Residue fields of points

Main definitions

The following are in the AlgebraicGeometry.Scheme namespace:

• AlgebraicGeometry.Scheme.residueField: The residue field of the stalk at x.
• AlgebraicGeometry.Scheme.evaluation: For open subsets U of X containing x, the evaluation map from sections over U to the residue field at x.
• AlgebraicGeometry.Scheme.Hom.residueFieldMap: A morphism of schemes induce a homomorphism of residue fields.
• AlgebraicGeometry.Scheme.fromSpecResidueField: The canonical map Spec κ(x) ⟶ X.
• AlgebraicGeometry.Scheme.SpecToEquivOfField: morphisms Spec K ⟶ X for a field K correspond to pairs of x : X with embedding κ(x) ⟶ K.

def AlgebraicGeometry.Scheme.residueField (X : Scheme) (x : ↥X) : CommRingCat

The residue field of X at a point x is the residue field of the stalk of X at x.

instance AlgebraicGeometry.Scheme.instFieldCarrierResidueField (X : Scheme) (x : ↥X) : Field ↑(X.residueField x)

instance AlgebraicGeometry.Scheme.instUniqueCarrierCarrierCommRingCatSpecResidueField (X : Scheme) (x : ↥X) : Unique ↥(Spec (X.residueField x))

def AlgebraicGeometry.Scheme.residue (X : Scheme) (x : ↥X) : X.presheaf.stalk x ⟶ X.residueField x

The residue map from the stalk to the residue field.

instance AlgebraicGeometry.Scheme.instIsPreimmersionMapResidue {X : Scheme} (x : ↥X) : IsPreimmersion (Spec.map (X.residue x))

See AlgebraicGeometry.IsClosedImmersion.Spec_map_residue for the stronger result that Spec.map (X.residue x) is a closed immersion.

@[simp]

theorem AlgebraicGeometry.Scheme.Spec_map_residue_apply {X : Scheme} (x : ↥X) (s : ↥(Spec (X.residueField x))) : (CategoryTheory.ConcreteCategory.hom (Spec.map (X.residue x)).base) s = IsLocalRing.closedPoint ↑(X.presheaf.stalk x)

theorem AlgebraicGeometry.Scheme.residue_surjective (X : Scheme) (x : ↥X) : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (X.residue x))

instance AlgebraicGeometry.Scheme.instEpiCommRingCatResidue (X : Scheme) (x : ↥X) : CategoryTheory.Epi (X.residue x)

def AlgebraicGeometry.Scheme.descResidueField {K : Type u} [Field K] {X : Scheme} {x : ↥X} (f : X.presheaf.stalk x ⟶ CommRingCat.of K) [IsLocalHom (CommRingCat.Hom.hom f)] : X.residueField x ⟶ CommRingCat.of K

If K is a field and f : 𝒪_{X, x} ⟶ K is a ring map, then this is the induced map κ(x) ⟶ K.

@[simp]

theorem AlgebraicGeometry.Scheme.residue_descResidueField {K : Type u} [Field K] {X : Scheme} {x : ↥X} (f : X.presheaf.stalk x ⟶ CommRingCat.of K) [IsLocalHom (CommRingCat.Hom.hom f)] : CategoryTheory.CategoryStruct.comp (X.residue x) (descResidueField f) = f

@[simp]

theorem AlgebraicGeometry.Scheme.residue_descResidueField_assoc {K : Type u} [Field K] {X : Scheme} {x : ↥X} (f : X.presheaf.stalk x ⟶ CommRingCat.of K) [IsLocalHom (CommRingCat.Hom.hom f)] {Z : CommRingCat} (h : CommRingCat.of K ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.residue x) (CategoryTheory.CategoryStruct.comp (descResidueField f) h) = CategoryTheory.CategoryStruct.comp f h

def AlgebraicGeometry.Scheme.evaluation (X : Scheme) (U : X.Opens) (x : ↥X) (hx : x ∈ U) : X.presheaf.obj (Opposite.op U) ⟶ X.residueField x

If U is an open of X containing x, we have a canonical ring map from the sections over U to the residue field of x.

