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[Merged by Bors] - chore(AlgebraicGeometry): add LocallyRingedSpace.residue #41087

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[Merged by Bors] - chore(AlgebraicGeometry): add LocallyRingedSpace.residue #41087

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b5c3e88
residue
chrisflav Jun 26, 2026
5ad1498
fix
chrisflav Jun 26, 2026
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2 changes: 1 addition & 1 deletion Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -99,7 +99,7 @@ instance : Quiver LocallyRingedSpace :=
/-- A morphism of locally ringed spaces `f : X ⟶ Y` induces
a local ring homomorphism from `Y.stalk (f x)` to `X.stalk x` for any `x : X`.
-/
noncomputable def Hom.stalkMap {X Y : LocallyRingedSpace.{u}} (f : Hom X Y) (x : X) :
noncomputable def Hom.stalkMap {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X) :
Y.presheaf.stalk (f.1.1 x) ⟶ X.presheaf.stalk x :=
f.toShHom.hom.stalkMap x

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19 changes: 13 additions & 6 deletions Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
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Expand Up @@ -48,6 +48,16 @@ def residueField (x : X) : CommRingCat :=
instance (x : X) : Field (X.residueField x) :=
inferInstanceAs <| Field (IsLocalRing.ResidueField (X.presheaf.stalk x))

/-- The residue map from the stalk to the residue field. -/
def residue (X : LocallyRingedSpace.{u}) (x : X) : X.presheaf.stalk x ⟶ X.residueField x :=
CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk x))

lemma residue_surjective (x : X) : Function.Surjective (X.residue x) :=
Ideal.Quotient.mk_surjective

instance (x : X) : Epi (X.residue x) :=
ConcreteCategory.epi_of_surjective _ (X.residue_surjective 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`.
Expand All @@ -56,9 +66,7 @@ If we interpret sections over `U` as functions of `X` defined on `U`, then this
corresponds to evaluation at `x`.
-/
def evaluation (x : U) : X.presheaf.obj (op U) ⟶ X.residueField x :=
-- TODO: make a new definition wrapping
-- `CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk _))`?
X.presheaf.germ U x.1 x.2 ≫ CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk _))
X.presheaf.germ U x.1 x.2 ≫ X.residue _

/-- The global evaluation map from `Γ(X, ⊤)` to the residue field at `x`. -/
def Γevaluation (x : X) : X.presheaf.obj (op ⊤) ⟶ X.residueField x :=
Expand Down Expand Up @@ -97,10 +105,9 @@ a morphism of residue fields in the other direction. -/
def residueFieldMap (x : X) : Y.residueField (f.base x) ⟶ X.residueField x :=
CommRingCat.ofHom (IsLocalRing.ResidueField.map (f.stalkMap x).hom)

@[reassoc]
lemma residue_comp_residueFieldMap_eq_stalkMap_comp_residue (x : X) :
CommRingCat.ofHom (IsLocalRing.residue (Y.presheaf.stalk (f.base x))) ≫
residueFieldMap f x = f.stalkMap x ≫
CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk x)) := by
Y.residue _ ≫ residueFieldMap f x = f.stalkMap x ≫ X.residue _ := by
simp [residueFieldMap]
rfl

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