fact
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stringclasses 32
values | imports
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| filename
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fromSpec_image_basicOpen :
hU.fromSpec ''ᵁ (PrimeSpectrum.basicOpen f) = X.basicOpen f := by
rw [← hU.fromSpec_preimage_basicOpen]
ext1
change hU.fromSpec.base '' (hU.fromSpec.base ⁻¹' (X.basicOpen f : Set X)) = _
rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left, hU.range_fromSpec]
exact Scheme.basicOpen_le _ _
@[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
|
fromSpec_image_basicOpen
| null |
basicOpen_fromSpec_app :
(Spec Γ(X, U)).basicOpen (hU.fromSpec.app U f) = PrimeSpectrum.basicOpen f := by
rw [← hU.fromSpec_preimage_basicOpen, Scheme.preimage_basicOpen]
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
|
basicOpen_fromSpec_app
| null |
basicOpen :
IsAffineOpen (X.basicOpen f) := by
rw [← hU.fromSpec_image_basicOpen, Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion]
convert isAffineOpen_opensRange
(Spec.map (CommRingCat.ofHom <| algebraMap Γ(X, U) (Localization.Away f)))
exact Opens.ext (PrimeSpectrum.localization_away_comap_range (Localization.Away f) f).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
|
basicOpen
| null |
Spec_basicOpen {R : CommRingCat} (f : R) :
IsAffineOpen (X := Spec R) (PrimeSpectrum.basicOpen f) :=
basicOpen_eq_of_affine f ▸ (isAffineOpen_top Spec(R)).basicOpen _
|
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_basicOpen
| null |
ι_basicOpen_preimage (r : Γ(X, ⊤)) :
IsAffineOpen ((X.basicOpen r).ι ⁻¹ᵁ U) := by
apply (X.basicOpen r).ι.isAffineOpen_iff_of_isOpenImmersion.mp
dsimp [Scheme.Hom.opensFunctor, LocallyRingedSpace.IsOpenImmersion.opensFunctor]
rw [Opens.functor_obj_map_obj, Opens.isOpenEmbedding_obj_top, inf_comm,
← Scheme.basicOpen_res _ _ (homOfLE le_top).op]
exact hU.basicOpen _
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
|
ι_basicOpen_preimage
| null |
exists_basicOpen_le {V : X.Opens} (x : V) (h : ↑x ∈ U) :
∃ f : Γ(X, U), X.basicOpen f ≤ V ∧ ↑x ∈ X.basicOpen f := by
have : IsAffine _ := hU
obtain ⟨_, ⟨_, ⟨r, rfl⟩, rfl⟩, h₁, h₂⟩ :=
(isBasis_basicOpen U).exists_subset_of_mem_open (x.2 : (⟨x, h⟩ : U) ∈ _)
((Opens.map U.inclusion').obj V).isOpen
have :
U.ι ''ᵁ (U.toScheme.basicOpen r) =
X.basicOpen (X.presheaf.map (eqToHom U.isOpenEmbedding_obj_top.symm).op r) := by
refine (Scheme.image_basicOpen U.ι r).trans ?_
rw [Scheme.basicOpen_res_eq]
simp only [Scheme.Opens.toScheme_presheaf_obj, Scheme.Opens.ι_appIso, Iso.refl_inv,
CommRingCat.id_apply]
use X.presheaf.map (eqToHom U.isOpenEmbedding_obj_top.symm).op r
rw [← this]
exact ⟨Set.image_subset_iff.mpr h₂, ⟨_, h⟩, h₁, 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
|
exists_basicOpen_le
| null |
@[simp]
algebraMap_Spec_obj {R : CommRingCat} {U} : algebraMap R Γ(Spec R, U) =
((Scheme.ΓSpecIso R).inv ≫ (Spec R).presheaf.map (homOfLE le_top).op).hom := 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
|
algebraMap_Spec_obj
| null |
basicOpenSectionsToAffine :
Γ(X, X.basicOpen f) ⟶ Γ(Spec Γ(X, U), PrimeSpectrum.basicOpen f) :=
hU.fromSpec.c.app (op <| X.basicOpen f) ≫
(Spec Γ(X, U)).presheaf.map (eqToHom <| (hU.fromSpec_preimage_basicOpen f).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
|
basicOpenSectionsToAffine
|
Given an affine open U and some `f : U`,
this is the canonical map `Γ(𝒪ₓ, D(f)) ⟶ Γ(Spec 𝒪ₓ(U), D(f))`
This is an isomorphism, as witnessed by an `IsIso` instance.
|
basicOpenSectionsToAffine_isIso :
IsIso (basicOpenSectionsToAffine hU f) := by
delta basicOpenSectionsToAffine
refine IsIso.comp_isIso' ?_ inferInstance
apply PresheafedSpace.IsOpenImmersion.isIso_of_subset
rw [hU.range_fromSpec]
exact RingedSpace.basicOpen_le _ _
include hU in
|
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
|
basicOpenSectionsToAffine_isIso
| null |
isLocalization_basicOpen :
IsLocalization.Away f Γ(X, X.basicOpen f) := by
apply
(IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
(asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
congr 1
dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra, ← CommRingCat.hom_comp,
basicOpenSectionsToAffine]
rw [hU.fromSpec.naturality_assoc, hU.fromSpec_app_self]
simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
exact CommRingCat.hom_ext_iff.mp (StructureSheaf.toOpen_res _ _ _ _)
|
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
|
isLocalization_basicOpen
| null |
_root_.AlgebraicGeometry.isLocalization_away_of_isAffine
[IsAffine X] (r : Γ(X, ⊤)) :
IsLocalization.Away r Γ(X, X.basicOpen r) :=
isLocalization_basicOpen (isAffineOpen_top X) 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
|
_root_.AlgebraicGeometry.isLocalization_away_of_isAffine
| null |
appLE_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (hU : IsAffineOpen U)
{V : X.Opens} (hV : IsAffineOpen V) (e) (r : Γ(Y, U)) :
letI := hU.isLocalization_basicOpen r
letI := hV.isLocalization_basicOpen (f.appLE U V e r)
f.appLE (Y.basicOpen r) (X.basicOpen (f.appLE U V e r)) (by simp [Scheme.Hom.appLE]) =
CommRingCat.ofHom (IsLocalization.Away.map _ _ (f.appLE U V e).hom r) := by
letI := hU.isLocalization_basicOpen r
letI := hV.isLocalization_basicOpen (f.appLE U V e r)
ext : 1
apply IsLocalization.ringHom_ext (.powers r)
rw [IsLocalization.Away.map, CommRingCat.hom_ofHom, IsLocalization.map_comp,
RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra, ← CommRingCat.hom_comp,
← CommRingCat.hom_comp, 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
|
appLE_eq_away_map
| null |
app_basicOpen_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens}
(hU : IsAffineOpen U) (h : IsAffineOpen (f ⁻¹ᵁ U)) (r : Γ(Y, U)) :
haveI := hU.isLocalization_basicOpen r
haveI := h.isLocalization_basicOpen (f.app U r)
f.app (Y.basicOpen r) =
(CommRingCat.ofHom
(IsLocalization.Away.map Γ(Y, Y.basicOpen r) Γ(X, X.basicOpen (f.app U r)) (f.app U).hom r)
≫ X.presheaf.map (eqToHom (by simp)).op) := by
haveI := hU.isLocalization_basicOpen r
haveI := h.isLocalization_basicOpen (f.app U r)
ext : 1
apply IsLocalization.ringHom_ext (.powers r)
rw [IsLocalization.Away.map, CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom,
IsLocalization.map_comp, RingHom.algebraMap_toAlgebra,
RingHom.algebraMap_toAlgebra, ← RingHom.comp_assoc, ← CommRingCat.hom_comp,
← CommRingCat.hom_comp, ← X.presheaf.map_comp]
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
|
app_basicOpen_eq_away_map
| null |
appBasicOpenIsoAwayMap {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens}
(hU : IsAffineOpen U) (h : IsAffineOpen (f ⁻¹ᵁ U)) (r : Γ(Y, U)) :
haveI := hU.isLocalization_basicOpen r
haveI := h.isLocalization_basicOpen (f.app U r)
Arrow.mk (f.app (Y.basicOpen r)) ≅
Arrow.mk (CommRingCat.ofHom (IsLocalization.Away.map Γ(Y, Y.basicOpen r)
Γ(X, X.basicOpen (f.app U r)) (f.app U).hom r)) :=
Arrow.isoMk (Iso.refl _) (X.presheaf.mapIso (eqToIso (by simp)).op) <| by
simp [hU.app_basicOpen_eq_away_map f h]
include hU in
|
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
|
appBasicOpenIsoAwayMap
|
`f.app (Y.basicOpen r)` is isomorphic to map induced on localizations