If we interpret sections over U as functions of X defined on U, then this ring map corresponds to evaluation at x.

theorem AlgebraicGeometry.Scheme.germ_residue (X : Scheme) {U : X.Opens} (x : ↥X) (hx : x ∈ U) : CategoryTheory.CategoryStruct.comp (X.presheaf.germ U x hx) (X.residue x) = X.evaluation U x hx

theorem AlgebraicGeometry.Scheme.germ_residue_assoc (X : Scheme) {U : X.Opens} (x : ↥X) (hx : x ∈ U) {Z : CommRingCat} (h : X.residueField x ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.presheaf.germ U x hx) (CategoryTheory.CategoryStruct.comp (X.residue x) h) = CategoryTheory.CategoryStruct.comp (X.evaluation U x hx) h

@[reducible, inline]

abbrev AlgebraicGeometry.Scheme.Γevaluation (X : Scheme) (x : ↥X) : X.presheaf.obj (Opposite.op ⊤) ⟶ X.residueField x

The global evaluation map from Γ(X, ⊤) to the residue field at x.

@[simp]

theorem AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen (X : Scheme) {U : X.Opens} (x : ↥X) (hx : x ∈ U) (f : ↑(X.presheaf.obj (Opposite.op U))) : (CategoryTheory.ConcreteCategory.hom (X.evaluation U x hx)) f = 0 ↔ x ∉ X.basicOpen f

@[deprecated AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen (since := "2025-05-23")]

theorem AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_not_mem_basicOpen (X : Scheme) {U : X.Opens} (x : ↥X) (hx : x ∈ U) (f : ↑(X.presheaf.obj (Opposite.op U))) : (CategoryTheory.ConcreteCategory.hom (X.evaluation U x hx)) f = 0 ↔ x ∉ X.basicOpen f

Alias of AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen.

theorem AlgebraicGeometry.Scheme.evaluation_ne_zero_iff_mem_basicOpen (X : Scheme) {U : X.Opens} (x : ↥X) (hx : x ∈ U) (f : ↑(X.presheaf.obj (Opposite.op U))) : (CategoryTheory.ConcreteCategory.hom (X.evaluation U x hx)) f ≠ 0 ↔ x ∈ X.basicOpen f

theorem AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero (X : Scheme) {U : X.Opens} (f : ↑(X.presheaf.obj (Opposite.op U))) : X.basicOpen f = ⊥ ↔ ∀ (x : ↥U), (CategoryTheory.ConcreteCategory.hom (X.evaluation U ↑x ⋯)) f = 0

def AlgebraicGeometry.Scheme.Hom.residueFieldMap {X Y : Scheme} (f : X.Hom Y) (x : ↥X) : Y.residueField ((CategoryTheory.ConcreteCategory.hom f.base) x) ⟶ X.residueField x

If X ⟶ Y is a morphism of locally ringed spaces and x a point of X, we obtain a morphism of residue fields in the other direction.

theorem AlgebraicGeometry.Scheme.residue_residueFieldMap {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) : CategoryTheory.CategoryStruct.comp (Y.residue ((CategoryTheory.ConcreteCategory.hom f.base) x)) (Hom.residueFieldMap f x) = CategoryTheory.CategoryStruct.comp (Hom.stalkMap f x) (X.residue x)

theorem AlgebraicGeometry.Scheme.residue_residueFieldMap_assoc {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) {Z : CommRingCat} (h : X.residueField x ⟶ Z) : CategoryTheory.CategoryStruct.comp (Y.residue ((CategoryTheory.ConcreteCategory.hom f.base) x)) (CategoryTheory.CategoryStruct.comp (Hom.residueFieldMap f x) h) = CategoryTheory.CategoryStruct.comp (Hom.stalkMap f x) (CategoryTheory.CategoryStruct.comp (X.residue x) h)

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldMap_id {X : Scheme} (x : ↥X) : Hom.residueFieldMap (CategoryTheory.CategoryStruct.id X) x = CategoryTheory.CategoryStruct.id (X.residueField x)

