this
have := (Scheme.Γ.map_inv hU.isoSpec.inv.op).trans this
rwa [← op_inv, IsIso.Iso.inv_inv] at this
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_hom_appTop
| null |
fromSpec :
Spec Γ(X, U) ⟶ X :=
haveI : IsAffine U := hU
hU.isoSpec.inv ≫ U.ι
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec
|
The open immersion `Spec Γ(X, U) ⟶ X` for an affine `U`.
|
isOpenImmersion_fromSpec :
IsOpenImmersion hU.fromSpec := by
delta fromSpec
infer_instance
@[reassoc (attr := simp)]
|
instance
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isOpenImmersion_fromSpec
| null |
isoSpec_inv_ι : hU.isoSpec.inv ≫ U.ι = hU.fromSpec := rfl
@[reassoc (attr := simp)]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isoSpec_inv_ι
| null |
toSpecΓ_fromSpec : U.toSpecΓ ≫ hU.fromSpec = U.ι := toSpecΓ_isoSpec_inv_assoc _ _
@[simp]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
toSpecΓ_fromSpec
| null |
range_fromSpec :
Set.range hU.fromSpec.base = (U : Set X) := by
delta IsAffineOpen.fromSpec; dsimp [IsAffineOpen.isoSpec_inv]
rw [Set.range_comp, Set.range_eq_univ.mpr, Set.image_univ]
· exact Subtype.range_coe
rw [← TopCat.coe_comp, ← TopCat.epi_iff_surjective]
infer_instance
@[simp]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
range_fromSpec
| null |
opensRange_fromSpec : hU.fromSpec.opensRange = U := Opens.ext (range_fromSpec hU)
@[reassoc (attr := simp)]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
opensRange_fromSpec
| null |
map_fromSpec {V : X.Opens} (hV : IsAffineOpen V) (f : op U ⟶ op V) :
Spec.map (X.presheaf.map f) ≫ hU.fromSpec = hV.fromSpec := by
have : IsAffine U := hU
haveI : IsAffine _ := hV
conv_rhs =>
rw [fromSpec, ← X.homOfLE_ι (V := U) f.unop.le, isoSpec_inv, Category.assoc,
← Scheme.isoSpec_inv_naturality_assoc,
← Spec.map_comp_assoc, Scheme.homOfLE_appTop, ← Functor.map_comp]
rw [fromSpec, isoSpec_inv, Category.assoc, ← Spec.map_comp_assoc, ← Functor.map_comp]
rfl
@[reassoc]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
map_fromSpec
| null |
Spec_map_appLE_fromSpec (f : X ⟶ Y) {V : X.Opens} {U : Y.Opens}
(hU : IsAffineOpen U) (hV : IsAffineOpen V) (i : V ≤ f ⁻¹ᵁ U) :
Spec.map (f.appLE U V i) ≫ hU.fromSpec = hV.fromSpec ≫ f := by
have : IsAffine U := hU
simp only [IsAffineOpen.fromSpec, Category.assoc, isoSpec_inv]
simp_rw [← Scheme.homOfLE_ι _ i]
rw [Category.assoc, ← morphismRestrict_ι,
← Category.assoc _ (f ∣_ U) U.ι, ← @Scheme.isoSpec_inv_naturality_assoc,
← Spec.map_comp_assoc, ← Spec.map_comp_assoc, Scheme.comp_appTop, morphismRestrict_appTop,
Scheme.homOfLE_appTop, Scheme.Hom.app_eq_appLE, Scheme.Hom.appLE_map,
Scheme.Hom.appLE_map, Scheme.Hom.appLE_map, Scheme.Hom.map_appLE]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
Spec_map_appLE_fromSpec
| null |
fromSpec_top [IsAffine X] : (isAffineOpen_top X).fromSpec = X.isoSpec.inv := by
rw [fromSpec, isoSpec_inv, Category.assoc, ← @Scheme.isoSpec_inv_naturality,
← Spec.map_comp_assoc, Scheme.Opens.ι_appTop, ← X.presheaf.map_comp, ← op_comp,