`Γ(Y, Y.basicOpen r) ⟶ Γ(X, X.basicOpen (f.app U r))`
|
isLocalization_of_eq_basicOpen {V : X.Opens} (i : V ⟶ U) (e : V = X.basicOpen f) :
@IsLocalization.Away _ _ f Γ(X, V) _ (X.presheaf.map i.op).hom.toAlgebra := by
subst e; convert isLocalization_basicOpen hU f using 3
|
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
|
isLocalization_of_eq_basicOpen
| null |
_root_.AlgebraicGeometry.Γ_restrict_isLocalization
(X : Scheme.{u}) [IsAffine X] (r : Γ(X, ⊤)) :
IsLocalization.Away r Γ(X.basicOpen r, ⊤) :=
(isAffineOpen_top X).isLocalization_of_eq_basicOpen r _ (Opens.isOpenEmbedding_obj_top _)
include hU in
|
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
|
_root_.AlgebraicGeometry.Γ_restrict_isLocalization
| null |
basicOpen_basicOpen_is_basicOpen (g : Γ(X, X.basicOpen f)) :
∃ f' : Γ(X, U), X.basicOpen f' = X.basicOpen g := by
have := isLocalization_basicOpen hU f
obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
use f * x
rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra,
Scheme.basicOpen_res]
refine (inf_eq_left.mpr (inf_le_left.trans_eq (Scheme.basicOpen_of_isUnit _ ?_).symm)).symm
exact
Submonoid.leftInv_le_isUnit _
(IsLocalization.toInvSubmonoid (Submonoid.powers f) (Γ(X, X.basicOpen f))
_).prop
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
|
basicOpen_basicOpen_is_basicOpen
| null |
_root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
{V : X.Opens} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
∃ (f : Γ(X, U)) (g : Γ(X, V)), X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f := by
obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basicOpen_le ⟨x, hx.2⟩ hx.1
obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basicOpen_le ⟨x, hf₂⟩ hx.2
obtain ⟨f', hf'⟩ :=
basicOpen_basicOpen_is_basicOpen hU f (X.presheaf.map (homOfLE hf₁ : _ ⟶ V).op g)
replace hf' := (hf'.trans (RingedSpace.basicOpen_res _ _ _)).trans (inf_eq_right.mpr hg₁)
exact ⟨f', g, hf', hf'.symm ▸ hg₂⟩
|
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.exists_basicOpen_le_affine_inter
| null |
noncomputable primeIdealOf (x : U) :
PrimeSpectrum Γ(X, U) :=
hU.isoSpec.hom.base x
|
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
|
primeIdealOf
|
The prime ideal of `𝒪ₓ(U)` corresponding to a point `x : U`.
|
fromSpec_primeIdealOf (x : U) :
hU.fromSpec.base (hU.primeIdealOf x) = x.1 := by
dsimp only [IsAffineOpen.fromSpec, Subtype.coe_mk, IsAffineOpen.primeIdealOf]
rw [← Scheme.comp_base_apply, Iso.hom_inv_id_assoc]
rfl
open IsLocalRing 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
|
fromSpec_primeIdealOf
| null |
primeIdealOf_eq_map_closedPoint (x : U) :
hU.primeIdealOf x = (Spec.map (X.presheaf.germ _ x x.2)).base (closedPoint _) :=
hU.isoSpec_hom_base_apply _
|
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
|
primeIdealOf_eq_map_closedPoint
| null |
primeIdealOf_isMaximal_of_isClosed (x : U) (hx : IsClosed {(x : X)}) :
(hU.primeIdealOf x).asIdeal.IsMaximal := by
have hx₀ : IsClosed {x} := by
simpa [← Set.image_singleton, Set.preimage_image_eq _ Subtype.val_injective]
using hx.preimage U.isOpenEmbedding'.continuous
apply (hU.primeIdealOf x).isClosed_singleton_iff_isMaximal.mp
rw [primeIdealOf, ← Set.image_singleton]
refine (Topology.IsClosedEmbedding.isClosed_iff_image_isClosed <|
IsHomeomorph.isClosedEmbedding ?_).mp hx₀
apply (TopCat.isIso_iff_isHomeomorph _).mp
infer_instance
|
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
|
primeIdealOf_isMaximal_of_isClosed
|
If a point `x : U` is a closed point, then its corresponding prime ideal is maximal.
|
isLocalization_stalk' (y : PrimeSpectrum Γ(X, U)) (hy : hU.fromSpec.base y ∈ U) :
@IsLocalization.AtPrime
(R := Γ(X, U))
(S := X.presheaf.stalk <| hU.fromSpec.base y) _ _
((TopCat.Presheaf.algebra_section_stalk X.presheaf _)) y.asIdeal _ := by
apply
(@IsLocalization.isLocalization_iff_of_ringEquiv (R := Γ(X, U))
(S := X.presheaf.stalk (hU.fromSpec.base y)) _ y.asIdeal.primeCompl _
(TopCat.Presheaf.algebra_section_stalk X.presheaf ⟨hU.fromSpec.base y, hy⟩) _ _
(asIso <| hU.fromSpec.stalkMap y).commRingCatIsoToRingEquiv).mpr
convert StructureSheaf.IsLocalization.to_stalk Γ(X, U) y using 1
delta IsLocalization.AtPrime StructureSheaf.stalkAlgebra
rw [RingHom.algebraMap_toAlgebra, RingEquiv.toRingHom_eq_coe,
CategoryTheory.Iso.commRingCatIsoToRingEquiv_toRingHom, asIso_hom, ← CommRingCat.hom_comp,
Scheme.stalkMap_germ, IsAffineOpen.fromSpec_app_self, Category.assoc, TopCat.Presheaf.germ_res]
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
|
isLocalization_stalk'
| null |
isLocalization_stalk (x : U) :
IsLocalization.AtPrime (X.presheaf.stalk x) (hU.primeIdealOf x).asIdeal := by
rcases x with ⟨x, hx⟩
set y := hU.primeIdealOf ⟨x, hx⟩ with hy
have : hU.fromSpec.base y = x := hy ▸ hU.fromSpec_primeIdealOf ⟨x, hx⟩
clear_value y
subst this
exact hU.isLocalization_stalk' y hx
|
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
|
isLocalization_stalk
| null |
stalkMap_injective (f : X ⟶ Y) {U : Opens Y} (hU : IsAffineOpen U) (x : X)
(hx : f.base x ∈ U)
(h : ∀ g, f.stalkMap x (Y.presheaf.germ U (f.base x) hx g) = 0 →
Y.presheaf.germ U (f.base x) hx g = 0) :
Function.Injective (f.stalkMap x) := by
letI := Y.presheaf.algebra_section_stalk ⟨f.base x, hx⟩
apply (hU.isLocalization_stalk ⟨f.base x, hx⟩).injective_of_map_algebraMap_zero
exact h
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
|
stalkMap_injective
| null |
mem_ideal_iff {s : Γ(X, U)} {I : Ideal Γ(X, U)} :
s ∈ I ↔ ∀ (x : X) (h : x ∈ U), X.presheaf.germ U x h s ∈ I.map (X.presheaf.germ U x h).hom := by
refine ⟨fun hs x hxU ↦ Ideal.mem_map_of_mem _ hs, fun H ↦ ?_⟩
letI (x : _) : Algebra Γ(X, U) (X.presheaf.stalk (hU.fromSpec.base x)) :=
TopCat.Presheaf.algebra_section_stalk X.presheaf _
have (P : Ideal Γ(X, U)) [hP : P.IsPrime] : IsLocalization.AtPrime _ P :=
hU.isLocalization_stalk' ⟨P, hP⟩ (hU.isoSpec.inv.base _).2
refine Submodule.mem_of_localization_maximal
(fun P hP ↦ X.presheaf.stalk (hU.fromSpec.base ⟨P, hP.isPrime⟩))
(fun P hP ↦ Algebra.linearMap _ _) _ _ ?_
intro P hP
rw [Ideal.localized₀_eq_restrictScalars_map]
exact H _ _
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
|
mem_ideal_iff
| null |
ideal_le_iff {I J : Ideal Γ(X, U)} :
I ≤ J ↔ ∀ (x : X) (h : x ∈ U),
I.map (X.presheaf.germ U x h).hom ≤ J.map (X.presheaf.germ U x h).hom :=
⟨fun h _ _ ↦ Ideal.map_mono h,
fun H _ hs ↦ hU.mem_ideal_iff.mpr fun x hx ↦ H x hx (Ideal.mem_map_of_mem _ hs)⟩
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
|
ideal_le_iff
| null |
ideal_ext_iff {I J : Ideal Γ(X, U)} :
I = J ↔ ∀ (x : X) (h : x ∈ U),
I.map (X.presheaf.germ U x h).hom = J.map (X.presheaf.germ U x h).hom := by
simp_rw [le_antisymm_iff, hU.ideal_le_iff, forall_and]
|
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
|
ideal_ext_iff
| null |
@[simps]
_root_.AlgebraicGeometry.Scheme.affineBasicOpen
(X : Scheme) {U : X.affineOpens} (f : Γ(X, U)) : X.affineOpens :=
⟨X.basicOpen f, U.prop.basicOpen f⟩
|
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.Scheme.affineBasicOpen
|
The basic open set of a section `f` on an affine open as an `X.affineOpens`.