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldMap_comp {X Y : Scheme} (f : X ⟶ Y) {Z : Scheme} (g : Y ⟶ Z) (x : ↥X) : Hom.residueFieldMap (CategoryTheory.CategoryStruct.comp f g) x = CategoryTheory.CategoryStruct.comp (Hom.residueFieldMap g ((CategoryTheory.ConcreteCategory.hom f.base) x)) (Hom.residueFieldMap f x)

theorem AlgebraicGeometry.Scheme.evaluation_naturality {X Y : Scheme} (f : X ⟶ Y) {V : Y.Opens} (x : ↥X) (hx : (CategoryTheory.ConcreteCategory.hom f.base) x ∈ V) : CategoryTheory.CategoryStruct.comp (Y.evaluation V ((CategoryTheory.ConcreteCategory.hom f.base) x) hx) (Hom.residueFieldMap f x) = CategoryTheory.CategoryStruct.comp (Hom.app f V) (X.evaluation ((TopologicalSpace.Opens.map f.base).obj V) x hx)

theorem AlgebraicGeometry.Scheme.evaluation_naturality_assoc {X Y : Scheme} (f : X ⟶ Y) {V : Y.Opens} (x : ↥X) (hx : (CategoryTheory.ConcreteCategory.hom f.base) x ∈ V) {Z : CommRingCat} (h : X.residueField x ⟶ Z) : CategoryTheory.CategoryStruct.comp (Y.evaluation V ((CategoryTheory.ConcreteCategory.hom f.base) x) hx) (CategoryTheory.CategoryStruct.comp (Hom.residueFieldMap f x) h) = CategoryTheory.CategoryStruct.comp (Hom.app f V) (CategoryTheory.CategoryStruct.comp (X.evaluation ((TopologicalSpace.Opens.map f.base).obj V) x hx) h)

theorem AlgebraicGeometry.Scheme.evaluation_naturality_apply {X Y : Scheme} (f : X ⟶ Y) {V : Y.Opens} (x : ↥X) (hx : (CategoryTheory.ConcreteCategory.hom f.base) x ∈ V) (s : ↑(Y.presheaf.obj (Opposite.op V))) : (CategoryTheory.ConcreteCategory.hom (Hom.residueFieldMap f x)) ((CategoryTheory.ConcreteCategory.hom (Y.evaluation V ((CategoryTheory.ConcreteCategory.hom f.base) x) hx)) s) = (CategoryTheory.ConcreteCategory.hom (X.evaluation ((TopologicalSpace.Opens.map f.base).obj V) x hx)) ((CategoryTheory.ConcreteCategory.hom (Hom.app f V)) s)

theorem AlgebraicGeometry.Scheme.Γevaluation_naturality {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) : CategoryTheory.CategoryStruct.comp (Y.Γevaluation ((CategoryTheory.ConcreteCategory.hom f.base) x)) (Hom.residueFieldMap f x) = CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op ⊤)) (X.Γevaluation x)

theorem AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) {Z : CommRingCat} (h : X.residueField x ⟶ Z) : CategoryTheory.CategoryStruct.comp (Y.Γevaluation ((CategoryTheory.ConcreteCategory.hom f.base) x)) (CategoryTheory.CategoryStruct.comp (Hom.residueFieldMap f x) h) = CategoryTheory.CategoryStruct.comp (f.c.app (Opposite.op ⊤)) (CategoryTheory.CategoryStruct.comp (X.Γevaluation x) h)

theorem AlgebraicGeometry.Scheme.Γevaluation_naturality_apply {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) (a : ↑(Y.presheaf.obj (Opposite.op ⊤))) : (CategoryTheory.ConcreteCategory.hom (Hom.residueFieldMap f x)) ((CategoryTheory.ConcreteCategory.hom (Y.Γevaluation ((CategoryTheory.ConcreteCategory.hom f.base) x))) a) = (CategoryTheory.ConcreteCategory.hom (X.Γevaluation x)) ((CategoryTheory.ConcreteCategory.hom (f.c.app (Opposite.op ⊤))) a)

instance AlgebraicGeometry.Scheme.instIsIsoCommRingCatResidueFieldMapOfIsOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [IsOpenImmersion f] (x : ↥X) : CategoryTheory.IsIso (Hom.residueFieldMap f x)

def AlgebraicGeometry.Scheme.residueFieldCongr {X : Scheme} {x y : ↥X} (h : x = y) : X.residueField x ≅ X.residueField y

The isomorphism between residue fields of equal points.