eqToHom_comp_homOfLE, ← eqToHom_eq_homOfLE rfl, eqToHom_refl, op_id, X.presheaf.map_id,
Spec.map_id, Category.id_comp]
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_top
| null |
fromSpec_app_of_le (V : X.Opens) (h : U ≤ V) :
hU.fromSpec.app V = X.presheaf.map (homOfLE h).op ≫
(Scheme.ΓSpecIso Γ(X, U)).inv ≫ (Spec _).presheaf.map (homOfLE le_top).op := by
have : U.ι ⁻¹ᵁ V = ⊤ := eq_top_iff.mpr fun x _ ↦ h x.2
rw [IsAffineOpen.fromSpec, Scheme.comp_app, Scheme.Opens.ι_app, Scheme.app_eq _ this,
← Scheme.Hom.appTop, IsAffineOpen.isoSpec_inv_appTop]
simp only [Scheme.Opens.toScheme_presheaf_map, Scheme.Opens.topIso_hom,
Category.assoc, ← X.presheaf.map_comp_assoc]
rfl
include hU in
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_app_of_le
| null |
protected isCompact :
IsCompact (U : Set X) := by
convert @IsCompact.image _ _ _ _ Set.univ hU.fromSpec.base PrimeSpectrum.compactSpace.1
(by fun_prop)
convert hU.range_fromSpec.symm
exact Set.image_univ
include hU in
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
isCompact
| null |
image_of_isOpenImmersion (f : X ⟶ Y) [H : IsOpenImmersion f] :
IsAffineOpen (f ''ᵁ U) := by
have : IsAffine _ := hU
convert isAffineOpen_opensRange (U.ι ≫ f)
ext1
exact Set.image_eq_range _ _
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
image_of_isOpenImmersion
| null |
preimage_of_isIso {U : Y.Opens} (hU : IsAffineOpen U) (f : X ⟶ Y) [IsIso f] :
IsAffineOpen (f ⁻¹ᵁ U) :=
haveI : IsAffine _ := hU
.of_isIso (f ∣_ U)
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
preimage_of_isIso
| null |
_root_.AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
(f : AlgebraicGeometry.Scheme.Hom X Y) [H : IsOpenImmersion f] {U : X.Opens} :
IsAffineOpen (f ''ᵁ U) ↔ IsAffineOpen U where
mp hU := by
refine .of_isIso (IsOpenImmersion.isoOfRangeEq (X.ofRestrict U.isOpenEmbedding ≫ f)
(Y.ofRestrict _) ?_).hom (h := hU)
rw [Scheme.comp_base, TopCat.coe_comp, Set.range_comp]
dsimp [Opens.coe_inclusion', Scheme.restrict]
rw [Subtype.range_coe, Subtype.range_coe]
rfl
mpr hU := hU.image_of_isOpenImmersion f
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
_root_.AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
| null |
@[simps]
_root_.AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv (f : X ⟶ Y) [H : IsOpenImmersion f] :
X.affineOpens ≃ { U : Y.affineOpens // U ≤ f.opensRange } where
toFun U := ⟨⟨f ''ᵁ U, U.2.image_of_isOpenImmersion f⟩, Set.image_subset_range _ _⟩
invFun U := ⟨f ⁻¹ᵁ U, f.isAffineOpen_iff_of_isOpenImmersion.mp (by
rw [show f ''ᵁ f ⁻¹ᵁ U = U from Opens.ext (Set.image_preimage_eq_of_subset U.2)]; exact U.1.2)⟩
left_inv _ := Subtype.ext (Opens.ext (Set.preimage_image_eq _ H.base_open.injective))
right_inv U := Subtype.ext (Subtype.ext (Opens.ext (Set.image_preimage_eq_of_subset U.2)))
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
_root_.AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv
|
The affine open sets of an open subscheme corresponds to
the affine open sets containing in the image.