|
_root_.AlgebraicGeometry.Scheme.affineBasicOpen_le
(X : Scheme) {V : X.affineOpens} (f : Γ(X, V.1)) : X.affineBasicOpen f ≤ V :=
X.basicOpen_le f
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
|
_root_.AlgebraicGeometry.Scheme.affineBasicOpen_le
| null |
basicOpen_union_eq_self_iff (s : Set Γ(X, U)) :
⨆ f : s, X.basicOpen (f : Γ(X, U)) = U ↔ Ideal.span s = ⊤ := by
trans ⋃ i : s, (PrimeSpectrum.basicOpen i.1).1 = Set.univ
· trans
hU.fromSpec.base ⁻¹' (⨆ f : s, X.basicOpen (f : Γ(X, U))).1 =
hU.fromSpec.base ⁻¹' U.1
· refine ⟨fun h => by rw [h], ?_⟩
intro h
apply_fun Set.image hU.fromSpec.base at h
rw [Set.image_preimage_eq_inter_range, Set.image_preimage_eq_inter_range, hU.range_fromSpec]
at h
simp only [Set.inter_self, Opens.carrier_eq_coe, Set.inter_eq_right] at h
ext1
refine Set.Subset.antisymm ?_ h
simp only [Set.iUnion_subset_iff, SetCoe.forall, Opens.coe_iSup]
intro x _
exact X.basicOpen_le x
· simp only [Opens.iSup_def, Set.preimage_iUnion]
congr! 1
· refine congr_arg (Set.iUnion ·) ?_
ext1 x
exact congr_arg Opens.carrier (hU.fromSpec_preimage_basicOpen _)
· exact congr_arg Opens.carrier hU.fromSpec_preimage_self
· simp only [Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]
rw [← Set.compl_iInter, Set.compl_univ_iff, ← PrimeSpectrum.zeroLocus_iUnion, ←
PrimeSpectrum.zeroLocus_empty_iff_eq_top, PrimeSpectrum.zeroLocus_span]
simp only [Set.iUnion_singleton_eq_range, Subtype.range_val_subtype, Set.setOf_mem_eq]
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
|
basicOpen_union_eq_self_iff
|
In an affine open set `U`, a family of basic open covers `U` iff the sections span `Γ(X, U)`.
See `iSup_basicOpen_of_span_eq_top` for the inverse direction without the affine-ness assumption.
|
self_le_basicOpen_union_iff (s : Set Γ(X, U)) :
(U ≤ ⨆ f : s, X.basicOpen f.1) ↔ Ideal.span s = ⊤ := by
rw [← hU.basicOpen_union_eq_self_iff, @comm _ Eq]
refine ⟨fun h => le_antisymm h ?_, le_of_eq⟩
simp only [iSup_le_iff, SetCoe.forall]
intro x _
exact X.basicOpen_le 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
|
self_le_basicOpen_union_iff
| null |
noncomputable
SpecMapRestrictBasicOpenIso {R S : CommRingCat} (f : R ⟶ S) (r : R) :
Arrow.mk (Spec.map f ∣_ (PrimeSpectrum.basicOpen r)) ≅
Arrow.mk (Spec.map <| CommRingCat.ofHom (Localization.awayMap f.hom r)) := by
letI e₁ : Localization.Away r ≃ₐ[R] Γ(Spec R, basicOpen r) :=
IsLocalization.algEquiv (Submonoid.powers r) _ _
letI e₂ : Localization.Away (f.hom r) ≃ₐ[S] Γ(Spec S, basicOpen (f.hom r)) :=
IsLocalization.algEquiv (Submonoid.powers (f.hom r)) _ _
refine Arrow.isoMk ?_ ?_ ?_
· exact Spec(S).isoOfEq (comap_basicOpen _ _) ≪≫
(IsAffineOpen.Spec_basicOpen (f.hom r)).isoSpec ≪≫ Scheme.Spec.mapIso e₂.toCommRingCatIso.op
· exact (IsAffineOpen.Spec_basicOpen r).isoSpec ≪≫ Scheme.Spec.mapIso e₁.toCommRingCatIso.op
· have := AlgebraicGeometry.IsOpenImmersion.of_isLocalization
(S := (Localization.Away r)) r
rw [← cancel_mono (Spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away r))))]
simp only [Arrow.mk_left, Arrow.mk_right, Functor.id_obj, Scheme.isoOfEq_rfl, Iso.refl_trans,
Iso.trans_hom, Functor.mapIso_hom, Iso.op_hom, Scheme.Spec_map, Quiver.Hom.unop_op,
Arrow.mk_hom, Category.assoc, ← Spec.map_comp]
conv =>
congr
· enter [2, 1]; tactic =>
change _ =
(f ≫ (Scheme.ΓSpecIso S).inv ≫ (Spec S).presheaf.map (homOfLE le_top).op)
ext
simp only [Localization.awayMap, IsLocalization.Away.map, AlgEquiv.toRingEquiv_eq_coe,
RingEquiv.toCommRingCatIso_hom, AlgEquiv.toRingEquiv_toRingHom, CommRingCat.hom_comp,
CommRingCat.hom_ofHom, RingHom.comp_apply, IsLocalization.map_eq, RingHom.coe_coe,
AlgEquiv.commutes, IsAffineOpen.algebraMap_Spec_obj]
· enter [2, 2, 1]; tactic =>
change _ = (Scheme.ΓSpecIso R).inv ≫ (Spec R).presheaf.map (homOfLE le_top).op
ext
simp only [AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toCommRingCatIso_hom,
AlgEquiv.toRingEquiv_toRingHom, CommRingCat.hom_comp, CommRingCat.hom_ofHom,
RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgEquiv.commutes,
IsAffineOpen.algebraMap_Spec_obj, homOfLE_leOfHom]
simp only [IsAffineOpen.isoSpec_hom, homOfLE_leOfHom, Spec.map_comp, Category.assoc,
Scheme.Opens.toSpecΓ_SpecMap_map_assoc, Scheme.Opens.toSpecΓ_top, Scheme.homOfLE_ι_assoc,
morphismRestrict_ι_assoc]
simp only [← SpecMap_ΓSpecIso_hom, ← Spec.map_comp, Category.assoc, Iso.inv_hom_id,
Category.comp_id, Category.id_comp]
rfl
|
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
|
SpecMapRestrictBasicOpenIso
|
The restriction of `Spec.map f` to a basic open `D(r)` is isomorphic to `Spec.map` of the
localization of `f` away from `r`.
|
stalkMap_injective_of_isAffine {X Y : Scheme} (f : X ⟶ Y) [IsAffine Y] (x : X)
(h : ∀ g, f.stalkMap x (Y.presheaf.Γgerm (f.base x) g) = 0 →
Y.presheaf.Γgerm (f.base x) g = 0) :
Function.Injective (f.stalkMap x) :=
(isAffineOpen_top Y).stalkMap_injective f x trivial h
|
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
|
stalkMap_injective_of_isAffine
| null |
iSup_basicOpen_of_span_eq_top {X : Scheme} (U) (s : Set Γ(X, U))
(hs : Ideal.span s = ⊤) : (⨆ i ∈ s, X.basicOpen i) = U := by
apply le_antisymm
· rw [iSup₂_le_iff]
exact fun i _ ↦ X.basicOpen_le i
· intro x hx
obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := (isBasis_affine_open X).exists_subset_of_mem_open hx U.2
refine SetLike.mem_of_subset ?_ hxV
rw [← (hV.basicOpen_union_eq_self_iff (X.presheaf.map (homOfLE hVU).op '' s)).mpr
(by rw [← Ideal.map_span, hs, Ideal.map_top])]
simp only [Opens.iSup_mk, Opens.carrier_eq_coe, Set.iUnion_coe_set, Set.mem_image,
Set.iUnion_exists, Set.biUnion_and', Set.iUnion_iUnion_eq_right, Scheme.basicOpen_res,
Opens.coe_inf, Opens.coe_mk, Set.iUnion_subset_iff]
exact fun i hi ↦ (Set.inter_subset_right.trans
(Set.subset_iUnion₂ (s := fun x _ ↦ (X.basicOpen x : Set X)) i hi))
|
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
|
iSup_basicOpen_of_span_eq_top
|
Given a spanning set of `Γ(X, U)`, the corresponding basic open sets cover `U`.
