The singular locus along étale morphisms

Let g : X' ⟶ X be an étale morphism of relative curves over Spec R. Then the relative singular locus of X' is the inverse image of that of X: Sing(X'/R) = g⁻¹ Sing(X/R), as ideal sheaves and hence as closed subschemes. On affine charts V ⊆ g⁻¹ U, the ring map Γ(X, U) → Γ(X', V) is étale, so Ω[Γ(X', V)⁄R] is the base change of Ω[Γ(X, U)⁄R], and first Fitting ideals commute with base change.

Open immersions are étale, so this includes the restriction of the singular locus to open subschemes. Étale neighbourhoods are the charts in which the local normal form of a node is stated.

Main results

• AlgebraicGeometry.Scheme.singularLocus_ideal_eq_map_of_etale: on affine charts V ⊆ g⁻¹ U, the singular ideal of X' is the extension of that of X.
• AlgebraicGeometry.Scheme.singularLocus_eq_comap_of_etale: Sing(X'/R) = g⁻¹ Sing(X/R).
• AlgebraicGeometry.Scheme.isPullback_singularLocus_subschemeι_of_etale: the singular subscheme of X' is the base change Sing(X/R) ×_X X'.

References

• Stacks Project, Tag 0C3C: Fitting ideals of quasi-coherent modules and their pullback.
• Stacks Project, Tag 0C58: families of nodal curves.

theorem AlgebraicGeometry.Scheme.singularLocus_ideal_eq_map_of_etale {R : Type u} [CommRing R] {X X' : Scheme} [X.Over (Spec ↧R)] [X'.Over (Spec ↧R)] (g : X' ⟶ X) [Etale g] [Hom.IsOver g (Spec ↧R)] [Flat (X ↘ Spec ↧R)] [LocallyOfFinitePresentation (X ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X ↘ Spec ↧R)] [Flat (X' ↘ Spec ↧R)] [LocallyOfFinitePresentation (X' ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X' ↘ Spec ↧R)] (U : ↑X.affineOpens) (V : ↑X'.affineOpens) (e : ↑V ≤ (TopologicalSpace.Opens.map g.base).obj ↑U) : (singularLocus R X').ideal V = Ideal.map (CommRingCat.Hom.hom (Hom.appLE g (↑U) (↑V) e)) ((singularLocus R X).ideal U)

For an étale morphism g : X' ⟶ X of relative curves over Spec R and affine opens V ⊆ g⁻¹ U, the singular ideal of X' on V is generated by the image of the singular ideal of X on U.

theorem AlgebraicGeometry.Scheme.singularLocus_eq_comap_of_etale {R : Type u} [CommRing R] {X X' : Scheme} [X.Over (Spec ↧R)] [X'.Over (Spec ↧R)] (g : X' ⟶ X) [Etale g] [Hom.IsOver g (Spec ↧R)] [Flat (X ↘ Spec ↧R)] [LocallyOfFinitePresentation (X ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X ↘ Spec ↧R)] [Flat (X' ↘ Spec ↧R)] [LocallyOfFinitePresentation (X' ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X' ↘ Spec ↧R)] : singularLocus R X' = (singularLocus R X).comap g

The singular locus commutes with étale morphisms. If g : X' ⟶ X is an étale morphism of relative curves over Spec R, then the singular locus of X' is the inverse image under g of the singular locus of X. For the closed subschemes, see AlgebraicGeometry.Scheme.isPullback_singularLocus_subschemeι_of_etale.

theorem AlgebraicGeometry.Scheme.isPullback_singularLocus_subschemeι_of_etale {R : Type u} [CommRing R] {X X' : Scheme} [X.Over (Spec ↧R)] [X'.Over (Spec ↧R)] (g : X' ⟶ X) [Etale g] [Hom.IsOver g (Spec ↧R)] [Flat (X ↘ Spec ↧R)] [LocallyOfFinitePresentation (X ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X ↘ Spec ↧R)] [Flat (X' ↘ Spec ↧R)] [LocallyOfFinitePresentation (X' ↘ Spec ↧R)] [TauCeti.AlgebraicGeometry.PureRelativeDimension 1 (X' ↘ Spec ↧R)] : CategoryTheory.IsPullback (singularLocus R X').subschemeι ((singularLocus R X').subschemeMap (singularLocus R X) g ⋯) g (singularLocus R X).subschemeι

The singular subscheme commutes with étale morphisms. If g : X' ⟶ X is an étale morphism of relative curves over Spec R, then Sing(X'/R) is the base change Sing(X/R) ×_X X' of the closed subscheme Sing(X/R).