The cotangent space at a nodal point

Let A be a local algebra, essentially of finite type over a field K, and let I = Fitt₁(Ω[A⁄K]) be the first Fitting ideal of its differentials. This ideal cuts out the singular locus; the scheme-theoretic criterion for a nodal curve asks that A ⧸ I be unramified over K. If the closed point is singular, that is I ≠ A, then:

• I = 𝔪: an unramified local K-algebra is a field, so the singular locus is reduced (Ideal.eq_maximalIdeal_of_formallyUnramified_quotient);
• dim_κ κ ⊗[A] Ω[A⁄K] = 2: it is at least two because the point is singular, and at most two because Fitt₁(Ω[A⁄K]) ≤ 𝔪 ^ (dim_κ κ ⊗[A] Ω[A⁄K] - 1), while 𝔪 ≤ 𝔪² would force 𝔪 = 0 by Nakayama's lemma, and then Ω[A⁄K] = 0;
• dim_κ 𝔪/𝔪² = 2: the residue field is separable over K, so the conormal sequence identifies the Zariski cotangent space 𝔪/𝔪² with κ ⊗[A] Ω[A⁄K].

Since I = 𝔪, the last statement says that the conormal space I/I² of the singular locus at a node is a plane; the two branches through the node are lines in this plane. No syntomic or dimension hypothesis is needed for these statements.

Localizing, the same holds at every point of the singular locus of an algebra A essentially of finite type over K with A ⧸ I unramified over K: there the fibre of Ω[A⁄K] has dimension two. This is the form used for nodal curves.

Main results

• TauCeti.finrank_residueField_tensor_kaehlerDifferential_eq_two: dim_κ κ ⊗[A] Ω[A⁄K] = 2.
• TauCeti.formallyEtale_residueField_of_fittingIdeal_ne_top: the residue field is formally étale over K.
• TauCeti.finrank_cotangentSpace_eq_two: dim_κ 𝔪/𝔪² = 2.
• TauCeti.formallyUnramified_localization_quotient_fittingIdeal: the unramified singular locus localizes.
• TauCeti.finrank_tensor_kaehlerDifferential_eq_two: for A not necessarily local, the fibre of Ω[A⁄K] at every point of V(I) has dimension two.

References

• Stacks Project, Section 53.19, Tag 0C46: nodal curves; a node has a two-dimensional tangent space.
• Stacks Project, Lemma 53.20.1, Tag 0C58: the singular-locus criterion for families of nodal curves.

theorem TauCeti.finrank_residueField_tensor_kaehlerDifferential_eq_two {K : Type u_1} {A : Type u_2} [Field K] [CommRing A] [Algebra K A] [IsLocalRing A] [Algebra.EssFiniteType K A] [Algebra.FormallyUnramified K (A ⧸ fittingIdeal A Ω[A⁄K] 1)] (h : fittingIdeal A Ω[A⁄K] 1 ≠ ⊤) : Module.finrank (IsLocalRing.ResidueField A) (TensorProduct A (IsLocalRing.ResidueField A) Ω[A⁄K]) = 2

The differentials at a node need two generators. If the first Fitting ideal of Ω[A⁄K] is a proper ideal of the local K-algebra A with unramified quotient, the fibre of Ω[A⁄K] at the closed point is a plane.

theorem TauCeti.formallyEtale_residueField_of_fittingIdeal_ne_top {K : Type u_1} {A : Type u_2} [Field K] [CommRing A] [Algebra K A] [IsLocalRing A] [Algebra.EssFiniteType K A] [Algebra.FormallyUnramified K (A ⧸ fittingIdeal A Ω[A⁄K] 1)] (h : fittingIdeal A Ω[A⁄K] 1 ≠ ⊤) : Algebra.FormallyEtale K (IsLocalRing.ResidueField A)

The residue field at a node is étale. If the first Fitting ideal of Ω[A⁄K] is a proper ideal of the local K-algebra A with unramified quotient, then the residue field of A, a quotient of A ⧸ Fitt₁(Ω[A⁄K]), is separable and hence formally étale over K.

theorem TauCeti.finrank_cotangentSpace_eq_two {K : Type u_1} {A : Type u_2} [Field K] [CommRing A] [Algebra K A] [IsLocalRing A] [Algebra.EssFiniteType K A] [Algebra.FormallyUnramified K (A ⧸ fittingIdeal A Ω[A⁄K] 1)] (h : fittingIdeal A Ω[A⁄K] 1 ≠ ⊤) : Module.finrank (IsLocalRing.ResidueField A) (IsLocalRing.CotangentSpace A) = 2

The tangent space at a node is a plane. If the first Fitting ideal of Ω[A⁄K] is a proper ideal of the local K-algebra A with unramified quotient, the Zariski cotangent space 𝔪/𝔪² has dimension two.

theorem TauCeti.formallyUnramified_localization_quotient_fittingIdeal {K : Type u_1} {A : Type u_2} [CommRing K] [CommRing A] [Algebra K A] [Algebra.EssFiniteType K A] [Algebra.FormallyUnramified K (A ⧸ fittingIdeal A Ω[A⁄K] 1)] (M : Submonoid A) : Algebra.FormallyUnramified K (Localization M ⧸ fittingIdeal (Localization M) Ω[Localization M⁄K] 1)

The unramified singular locus localizes. Let A be essentially of finite type over a ring K, with A ⧸ Fitt₁(Ω[A⁄K]) unramified over K. Then the same holds for every localization of A: its singular locus is the localization of that of A.

theorem TauCeti.finrank_tensor_kaehlerDifferential_eq_two {K : Type u_1} {A : Type u_2} [Field K] [CommRing A] [Algebra K A] [Algebra.EssFiniteType K A] [Algebra.FormallyUnramified K (A ⧸ fittingIdeal A Ω[A⁄K] 1)] {L : Type u_3} [Field L] [Algebra A L] (h : fittingIdeal A Ω[A⁄K] 1 ≤ RingHom.ker (algebraMap A L)) : Module.finrank L (TensorProduct A L Ω[A⁄K]) = 2

The differentials along an unramified singular locus. Let A be essentially of finite type over a field K, with A ⧸ Fitt₁(Ω[A⁄K]) unramified over K. At every point of the singular locus V(Fitt₁(Ω[A⁄K])), given by a field L over A, the fibre of Ω[A⁄K] has dimension two.