See `IsAffineOpen.basicOpen_union_eq_self_iff` for the inverse direction for affine open sets.
|
@[elab_as_elim]
of_affine_open_cover {X : Scheme} {P : X.affineOpens → Prop}
{ι} (U : ι → X.affineOpens) (iSup_U : (⨆ i, U i : X.Opens) = ⊤)
(V : X.affineOpens)
(basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)), P U → P (X.affineBasicOpen f))
(openCover :
∀ (U : X.affineOpens) (s : Finset (Γ(X, U)))
(_ : Ideal.span (s : Set (Γ(X, U))) = ⊤),
(∀ f : s, P (X.affineBasicOpen f.1)) → P U)
(hU : ∀ i, P (U i)) : P V := by
classical
have : ∀ (x : V.1), ∃ f : Γ(X, V), ↑x ∈ X.basicOpen f ∧ P (X.affineBasicOpen f) := by
intro x
obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (show x.1 ∈ (⨆ i, U i : X.Opens) from iSup_U ▸ trivial)
obtain ⟨f, g, e, hf⟩ := exists_basicOpen_le_affine_inter V.prop (U i).prop x ⟨x.prop, hi⟩
refine ⟨f, hf, ?_⟩
convert basicOpen _ g (hU i) using 1
ext1
exact e
choose f hf₁ hf₂ using this
suffices Ideal.span (Set.range f) = ⊤ by
obtain ⟨t, ht₁, ht₂⟩ := (Ideal.span_eq_top_iff_finite _).mp this
apply openCover V t ht₂
rintro ⟨i, hi⟩
obtain ⟨x, rfl⟩ := ht₁ hi
exact hf₂ x
rw [← V.prop.self_le_basicOpen_union_iff]
intro x hx
rw [iSup_range', SetLike.mem_coe, Opens.mem_iSup]
exact ⟨_, hf₁ ⟨x, hx⟩⟩
|
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
|
of_affine_open_cover
|
Let `P` be a predicate on the affine open sets of `X` satisfying
1. If `P` holds on `U`, then `P` holds on the basic open set of every section on `U`.
2. If `P` holds for a family of basic open sets covering `U`, then `P` holds for `U`.
3. There exists an affine open cover of `X` each satisfying `P`.
Then `P` holds for every affine open of `X`.
This is also known as the **Affine communication lemma** in [*The rising sea*][RisingSea].
|
toSpecΓ_preimage_zeroLocus (s : Set Γ(X, ⊤)) :
X.toSpecΓ.base ⁻¹' PrimeSpectrum.zeroLocus s = X.zeroLocus s :=
LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq s
|
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_preimage_zeroLocus
|
On a scheme `X`, the preimage of the zero locus of the prime spectrum
of `Γ(X, ⊤)` under `X.toSpecΓ : X ⟶ Spec Γ(X, ⊤)` agrees with the associated zero locus on `X`.
|
isoSpec_image_zeroLocus [IsAffine X]
(s : Set Γ(X, ⊤)) :
X.isoSpec.hom.base '' X.zeroLocus s = PrimeSpectrum.zeroLocus s := by
rw [← X.toSpecΓ_preimage_zeroLocus]
erw [Set.image_preimage_eq]
exact (bijective_of_isIso X.isoSpec.hom.base).surjective
|
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_image_zeroLocus
|
If `X` is affine, the image of the zero locus of global sections of `X` under `X.isoSpec`
is the zero locus in terms of the prime spectrum of `Γ(X, ⊤)`.
|
toSpecΓ_image_zeroLocus [IsAffine X] (s : Set Γ(X, ⊤)) :
X.toSpecΓ.base '' X.zeroLocus s = PrimeSpectrum.zeroLocus s :=
X.isoSpec_image_zeroLocus _
|
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_image_zeroLocus
| null |
isoSpec_inv_preimage_zeroLocus [IsAffine X] (s : Set Γ(X, ⊤)) :
X.isoSpec.inv.base ⁻¹' X.zeroLocus s = PrimeSpectrum.zeroLocus s := by
rw [← toSpecΓ_preimage_zeroLocus, ← Set.preimage_comp, ← TopCat.coe_comp, ← Scheme.comp_base,
X.isoSpec_inv_toSpecΓ]
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
|
isoSpec_inv_preimage_zeroLocus
| null |
isoSpec_inv_image_zeroLocus [IsAffine X] (s : Set Γ(X, ⊤)) :
X.isoSpec.inv.base '' PrimeSpectrum.zeroLocus s = X.zeroLocus s := by
rw [← isoSpec_inv_preimage_zeroLocus, Set.image_preimage_eq]
exact (bijective_of_isIso X.isoSpec.inv.base).surjective
|
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_image_zeroLocus
| null |
eq_zeroLocus_of_isClosed_of_isAffine [IsAffine X] (s : Set X) :
IsClosed s ↔ ∃ I : Ideal (Γ(X, ⊤)), s = X.zeroLocus (I : Set Γ(X, ⊤)) := by
refine ⟨fun hs ↦ ?_, ?_⟩
· let Z : Set (Spec <| Γ(X, ⊤)) := X.toΓSpecFun '' s
have hZ : IsClosed Z := (X.isoSpec.hom.homeomorph).isClosedMap _ hs
obtain ⟨I, (hI : Z = _)⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal _).mp hZ
use I
simp only [← Scheme.toSpecΓ_preimage_zeroLocus, ← hI, Z]
symm
exact Set.preimage_image_eq _ (bijective_of_isIso X.isoSpec.hom.base).injective
· rintro ⟨I, rfl⟩
exact zeroLocus_isClosed X I.carrier
open Set.Notation 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
|
eq_zeroLocus_of_isClosed_of_isAffine
|
If `X` is an affine scheme, every closed set of `X` is the zero locus
of a set of global sections.
|
Opens.toSpecΓ_preimage_zeroLocus {X : Scheme.{u}} (U : X.Opens)
(s : Set Γ(X, U)) :
U.toSpecΓ.base ⁻¹' PrimeSpectrum.zeroLocus s = U.1 ↓∩ X.zeroLocus s := by
rw [toSpecΓ, Scheme.comp_base, TopCat.coe_comp, Set.preimage_comp, Spec.map_base, hom_ofHom]
erw [PrimeSpectrum.preimage_comap_zeroLocus]
rw [Scheme.toSpecΓ_preimage_zeroLocus]
change _ = U.ι.base ⁻¹' (X.zeroLocus s)
rw [Scheme.preimage_zeroLocus, U.ι_app_self, ← zeroLocus_map_of_eq _ U.ι_preimage_self,
← Set.image_comp, ← RingHom.coe_comp, ← CommRingCat.hom_comp]
congr!