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_refl {X : Scheme} {x : ↥X} : residueFieldCongr ⋯ = CategoryTheory.Iso.refl (X.residueField x)

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_symm {X : Scheme} {x y : ↥X} (e : x = y) : (residueFieldCongr e).symm = residueFieldCongr ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_inv {X : Scheme} {x y : ↥X} (e : x = y) : (residueFieldCongr e).inv = (residueFieldCongr ⋯).hom

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_trans {X : Scheme} {x y z : ↥X} (e : x = y) (e' : y = z) : residueFieldCongr e ≪≫ residueFieldCongr e' = residueFieldCongr ⋯

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_trans_hom (X : Scheme) {x y z : ↥X} (e : x = y) (e' : y = z) : CategoryTheory.CategoryStruct.comp (residueFieldCongr e).hom (residueFieldCongr e').hom = (residueFieldCongr ⋯).hom

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_trans_hom_assoc (X : Scheme) {x y z : ↥X} (e : x = y) (e' : y = z) {Z : CommRingCat} (h : X.residueField z ⟶ Z) : CategoryTheory.CategoryStruct.comp (residueFieldCongr e).hom (CategoryTheory.CategoryStruct.comp (residueFieldCongr e').hom h) = CategoryTheory.CategoryStruct.comp (residueFieldCongr ⋯).hom h

theorem AlgebraicGeometry.Scheme.residue_residueFieldCongr (X : Scheme) {x y : ↥X} (h : x = y) : CategoryTheory.CategoryStruct.comp (X.residue x) (residueFieldCongr h).hom = CategoryTheory.CategoryStruct.comp (X.presheaf.stalkCongr ⋯).hom (X.residue y)

theorem AlgebraicGeometry.Scheme.residue_residueFieldCongr_assoc (X : Scheme) {x y : ↥X} (h : x = y) {Z : CommRingCat} (h✝ : X.residueField y ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.residue x) (CategoryTheory.CategoryStruct.comp (residueFieldCongr h).hom h✝) = CategoryTheory.CategoryStruct.comp (X.presheaf.stalkCongr ⋯).hom (CategoryTheory.CategoryStruct.comp (X.residue y) h✝)

theorem AlgebraicGeometry.Scheme.Hom.residueFieldMap_congr {X Y : Scheme} {f g : X ⟶ Y} (e : f = g) (x : ↥X) : residueFieldMap f x = CategoryTheory.CategoryStruct.comp (residueFieldCongr ⋯).hom (residueFieldMap g x)

def AlgebraicGeometry.Scheme.fromSpecResidueField (X : Scheme) (x : ↥X) : Spec (X.residueField x) ⟶ X

The canonical map Spec κ(x) ⟶ X.

instance AlgebraicGeometry.Scheme.instIsPreimmersionFromSpecResidueField {X : Scheme} (x : ↥X) : IsPreimmersion (X.fromSpecResidueField x)

noncomputable instance AlgebraicGeometry.Scheme.instOverSpecResidueField {X : Scheme} (x : ↥X) : (Spec (X.residueField x)).Over X

@[simp]

theorem AlgebraicGeometry.Scheme.over_def {X : Scheme} (x : ↥X) : Spec (X.residueField x) ↘ X = X.fromSpecResidueField x

noncomputable instance AlgebraicGeometry.Scheme.instCanonicallyOverSpecResidueField {X : Scheme} (x : ↥X) : (Spec (X.residueField x)).CanonicallyOver X

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField {X : Scheme} {x y : ↥X} (h : x = y) : CategoryTheory.CategoryStruct.comp (Spec.map (residueFieldCongr h).hom) (X.fromSpecResidueField x) = X.fromSpecResidueField y

