Finiteness of the relative normalization

Let f : X ⟶ Y be quasi-compact and quasi-separated. Over an affine open U of Y, the normalization of Y in X is the spectrum of the integral closure of Γ(Y, U) in Γ(X, f ⁻¹ᵁ U). Since the normalization is an affine morphism, it is finite over Y exactly when each of these integral closures is a finite Γ(Y, U)-module.

For an integral scheme Y and a finite extension L of its function field k(Y), consider the normalization of Y in Spec L, along Spec L ⟶ Spec k(Y) ⟶ Y. Over a nonempty affine open U, the sections of Spec L are L, and k(Y) is the fraction field of Γ(Y, U). So this normalization is finite when the rings Γ(Y, U) are Japanese; for instance when they are Nagata, as a Nagata domain is Japanese. For L = k(Y) itself, the normalization of Y in its function field is finite exactly when the rings Γ(Y, U) are N-1. This supplies the finiteness hypothesis of resolution of curves by normalization (TauCeti.AlgebraicGeometry.isRegular_normalization_of_field) for curves whose affine opens have Japanese rings.

Main results

• AlgebraicGeometry.Scheme.Hom.isFinite_fromNormalization_iff: the normalization of Y in X is finite over Y if and only if, over every affine open U of Y, the integral closure of Γ(Y, U) in Γ(X, f ⁻¹ᵁ U) is a finite Γ(Y, U)-module.
• TauCeti.AlgebraicGeometry.isFinite_fromNormalization_fromSpecStalk_genericPoint_iff: the normalization of an integral scheme in its function field is finite if and only if the ring of sections of every nonempty affine open is N-1.
• TauCeti.AlgebraicGeometry.isFinite_fromNormalization_of_isJapanese: if the ring of sections of every nonempty affine open of an integral scheme Y is Japanese, the normalization of Y in a finite extension of its function field is finite over Y (Stacks, Lemma 29.54.14 (0AVK), with the hypothesis on every affine open).

References

• Stacks Project, Lemma 29.54.14.
• Stacks Project, Tag 035H, for the relative normalization over affine opens.

theorem AlgebraicGeometry.Scheme.Hom.isFinite_fromNormalization_iff {X Y : Scheme} (f : X ⟶ Y) [QuasiCompact f] [QuasiSeparated f] : IsFinite (fromNormalization f) ↔ ∀ (U : Y.Opens), IsAffineOpen U → Module.Finite ↑(Y.presheaf.obj (Opposite.op U)) ↥(integralClosure ↑(Y.presheaf.obj (Opposite.op U)) ↑(X.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map f.base).obj U))))

The normalization of Y in X, along f, is finite over Y if and only if, over every affine open U of Y, the integral closure of Γ(Y, U) in Γ(X, f ⁻¹ᵁ U) is a finite Γ(Y, U)-module.

theorem TauCeti.AlgebraicGeometry.isFinite_fromNormalization_fromSpecStalk_genericPoint_iff {Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral Y] : AlgebraicGeometry.IsFinite (AlgebraicGeometry.Scheme.Hom.fromNormalization (Y.fromSpecStalk (genericPoint ↥Y))) ↔ ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → Nonempty ↥↑U → Ring.IsN1 ↑(Y.presheaf.obj (Opposite.op U))

Finite normalization and N-1 rings. The normalization of an integral scheme Y in its function field, along Spec k(Y) ⟶ Y, is finite over Y if and only if the ring of sections of every nonempty affine open of Y is N-1: over such an open U the normalization is the spectrum of the integral closure of Γ(Y, U) in its fraction field k(Y).

theorem TauCeti.AlgebraicGeometry.isFinite_fromNormalization_of_isJapanese {Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral Y] (L : Type u) [Field L] [Algebra (↑Y.functionField) L] (hY : ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → ∀ [inst : Nonempty ↥↑U], Ring.IsJapanese ↑(Y.presheaf.obj (Opposite.op U))) [FiniteDimensional (↑Y.functionField) L] : AlgebraicGeometry.IsFinite (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap (↑Y.functionField) L))) (Y.fromSpecStalk (genericPoint ↥Y))))

Finite normalization in a finite extension of the function field. Let Y be an integral scheme such that the ring of sections of every nonempty affine open is Japanese, for instance Nagata. Then for every finite extension L of the function field k(Y), the normalization of Y in Spec L, along Spec L ⟶ Spec k(Y) ⟶ Y, is finite over Y. Stacks, Lemma 29.54.14 (0AVK), with the hypothesis imposed on every nonempty affine open.