simp [← Functor.map_comp]
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
|
Opens.toSpecΓ_preimage_zeroLocus
| null |
IsAffineOpen.fromSpec_preimage_zeroLocus {X : Scheme.{u}} {U : X.Opens}
(hU : IsAffineOpen U) (s : Set Γ(X, U)) :
hU.fromSpec.base ⁻¹' X.zeroLocus s = PrimeSpectrum.zeroLocus s := by
ext x
suffices (∀ f ∈ s, ¬f ∉ x.asIdeal) ↔ s ⊆ x.asIdeal by
simpa [← hU.fromSpec_image_basicOpen, -not_not] using this
simp_rw [not_not]
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
|
IsAffineOpen.fromSpec_preimage_zeroLocus
| null |
IsAffineOpen.fromSpec_image_zeroLocus {X : Scheme.{u}} {U : X.Opens}
(hU : IsAffineOpen U) (s : Set Γ(X, U)) :
hU.fromSpec.base '' PrimeSpectrum.zeroLocus s = X.zeroLocus s ∩ U := by
rw [← hU.fromSpec_preimage_zeroLocus, Set.image_preimage_eq_inter_range, range_fromSpec]
open Set.Notation 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
|
IsAffineOpen.fromSpec_image_zeroLocus
| null |
Scheme.zeroLocus_inf (X : Scheme.{u}) {U : X.Opens} (I J : Ideal Γ(X, U)) :
X.zeroLocus (U := U) ↑(I ⊓ J) = X.zeroLocus (U := U) I ∪ X.zeroLocus (U := U) J := by
suffices U.1 ↓∩ (X.zeroLocus (U := U) ↑(I ⊓ J)) =
U.1 ↓∩ (X.zeroLocus (U := U) I ∪ X.zeroLocus (U := U) J) by
ext x
by_cases hxU : x ∈ U
· simpa [hxU] using congr(⟨x, hxU⟩ ∈ $this)
· simp only [Submodule.coe_inf, Set.mem_union,
codisjoint_iff_compl_le_left.mp (X.codisjoint_zeroLocus (U := U) (I ∩ J)) hxU,
codisjoint_iff_compl_le_left.mp (X.codisjoint_zeroLocus (U := U) I) hxU, true_or]
simp only [← U.toSpecΓ_preimage_zeroLocus, PrimeSpectrum.zeroLocus_inf I J,
Set.preimage_union]
|
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.zeroLocus_inf
| null |
Scheme.zeroLocus_biInf
{X : Scheme.{u}} {U : X.Opens} {ι : Type*}
(I : ι → Ideal Γ(X, U)) {t : Set ι} (ht : t.Finite) :
X.zeroLocus (U := U) ↑(⨅ i ∈ t, I i) = (⋃ i ∈ t, X.zeroLocus (U := U) (I i)) ∪ (↑U)ᶜ := by
refine ht.induction_on _ (by simp) fun {i t} hit ht IH ↦ ?_
simp only [Set.mem_insert_iff, Set.iUnion_iUnion_eq_or_left, ← IH, ← zeroLocus_inf,
Submodule.coe_inf, Set.union_assoc]
congr!
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.zeroLocus_biInf
| null |
Scheme.zeroLocus_biInf_of_nonempty
{X : Scheme.{u}} {U : X.Opens} {ι : Type*}
(I : ι → Ideal Γ(X, U)) {t : Set ι} (ht : t.Finite) (ht' : t.Nonempty) :
X.zeroLocus (U := U) ↑(⨅ i ∈ t, I i) = ⋃ i ∈ t, X.zeroLocus (U := U) (I i) := by
rw [zeroLocus_biInf I ht, Set.union_eq_left]
obtain ⟨i, hi⟩ := ht'
exact fun x hx ↦ Set.mem_iUnion₂_of_mem hi
(codisjoint_iff_compl_le_left.mp (X.codisjoint_zeroLocus (U := U) (I i)) hx)
|
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.zeroLocus_biInf_of_nonempty
| null |
Scheme.zeroLocus_iInf
{X : Scheme.{u}} {U : X.Opens} {ι : Type*}
(I : ι → Ideal Γ(X, U)) [Finite ι] :
X.zeroLocus (U := U) ↑(⨅ i, I i) = (⋃ i, X.zeroLocus (U := U) (I i)) ∪ (↑U)ᶜ := by
simpa using zeroLocus_biInf I Set.finite_univ
|
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.zeroLocus_iInf
| null |
Scheme.zeroLocus_iInf_of_nonempty
{X : Scheme.{u}} {U : X.Opens} {ι : Type*}
(I : ι → Ideal Γ(X, U)) [Finite ι] [Nonempty ι] :
X.zeroLocus (U := U) ↑(⨅ i, I i) = ⋃ i, X.zeroLocus (U := U) (I i) := by
simpa using zeroLocus_biInf_of_nonempty I Set.finite_univ
|
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.zeroLocus_iInf_of_nonempty
| null |
Scheme.Hom.liftQuotient (f : X.Hom (Spec A)) (I : Ideal A)
(hI : I ≤ RingHom.ker ((Scheme.ΓSpecIso A).inv ≫ f.appTop).hom) :
X ⟶ Spec(A ⧸ I) :=
X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom
(Ideal.Quotient.lift _ ((Scheme.ΓSpecIso _).inv ≫ f.appTop).hom hI))
@[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.Hom.liftQuotient
|
Given `f : X ⟶ Spec A` and some ideal `I ≤ ker(A ⟶ Γ(X, ⊤))`,
this is the lift to `X ⟶ Spec (A ⧸ I)`.
|
Scheme.Hom.liftQuotient_comp (f : X.Hom (Spec A)) (I : Ideal A)
(hI : I ≤ RingHom.ker ((Scheme.ΓSpecIso A).inv ≫ f.appTop).hom) :
f.liftQuotient I hI ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk _)) = f := by
rw [Scheme.Hom.liftQuotient, Category.assoc, ← Spec.map_comp, ← CommRingCat.ofHom_comp,
Ideal.Quotient.lift_comp_mk]
simp only [CommRingCat.hom_comp, CommRingCat.ofHom_comp, CommRingCat.ofHom_hom, Spec.map_comp, ←
Scheme.toSpecΓ_naturality_assoc, ← SpecMap_ΓSpecIso_hom]
simp only [← Spec.map_comp, Iso.inv_hom_id, Spec.map_id, Category.comp_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.Hom.liftQuotient_comp
| null |
specTargetImageIdeal (f : X ⟶ Spec A) : Ideal A :=
(RingHom.ker <| (((ΓSpec.adjunction).homEquiv X (op A)).symm f).unop.hom)
|
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
|
specTargetImageIdeal
|
If `X ⟶ Spec A` is a morphism of schemes, then `Spec` of `A ⧸ specTargetImage f`
is the scheme-theoretic image of `f`. For this quotient as an object of `CommRingCat` see
`specTargetImage` below.
|
specTargetImage (f : X ⟶ Spec A) : CommRingCat :=
CommRingCat.of (A ⧸ specTargetImageIdeal f)
|
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
|
specTargetImage
|
If `X ⟶ Spec A` is a morphism of schemes, then `Spec` of `specTargetImage f` is the
scheme-theoretic image of `f` and `f` factors as
`specTargetImageFactorization f ≫ Spec.map (specTargetImageRingHom f)`
(see `specTargetImageFactorization_comp`).
|
specTargetImageFactorization (f : X ⟶ Spec A) : X ⟶ Spec (specTargetImage f) :=
f.liftQuotient _ le_rfl
|
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
|
specTargetImageFactorization
|
If `f : X ⟶ Spec A` is a morphism of schemes, then `f` factors via
the inclusion of `Spec (specTargetImage f)` into `X`.
|
specTargetImageRingHom (f : X ⟶ Spec A) : A ⟶ specTargetImage f :=
CommRingCat.ofHom (Ideal.Quotient.mk (specTargetImageIdeal f))
variable (f : X ⟶ Spec A)
|
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
|
specTargetImageRingHom
|
If `f : X ⟶ Spec A` is a morphism of schemes, the induced morphism on spectra of
`specTargetImageRingHom f` is the inclusion of the scheme-theoretic image of `f` into `Spec A`.
|
specTargetImageRingHom_surjective : Function.Surjective (specTargetImageRingHom f) :=
Ideal.Quotient.mk_surjective
|
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
|
specTargetImageRingHom_surjective
| null |
specTargetImageFactorization_app_injective :
Function.Injective <| (specTargetImageFactorization f).appTop := by
let φ : A ⟶ Γ(X, ⊤) := (((ΓSpec.adjunction).homEquiv X (op A)).symm f).unop
let φ' : specTargetImage f ⟶ Scheme.Γ.obj (op X) := CommRingCat.ofHom (RingHom.kerLift φ.hom)
change Function.Injective <| ((ΓSpec.adjunction.homEquiv X _) φ'.op).appTop
rw [ΓSpec_adjunction_homEquiv_eq]
apply (RingHom.kerLift_injective φ.hom).comp
exact ((ConcreteCategory.isIso_iff_bijective (Scheme.ΓSpecIso _).hom).mp inferInstance).injective
@[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
|
specTargetImageFactorization_app_injective
| null |
specTargetImageFactorization_comp :
specTargetImageFactorization f ≫ Spec.map (specTargetImageRingHom f) = f :=
f.liftQuotient_comp _ _
open RingHom
|
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
|
specTargetImageFactorization_comp
| null |
@[elementwise]
Scheme.localRingHom_comp_stalkIso {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
(StructureSheaf.stalkIso R (PrimeSpectrum.comap f.hom p)).hom ≫
(CommRingCat.ofHom <| Localization.localRingHom
(PrimeSpectrum.comap f.hom p).asIdeal p.asIdeal f.hom rfl) ≫
(StructureSheaf.stalkIso S p).inv = (Spec.map f).stalkMap p :=
AlgebraicGeometry.localRingHom_comp_stalkIso f p
|
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.localRingHom_comp_stalkIso
|
Variant of `AlgebraicGeometry.localRingHom_comp_stalkIso` for `Spec.map`.