@[simp]

theorem AlgebraicGeometry.Scheme.residueFieldCongr_fromSpecResidueField_assoc {X : Scheme} {x y : ↥X} (h : x = y) {Z : Scheme} (h✝ : X ⟶ Z) : CategoryTheory.CategoryStruct.comp (Spec.map (residueFieldCongr h).hom) (CategoryTheory.CategoryStruct.comp (X.fromSpecResidueField x) h✝) = CategoryTheory.CategoryStruct.comp (X.fromSpecResidueField y) h✝

instance AlgebraicGeometry.Scheme.instIsOverMapHomCommRingCatResidueFieldCongr {X : Scheme} {x y : ↥X} (h : x = y) : Hom.IsOver (Spec.map (residueFieldCongr h).hom) X

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) : CategoryTheory.CategoryStruct.comp (Spec.map (residueFieldMap f x)) (Y.fromSpecResidueField ((CategoryTheory.ConcreteCategory.hom f.base) x)) = CategoryTheory.CategoryStruct.comp (X.fromSpecResidueField x) f

@[simp]

theorem AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc {X Y : Scheme} (f : X ⟶ Y) (x : ↥X) {Z : Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (Spec.map (residueFieldMap f x)) (CategoryTheory.CategoryStruct.comp (Y.fromSpecResidueField ((CategoryTheory.ConcreteCategory.hom f.base) x)) h) = CategoryTheory.CategoryStruct.comp (X.fromSpecResidueField x) (CategoryTheory.CategoryStruct.comp f h)

instance AlgebraicGeometry.Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass {X Y : Scheme} [X.Over Y] (x : ↥X) : Hom.IsOver (Spec.map (Hom.residueFieldMap (X ↘ Y) x)) Y

@[simp]

theorem AlgebraicGeometry.Scheme.fromSpecResidueField_apply {X : Scheme} (x : ↥X) (s : ↥(Spec (X.residueField x))) : (CategoryTheory.ConcreteCategory.hom (X.fromSpecResidueField x).base) s = x

theorem AlgebraicGeometry.Scheme.range_fromSpecResidueField {X : Scheme} (x : ↥X) : Set.range ⇑(CategoryTheory.ConcreteCategory.hom (X.fromSpecResidueField x).base) = {x}

theorem AlgebraicGeometry.Scheme.descResidueField_fromSpecResidueField {K : Type u_1} [Field K] (X : Scheme) {x : ↥X} (f : X.presheaf.stalk x ⟶ CommRingCat.of K) [IsLocalHom (CommRingCat.Hom.hom f)] : CategoryTheory.CategoryStruct.comp (Spec.map (descResidueField f)) (X.fromSpecResidueField x) = CategoryTheory.CategoryStruct.comp (Spec.map f) (X.fromSpecStalk x)

theorem AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField (K : Type u) [Field K] (X : Scheme) (f : Spec (CommRingCat.of K) ⟶ X) : CategoryTheory.CategoryStruct.comp (Spec.map (descResidueField (stalkClosedPointTo f))) (X.fromSpecResidueField ((CategoryTheory.ConcreteCategory.hom f.base) (IsLocalRing.closedPoint K))) = f

theorem AlgebraicGeometry.Scheme.SpecToEquivOfField_eq_iff {K : Type u_1} [Field K] {X : Scheme} {f₁ f₂ : (x : ↥X) × (X.residueField x ⟶ CommRingCat.of K)} : f₁ = f₂ ↔ ∃ (e : f₁.fst = f₂.fst), f₁.snd = CategoryTheory.CategoryStruct.comp (residueFieldCongr e).hom f₂.snd

A helper lemma to work with AlgebraicGeometry.Scheme.SpecToEquivOfField.

def AlgebraicGeometry.Scheme.SpecToEquivOfField (K : Type u) [Field K] (X : Scheme) : (Spec (CommRingCat.of K) ⟶ X) ≃ (x : ↥X) × (X.residueField x ⟶ CommRingCat.of K)

For a field K and a scheme X, the morphisms Spec K ⟶ X bijectively correspond to pairs of points x of X and embeddings κ(x) ⟶ K.