|
@[simps! apply_coe_coe]
_root_.AlgebraicGeometry.affineOpensRestrict {X : Scheme.{u}} (U : X.Opens) :
U.toScheme.affineOpens ≃ { V : X.affineOpens // V ≤ U } :=
(IsOpenImmersion.affineOpensEquiv U.ι).trans (Equiv.subtypeEquivProp (by simp))
@[simp]
|
def
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
_root_.AlgebraicGeometry.affineOpensRestrict
|
The affine open sets of an open subscheme
corresponds to the affine open sets containing in the subset.
|
_root_.AlgebraicGeometry.affineOpensRestrict_symm_apply_coe
{X : Scheme.{u}} (U : X.Opens) (V) :
((affineOpensRestrict U).symm V).1 = U.ι ⁻¹ᵁ V := rfl
|
lemma
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
_root_.AlgebraicGeometry.affineOpensRestrict_symm_apply_coe
| null |
@[simp]
fromSpec_preimage_self :
hU.fromSpec ⁻¹ᵁ U = ⊤ := by
ext1
rw [Opens.map_coe, Opens.coe_top, ← hU.range_fromSpec, ← Set.image_univ]
exact Set.preimage_image_eq _ PresheafedSpace.IsOpenImmersion.base_open.injective
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_preimage_self
| null |
ΓSpecIso_hom_fromSpec_app :
(Scheme.ΓSpecIso Γ(X, U)).hom ≫ hU.fromSpec.app U =
(Spec Γ(X, U)).presheaf.map (eqToHom hU.fromSpec_preimage_self).op := by
simp only [fromSpec, Scheme.comp_coeBase, Opens.map_comp_obj, Scheme.comp_app,
Scheme.Opens.ι_app_self, eqToHom_op, Scheme.app_eq _ U.ι_preimage_self,
Scheme.Opens.toScheme_presheaf_map, eqToHom_unop, eqToHom_map U.ι.opensFunctor, Opens.map_top,
isoSpec_inv_appTop, Scheme.Opens.topIso_hom, Category.assoc, ← Functor.map_comp_assoc,
eqToHom_trans, eqToHom_refl, X.presheaf.map_id, Category.id_comp, Iso.hom_inv_id_assoc]
@[elementwise]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
ΓSpecIso_hom_fromSpec_app
| null |
fromSpec_app_self :
hU.fromSpec.app U = (Scheme.ΓSpecIso Γ(X, U)).inv ≫
(Spec Γ(X, U)).presheaf.map (eqToHom hU.fromSpec_preimage_self).op := by
rw [← hU.ΓSpecIso_hom_fromSpec_app, Iso.inv_hom_id_assoc]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_app_self
| null |
fromSpec_preimage_basicOpen' :
hU.fromSpec ⁻¹ᵁ X.basicOpen f = (Spec Γ(X, U)).basicOpen ((Scheme.ΓSpecIso Γ(X, U)).inv f) := by
rw [Scheme.preimage_basicOpen, hU.fromSpec_app_self]
exact Scheme.basicOpen_res_eq _ _ (eqToHom hU.fromSpec_preimage_self).op
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_preimage_basicOpen'
| null |
fromSpec_preimage_basicOpen :
hU.fromSpec ⁻¹ᵁ X.basicOpen f = PrimeSpectrum.basicOpen f := by
rw [fromSpec_preimage_basicOpen', ← basicOpen_eq_of_affine]
|
theorem
|
AlgebraicGeometry
|
[
"Mathlib.AlgebraicGeometry.Cover.Open",
"Mathlib.AlgebraicGeometry.GammaSpecAdjunction",
"Mathlib.AlgebraicGeometry.Restrict",
"Mathlib.CategoryTheory.Limits.Opposites",
"Mathlib.RingTheory.Localization.InvSubmonoid",
"Mathlib.RingTheory.LocalProperties.Basic",
"Mathlib.Topology.Sheaves.CommRingCat"
] |
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
fromSpec_preimage_basicOpen
| null |
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