|
Scheme.arrowStalkMapSpecIso {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
Arrow.mk ((Spec.map f).stalkMap p) ≅ Arrow.mk (CommRingCat.ofHom <| Localization.localRingHom
(PrimeSpectrum.comap f.hom p).asIdeal p.asIdeal f.hom rfl) := Arrow.isoMk
(StructureSheaf.stalkIso R (PrimeSpectrum.comap f.hom p))
(StructureSheaf.stalkIso S p) <| by
rw [← Scheme.localRingHom_comp_stalkIso]
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.arrowStalkMapSpecIso
|
Given a morphism of rings `f : R ⟶ S`, the stalk map of `Spec S ⟶ Spec R` at
a prime of `S` is isomorphic to the localized ring homomorphism.
|
AffineSpace (n : Type v) (S : Scheme.{max u v}) : Scheme.{max u v} :=
pullback (terminal.from S) (terminal.from (Spec ℤ[n].{u, v}))
|
def
|
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
|
AffineSpace
|
`𝔸(n; S)` is the affine `n`-space over `S`.
Note that `n` is an arbitrary index type (e.g. `Fin m`).
|
toSpecMvPoly : 𝔸(n; S) ⟶ Spec ℤ[n].{u, v} := pullback.snd _ _
variable {X : Scheme.{max u v}}
|
def
|
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
|
toSpecMvPoly
|
`𝔸(n; S)` is the affine `n`-space over `S`. -/
scoped [AlgebraicGeometry] notation "𝔸("n"; "S")" => AffineSpace n S
variable {n} in
lemma of_mvPolynomial_int_ext {R} {f g : ℤ[n] ⟶ R} (h : ∀ i, f (.X i) = g (.X i)) : f = g := by
suffices f.hom.comp (MvPolynomial.mapEquiv _ ULift.ringEquiv.symm).toRingHom =
g.hom.comp (MvPolynomial.mapEquiv _ ULift.ringEquiv.symm).toRingHom by
ext x
· obtain ⟨x⟩ := x
simpa [-map_intCast, -eq_intCast] using DFunLike.congr_fun this (C x)
· simpa [-map_intCast, -eq_intCast] using DFunLike.congr_fun this (X x)
ext1
· exact RingHom.ext_int _ _
· simpa using h _
@[simps -isSimp]
instance over : 𝔸(n; S).CanonicallyOver S where
hom := pullback.fst _ _
/-- The map from the affine `n`-space over `S` to the integral model `Spec ℤ[n]`.
|
@[simps]
toSpecMvPolyIntEquiv : (X ⟶ Spec ℤ[n]) ≃ (n → Γ(X, ⊤)) where
toFun f i := f.appTop ((Scheme.ΓSpecIso ℤ[n]).inv (.X i))
invFun v := X.toSpecΓ ≫ Spec.map
(CommRingCat.ofHom (MvPolynomial.eval₂Hom ((algebraMap ℤ _).comp ULift.ringEquiv.toRingHom) v))
left_inv f := by
apply (ΓSpec.adjunction.homEquiv _ _).symm.injective
apply Quiver.Hom.unop_inj
rw [Adjunction.homEquiv_symm_apply, Adjunction.homEquiv_symm_apply]
simp only [Functor.rightOp_obj, Scheme.Γ_obj, Scheme.Spec_obj, algebraMap_int_eq,
RingEquiv.toRingHom_eq_coe, TopologicalSpace.Opens.map_top, Functor.rightOp_map, op_comp,
Scheme.Γ_map, unop_comp, Quiver.Hom.unop_op, Scheme.comp_app, Scheme.toSpecΓ_appTop,
Scheme.ΓSpecIso_naturality, ΓSpec.adjunction_counit_app, Category.assoc,
Iso.cancel_iso_inv_left, ← Iso.eq_inv_comp]
apply of_mvPolynomial_int_ext
intro i
rw [ConcreteCategory.hom_ofHom, coe_eval₂Hom, eval₂_X]
rfl
right_inv v := by
ext i
simp only [algebraMap_int_eq, RingEquiv.toRingHom_eq_coe, TopologicalSpace.Opens.map_top,
Scheme.comp_app, Scheme.toSpecΓ_appTop, Scheme.ΓSpecIso_naturality, CommRingCat.comp_apply,
CommRingCat.coe_of]
rw [CommRingCat.hom_inv_apply]
simp
|
def
|
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
|
toSpecMvPolyIntEquiv
|
Morphisms into `Spec ℤ[n]` are equivalent the choice of `n` global sections.
Use `homOverEquiv` instead.
|
toSpecMvPolyIntEquiv_comp {X Y : Scheme} (f : X ⟶ Y) (g : Y ⟶ Spec ℤ[n]) (i) :
toSpecMvPolyIntEquiv n (f ≫ g) i = f.appTop (toSpecMvPolyIntEquiv n g i) := rfl
variable {n} in
|
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
|
toSpecMvPolyIntEquiv_comp
| null |
coord (i : n) : Γ(𝔸(n; S), ⊤) := toSpecMvPolyIntEquiv _ (toSpecMvPoly n S) i
|
def
|
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
|
coord
|
The standard coordinates of `𝔸(n; S)`.
|
homOfVector (f : X ⟶ S) (v : n → Γ(X, ⊤)) : X ⟶ 𝔸(n; S) :=
pullback.lift f ((toSpecMvPolyIntEquiv n).symm v) (by simp)
variable (f : X ⟶ S) (v : n → Γ(X, ⊤))
@[reassoc (attr := simp)]
|
def
|
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
|
homOfVector
|
The morphism `X ⟶ 𝔸(n; S)` given by a `X ⟶ S` and a choice of `n`-coordinate functions.
|
homOfVector_over : homOfVector f v ≫ 𝔸(n; S) ↘ S = f :=
pullback.lift_fst _ _ _
@[reassoc]
|
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
|
homOfVector_over
| null |
homOfVector_toSpecMvPoly :
homOfVector f v ≫ toSpecMvPoly n S = (toSpecMvPolyIntEquiv n).symm v :=
pullback.lift_snd _ _ _
@[simp]
|
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
|
homOfVector_toSpecMvPoly
| null |
homOfVector_appTop_coord (i) :
(homOfVector f v).appTop (coord S i) = v i := by
rw [coord, ← toSpecMvPolyIntEquiv_comp, homOfVector_toSpecMvPoly,
Equiv.apply_symm_apply]
@[ext 1100]
|
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
|
homOfVector_appTop_coord
| null |
hom_ext {f g : X ⟶ 𝔸(n; S)}
(h₁ : f ≫ 𝔸(n; S) ↘ S = g ≫ 𝔸(n; S) ↘ S)
(h₂ : ∀ i, f.appTop (coord S i) = g.appTop (coord S i)) : f = g := by
apply pullback.hom_ext h₁
change f ≫ toSpecMvPoly _ _ = g ≫ toSpecMvPoly _ _
apply (toSpecMvPolyIntEquiv n).injective
ext i
rw [toSpecMvPolyIntEquiv_comp, toSpecMvPolyIntEquiv_comp]
exact h₂ i
@[reassoc]
|
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
|
hom_ext
| null |
comp_homOfVector {X Y : Scheme} (v : n → Γ(Y, ⊤)) (f : X ⟶ Y) (g : Y ⟶ S) :
f ≫ homOfVector g v = homOfVector (f ≫ g) (f.appTop ∘ v) := by
ext1 <;> simp
|
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
|
comp_homOfVector
| null |
@[simps]
homOverEquiv : { f : X ⟶ 𝔸(n; S) // f.IsOver S } ≃ (n → Γ(X, ⊤)) where
toFun f i := f.1.appTop (coord S i)
invFun v := ⟨homOfVector (X ↘ S) v, inferInstance⟩
left_inv f := by
ext : 2
· simp [f.2.1]
· rw [homOfVector_appTop_coord]
right_inv v := by ext i; simp [-TopologicalSpace.Opens.map_top, homOfVector_appTop_coord]
variable (n) in
|
def
|
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
|
homOverEquiv
|
`S`-morphisms into `Spec 𝔸(n; S)` are equivalent to the choice of `n` global sections.
