fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
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.ba... | 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 (Localiz... | 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... | 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... | 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
dsim... | 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 [Schem... | 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.m... | 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 (IsLocalizati... | 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, Rin... | 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_basi... | 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).isClos... | 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_rin... | 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⟩
a... | 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_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 | 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],... | 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 (... | 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... | 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))... | 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 fo... |
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... | 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... | 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
... | 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
... | 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, Se... | 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'
exa... | 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... | 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 <... | 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) ≫
(StructureShea... | 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.ho... | 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.... |
@[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... | 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, toSpe... | 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_t... | 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.ΓSpecIs... | 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... | 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... | 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.m... | 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,
← S... | 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, ← toSpecMvP... | 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... | 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 ... | 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) ... | 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.... | 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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