|
@[simps -isSimp hom inv]
isoOfIsAffine [IsAffine S] :
𝔸(n; S) ≅ Spec(MvPolynomial n Γ(S, ⊤)) where
hom := 𝔸(n; S).toSpecΓ ≫ Spec.map (CommRingCat.ofHom
(eval₂Hom ((𝔸(n; S) ↘ S).appTop).hom (coord S)))
inv := homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S.isoSpec.inv)
((Scheme.ΓSpecIso (.of (MvPolynomial n Γ(S, ⊤)))).inv ∘ MvPolynomial.X)
hom_inv_id := by
ext1
· simp only [Category.assoc, homOfVector_over, Category.id_comp]
rw [← Spec.map_comp_assoc, ← CommRingCat.ofHom_comp, eval₂Hom_comp_C,
CommRingCat.ofHom_hom, ← Scheme.toSpecΓ_naturality_assoc]
simp [Scheme.isoSpec]
· simp only [Category.assoc, Scheme.comp_app, Scheme.comp_coeBase,
TopologicalSpace.Opens.map_comp_obj, TopologicalSpace.Opens.map_top,
Scheme.toSpecΓ_appTop, Scheme.ΓSpecIso_naturality, CommRingCat.comp_apply,
homOfVector_appTop_coord, Function.comp_apply, CommRingCat.coe_of, Scheme.id_app,
CommRingCat.id_apply]
rw [CommRingCat.hom_inv_apply]
exact eval₂_X _ _ _
inv_hom_id := by
apply ext_of_isAffine
simp only [Scheme.comp_coeBase, TopologicalSpace.Opens.map_comp_obj,
TopologicalSpace.Opens.map_top, Scheme.comp_app, Scheme.toSpecΓ_appTop,
Scheme.ΓSpecIso_naturality, Category.assoc, Scheme.id_app, ← Iso.eq_inv_comp,
Category.comp_id]
ext : 1
apply ringHom_ext'
· change _ = (CommRingCat.ofHom C ≫ _).hom
rw [CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom, eval₂Hom_comp_C,
← CommRingCat.hom_comp, ← CommRingCat.hom_ext_iff,
← cancel_mono (Scheme.ΓSpecIso _).hom]
rw [← Scheme.comp_appTop, homOfVector_over, Scheme.comp_appTop]
simp only [Category.assoc, Scheme.ΓSpecIso_naturality, CommRingCat.of_carrier,
← Scheme.toSpecΓ_appTop]
rw [← Scheme.comp_appTop_assoc, Scheme.isoSpec, asIso_inv, IsIso.hom_inv_id]
simp
· intro i
rw [CommRingCat.comp_apply, ConcreteCategory.hom_ofHom, coe_eval₂Hom]
simp only [eval₂_X]
exact homOfVector_appTop_coord _ _ _
@[simp]
|
def
|
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
|
isoOfIsAffine
|
The affine space over an affine base is isomorphic to the spectrum of the polynomial ring.
Also see `AffineSpace.SpecIso`.
|
isoOfIsAffine_hom_appTop [IsAffine S] :
(isoOfIsAffine n S).hom.appTop =
(Scheme.ΓSpecIso _).hom ≫ CommRingCat.ofHom
(eval₂Hom ((𝔸(n; S) ↘ S).appTop).hom (coord S)) := by
simp [isoOfIsAffine_hom]
@[simp]
|
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
|
isoOfIsAffine_hom_appTop
| null |
isoOfIsAffine_inv_appTop_coord [IsAffine S] (i) :
(isoOfIsAffine n S).inv.appTop (coord _ i) = (Scheme.ΓSpecIso (.of _)).inv (.X i) :=
homOfVector_appTop_coord _ _ _
@[reassoc (attr := simp)]
|
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
|
isoOfIsAffine_inv_appTop_coord
| null |
isoOfIsAffine_inv_over [IsAffine S] :
(isoOfIsAffine n S).inv ≫ 𝔸(n; S) ↘ S = Spec.map (CommRingCat.ofHom C) ≫ S.isoSpec.inv :=
pullback.lift_fst _ _ _
|
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
|
isoOfIsAffine_inv_over
| null |
SpecIso (R : CommRingCat.{max u v}) :
𝔸(n; Spec R) ≅ Spec(MvPolynomial n R) :=
isoOfIsAffine _ _ ≪≫ Scheme.Spec.mapIso (MvPolynomial.mapEquiv _
(Scheme.ΓSpecIso R).symm.commRingCatIsoToRingEquiv).toCommRingCatIso.op
@[simp]
|
def
|
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
|
SpecIso
|
The affine space over an affine base is isomorphic to the spectrum of the polynomial ring.
|
SpecIso_hom_appTop (R : CommRingCat.{max u v}) :
(SpecIso n R).hom.appTop = (Scheme.ΓSpecIso _).hom ≫
CommRingCat.ofHom (eval₂Hom ((Scheme.ΓSpecIso _).inv ≫
(𝔸(n; Spec R) ↘ Spec R).appTop).hom (coord (Spec R))) := by
simp only [SpecIso, Iso.trans_hom, Functor.mapIso_hom, Iso.op_hom,
Scheme.Spec_map, Quiver.Hom.unop_op, TopologicalSpace.Opens.map_top, Scheme.comp_app,
isoOfIsAffine_hom_appTop, Scheme.ΓSpecIso_naturality_assoc]
congr 1
ext : 1
apply ringHom_ext'
· ext; simp
· simp
@[simp]
|
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
|
SpecIso_hom_appTop
| null |
SpecIso_inv_appTop_coord (R : CommRingCat.{max u v}) (i) :
(SpecIso n R).inv.appTop (coord _ i) = (Scheme.ΓSpecIso (.of _)).inv (.X i) := by
simp only [SpecIso, Iso.trans_inv, Functor.mapIso_inv, Iso.op_inv, Scheme.Spec_map,
Quiver.Hom.unop_op, TopologicalSpace.Opens.map_top, Scheme.comp_app, CommRingCat.comp_apply]
rw [isoOfIsAffine_inv_appTop_coord, ← CommRingCat.comp_apply, ← Scheme.ΓSpecIso_inv_naturality,
CommRingCat.comp_apply]
congr 1
exact map_X _ _
@[reassoc (attr := simp)]
|
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
|
SpecIso_inv_appTop_coord
| null |
SpecIso_inv_over (R : CommRingCat.{max u v}) :
(SpecIso n R).inv ≫ 𝔸(n; Spec R) ↘ Spec R = Spec.map (CommRingCat.ofHom C) := by
simp only [SpecIso, Iso.trans_inv, Functor.mapIso_inv, Iso.op_inv, Scheme.Spec_map,
Quiver.Hom.unop_op, Category.assoc, isoOfIsAffine_inv_over, Scheme.isoSpec_Spec_inv,
← Spec.map_comp]
congr 1
rw [Iso.inv_comp_eq]
ext : 2
exact map_C _ _
|
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
|
SpecIso_inv_over
| null |
map {S T : Scheme.{max u v}} (f : S ⟶ T) : 𝔸(n; S) ⟶ 𝔸(n; T) :=
homOfVector (𝔸(n; S) ↘ S ≫ f) (coord S)
@[reassoc (attr := simp)]
|
def
|
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
|
map
|
`𝔸(n; S)` is functorial w.r.t. `S`.
|
map_over {S T : Scheme.{max u v}} (f : S ⟶ T) : map n f ≫ 𝔸(n; T) ↘ T = 𝔸(n; S) ↘ S ≫ f :=
pullback.lift_fst _ _ _
@[simp]
|
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
|
map_over
| null |
map_appTop_coord {S T : Scheme.{max u v}} (f : S ⟶ T) (i) :
(map n f).appTop (coord T i) = coord S i :=
homOfVector_appTop_coord _ _ _
@[reassoc (attr := simp)]
|
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
|
map_appTop_coord
| null |
map_toSpecMvPoly {S T : Scheme.{max u v}} (f : S ⟶ T) :
map n f ≫ toSpecMvPoly n T = toSpecMvPoly n S := by
apply (toSpecMvPolyIntEquiv _).injective
ext i
rw [toSpecMvPolyIntEquiv_comp, ← coord, map_appTop_coord, coord]
@[simp]
|
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
|
map_toSpecMvPoly
| null |
map_id : map n (𝟙 S) = 𝟙 𝔸(n; S) := by
ext1 <;> simp
@[reassoc, simp]
|
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
|
map_id
| null |
map_comp {S S' S'' : Scheme} (f : S ⟶ S') (g : S' ⟶ S'') :
map n (f ≫ g) = map n f ≫ map n g := by
ext1
· simp
· simp
|
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
|
map_comp
| null |
map_Spec_map {R S : CommRingCat.{max u v}} (φ : R ⟶ S) :
map n (Spec.map φ) =
(SpecIso n S).hom ≫ Spec.map (CommRingCat.ofHom (MvPolynomial.map φ.hom)) ≫
(SpecIso n R).inv := by
rw [← Iso.inv_comp_eq]
ext1
· simp only [map_over, Category.assoc, SpecIso_inv_over, SpecIso_inv_over_assoc,
← Spec.map_comp, ← CommRingCat.ofHom_comp]
rw [map_comp_C, CommRingCat.ofHom_comp, CommRingCat.ofHom_hom]
· simp only [TopologicalSpace.Opens.map_top, Scheme.comp_app, CommRingCat.comp_apply]
conv_lhs => enter[2]; tactic => exact map_appTop_coord _ _
conv_rhs => enter[2]; tactic => exact SpecIso_inv_appTop_coord _ _
rw [SpecIso_inv_appTop_coord, ← CommRingCat.comp_apply, ← Scheme.ΓSpecIso_inv_naturality,
CommRingCat.comp_apply, ConcreteCategory.hom_ofHom, map_X]
|
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
|
map_Spec_map
| null |
mapSpecMap {R S : CommRingCat.{max u v}} (φ : R ⟶ S) :
Arrow.mk (map n (Spec.map φ)) ≅
Arrow.mk (Spec.map (CommRingCat.ofHom (MvPolynomial.map (σ := n) φ.hom))) :=
Arrow.isoMk (SpecIso n S) (SpecIso n R) (by have := (SpecIso n R).inv_hom_id; simp [map_Spec_map])
|
def
|
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
|
mapSpecMap
|
The map between affine spaces over affine bases is
isomorphic to the natural map between polynomial rings.
|
isPullback_map {S T : Scheme.{max u v}} (f : S ⟶ T) :
IsPullback (map n f) (𝔸(n; S) ↘ S) (𝔸(n; T) ↘ T) f := by
refine (IsPullback.paste_horiz_iff (.flip <| .of_hasPullback _ _) (map_over f)).mp ?_
simp only [terminal.comp_from, ]
convert (IsPullback.of_hasPullback _ _).flip
rw [← toSpecMvPoly, ← toSpecMvPoly, map_toSpecMvPoly]
|
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
|
isPullback_map
| null |
reindex {n m : Type v} (i : m → n) (S : Scheme.{max u v}) : 𝔸(n; S) ⟶ 𝔸(m; S) :=
homOfVector (𝔸(n; S) ↘ S) (coord S ∘ i)
@[simp, reassoc]
|
def
|
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
|
reindex
|
`𝔸(n; S)` is functorial w.r.t. `n`.
|
reindex_over {n m : Type v} (i : m → n) (S : Scheme.{max u v}) :
reindex i S ≫ 𝔸(m; S) ↘ S = 𝔸(n; S) ↘ S :=
pullback.lift_fst _ _ _
@[simp]
|
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
|
reindex_over
| null |
reindex_appTop_coord {n m : Type v} (i : m → n) (S : Scheme.{max u v}) (j : m) :
(reindex i S).appTop (coord S j) = coord S (i j) :=
homOfVector_appTop_coord _ _ _
@[simp]
|
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
|
reindex_appTop_coord
| null |
reindex_id : reindex id S = 𝟙 𝔸(n; S) := by
ext1 <;> simp
@[simp, reassoc]
|
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
|
reindex_id
| null |
reindex_comp {n₁ n₂ n₃ : Type v} (i : n₁ → n₂) (j : n₂ → n₃) (S : Scheme.{max u v}) :
reindex (j ∘ i) S = reindex j S ≫ reindex i S := by
have H₁ : reindex (j ∘ i) S ≫ 𝔸(n₁; S) ↘ S = (reindex j S ≫ reindex i S) ≫ 𝔸(n₁; S) ↘ S := by
simp
have H₂ (k) : (reindex (j ∘ i) S).appTop (coord S k) =
(reindex j S).appTop ((reindex i S).appTop (coord S k)) := by
rw [reindex_appTop_coord, reindex_appTop_coord, reindex_appTop_coord]
rfl
exact hom_ext H₁ H₂
@[reassoc (attr := simp)]
|
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
|
reindex_comp
| null |
map_reindex {n₁ n₂ : Type v} (i : n₁ → n₂) {S T : Scheme.{max u v}} (f : S ⟶ T) :
map n₂ f ≫ reindex i T = reindex i S ≫ map n₁ f := by
apply hom_ext <;> simp
|
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
|
map_reindex
| null |
@[simps]
functor : (Type v)ᵒᵖ ⥤ Scheme.{max u v} ⥤ Scheme.{max u v} where
obj n := { obj := AffineSpace n.unop, map := map n.unop, map_id := map_id, map_comp := map_comp }
map {n m} i := { app := reindex i.unop, naturality := fun _ _ ↦ map_reindex i.unop }
map_id n := by ext: 2; exact reindex_id _
map_comp f g := by ext: 2; dsimp; exact reindex_comp _ _ _
|
def
|
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
|
functor
|
The affine space as a functor.
|
isOpenMap_over : IsOpenMap (𝔸(n; S) ↘ S).base := by
change topologically @IsOpenMap _
wlog hS : ∃ R, S = Spec R
· refine (IsLocalAtTarget.iff_of_openCover (P := topologically @IsOpenMap) S.affineCover).mpr ?_
intro i
have := this (n := n) (S.affineCover.X i) ⟨_, rfl⟩
rwa [← (isPullback_map (n := n) (S.affineCover.f i)).isoPullback_hom_snd,
MorphismProperty.cancel_left_of_respectsIso (P := topologically @IsOpenMap)] at this
obtain ⟨R, rfl⟩ := hS
rw [← MorphismProperty.cancel_left_of_respectsIso (P := topologically @IsOpenMap)
(SpecIso n R).inv, SpecIso_inv_over]
exact MvPolynomial.isOpenMap_comap_C
open MorphismProperty in
|
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
|
isOpenMap_over
| null |
isIntegralHom_over_iff_isEmpty : IsIntegralHom (𝔸(n; S) ↘ S) ↔ IsEmpty S ∨ IsEmpty n := by
constructor
· intro h
cases isEmpty_or_nonempty S
· exact .inl ‹_›
refine .inr ?_
wlog hS : ∃ R, S = Spec R
· obtain ⟨x⟩ := ‹Nonempty S›
obtain ⟨y, hy⟩ := S.affineCover.covers x
exact this (S.affineCover.X _) (MorphismProperty.IsStableUnderBaseChange.of_isPullback
(isPullback_map (S.affineCover.f _)) h) ⟨y⟩ ⟨_, rfl⟩
obtain ⟨R, rfl⟩ := hS
have : Nontrivial R := (subsingleton_or_nontrivial R).resolve_left fun H ↦
not_isEmpty_of_nonempty (Spec R) (inferInstanceAs (IsEmpty (PrimeSpectrum R)))
constructor
intro i
have := RingHom.toMorphismProperty_respectsIso_iff.mp RingHom.isIntegral_respectsIso.{max u v}
rw [← MorphismProperty.cancel_left_of_respectsIso @IsIntegralHom (SpecIso n R).inv,
SpecIso_inv_over, HasAffineProperty.iff_of_isAffine (P := @IsIntegralHom)] at h
obtain ⟨p : Polynomial R, hp, hp'⟩ :=
(MorphismProperty.arrow_mk_iso_iff (RingHom.toMorphismProperty RingHom.IsIntegral)
(arrowIsoΓSpecOfIsAffine _)).mpr h.2 (X i)
have : (rename fun _ ↦ i).comp (pUnitAlgEquiv.{_, v} _).symm.toAlgHom p = 0 := by
simp [← hp', ← algebraMap_eq]
rw [AlgHom.comp_apply, map_eq_zero_iff _ (rename_injective _ (fun _ _ _ ↦ rfl))] at this
simp only [AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_coe, EmbeddingLike.map_eq_zero_iff] at this
simp [this] at hp
· rintro (_ | _) <;> infer_instance
|
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
|
isIntegralHom_over_iff_isEmpty
| null |
not_isIntegralHom [Nonempty S] [Nonempty n] : ¬ IsIntegralHom (𝔸(n; S) ↘ S) := by
simp [isIntegralHom_over_iff_isEmpty]
|
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
|
not_isIntegralHom
| null |
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