The differentials of a hypersurface

Let P be a K-algebra whose module of differentials Ω[P⁄K] is free with a finite basis b indexed by ι, for instance a polynomial ring, or a localization of a smooth algebra on which Ω is free. For f ∈ P, the differentials of the hypersurface A = P ⧸ (f) are the quotient of A ⊗[P] Ω[P⁄K] ≅ Aⁿ, where n = |ι|, by the single relation 1 ⊗ df. Their (n - 1)-st Fitting ideal is therefore generated by the coordinates of df, the partial derivatives of f with respect to the dual basis. The results here are purely algebraic. When P is smooth of relative dimension n over a field K and f is a nonzero-divisor, so that A has relative dimension n - 1, this Fitting ideal cuts out the singular locus of A, and for n = 2 it is the Jacobian ideal (f, ∂f/∂s, ∂f/∂t) of a plane curve singularity, read in A.

Main results

• TauCeti.ker_mapBaseChange_quotient_span: for an ideal (s) of P, the kernel of (P ⧸ (s)) ⊗[P] Ω[P⁄K] → Ω[(P ⧸ (s))⁄K] is generated by the 1 ⊗ dg for g ∈ s; for A = P ⧸ (f) it is generated by 1 ⊗ df.
• TauCeti.span_one_tmul_D_eq_top_of_formallyUnramified: if P ⧸ (s) is unramified over K, the 1 ⊗ dg for g ∈ s span L ⊗[P] Ω[P⁄K] over every P-algebra L in which s vanishes, such as the residue field at a point of V(s).
• TauCeti.exists_basis_kaehlerDifferential_quotient: if the differentials of z : κ₁ ⊕ κ₂ → P form a basis of Ω[P⁄K], then those of the z (inr j) form a basis of the differentials of P ⧸ (z (inl i) : i ∈ κ₁).
• Module.Basis.mem_span_range_sq_of_D_mem: if the differentials of x form a basis of Ω[P⁄K], an element of (x) whose differential lies in (x) • Ω[P⁄K] lies in (x) ^ 2.
• TauCeti.fittingIdeal_kaehlerDifferential_quotient_span_range: the k-th Fitting ideal of the differentials of P ⧸ (g₁, …, g_c) is generated by the (n - k)-minors of the Jacobian matrix of the gⱼ.
• TauCeti.fittingIdeal_kaehlerDifferential_quotient_span_singleton: the (n - 1)-st Fitting ideal of Ω[A⁄K] is generated by the images of the coordinates of df.
• TauCeti.comap_fittingIdeal_kaehlerDifferential_quotient_span_singleton: its preimage in P is the Jacobian ideal, generated by f and the coordinates of df.

References

• Stacks Project, Tag 00RU: the conormal sequence for a quotient, here Mathlib's KaehlerDifferential.exact_kerCotangentToTensor_mapBaseChange.
• Stacks Project, Tag 07Z6: Fitting ideals.

theorem TauCeti.ker_mapBaseChange_quotient_span {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] (s : Set P) : (KaehlerDifferential.mapBaseChange K P (P ⧸ Ideal.span s)).ker = Submodule.span (P ⧸ Ideal.span s) ((fun (g : P) => 1 ⊗ₜ[P] (KaehlerDifferential.D K P) g) '' s)

The differentials of P ⧸ (s) are those of P, restricted to P ⧸ (s), modulo the differentials of the generators: the kernel of (P ⧸ (s)) ⊗[P] Ω[P⁄K] → Ω[(P ⧸ (s))⁄K] is generated by the elements 1 ⊗ dg for g ∈ s.

theorem TauCeti.span_one_tmul_D_eq_top_of_formallyUnramified {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] (s : Set P) [Algebra.FormallyUnramified K (P ⧸ Ideal.span s)] (L : Type u_3) [CommRing L] [Algebra P L] (hs : ∀ g ∈ s, (algebraMap P L) g = 0) : Submodule.span L ((fun (g : P) => 1 ⊗ₜ[P] (KaehlerDifferential.D K P) g) '' s) = ⊤

Generators of an unramified quotient have spanning differentials. If P ⧸ (s) is formally unramified over K, then for every P-algebra L in which the elements of s vanish, the elements 1 ⊗ dg for g ∈ s span L ⊗[P] Ω[P⁄K] over L. With L the residue field at a point of V(s), this says that the differentials of the generators span the fibre of Ω[P⁄K] there.

theorem TauCeti.exists_basis_kaehlerDifferential_quotient {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] {κ₁ : Type u_3} {κ₂ : Type u_4} (b : Module.Basis (κ₁ ⊕ κ₂) P Ω[P⁄K]) {z : κ₁ ⊕ κ₂ → P} (hb : ∀ (k : κ₁ ⊕ κ₂), b k = (KaehlerDifferential.D K P) (z k)) : ∃ (b' : Module.Basis κ₂ (P ⧸ Ideal.span (Set.range (z ∘ Sum.inl))) Ω[P ⧸ Ideal.span (Set.range (z ∘ Sum.inl))⁄K]), ∀ (j : κ₂), b' j = (KaehlerDifferential.D K (P ⧸ Ideal.span (Set.range (z ∘ Sum.inl)))) ((Ideal.Quotient.mk (Ideal.span (Set.range (z ∘ Sum.inl)))) (z (Sum.inr j)))

Quotient by some coordinates. If the differentials of z : κ₁ ⊕ κ₂ → P form a basis of Ω[P⁄K], then the differentials of the remaining elements z (inr j) form a basis of the differentials of P ⧸ (z (inl i) : i ∈ κ₁).

theorem Module.Basis.mem_span_range_sq_of_D_mem {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] {ι : Type u_3} {x : ι → P} (b : Basis ι P Ω[P⁄K]) (hb : ∀ (i : ι), b i = (KaehlerDifferential.D K P) (x i)) {g : P} (hg : g ∈ Ideal.span (Set.range x)) (hDg : (KaehlerDifferential.D K P) g ∈ Ideal.span (Set.range x) • ⊤) : g ∈ Ideal.span (Set.range x) ^ 2

Functions critical along a coordinate section vanish to second order. If the differentials of x : ι → P form a basis of Ω[P⁄K] and g ∈ (x) has differential in (x) • Ω[P⁄K], then g ∈ (x) ^ 2. Writing g = ∑ cᵢ xᵢ, the coefficient of dxⱼ in dg is cⱼ modulo (x).

theorem TauCeti.fittingIdeal_kaehlerDifferential_quotient_span_range {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] {ι : Type u_3} {σ : Type u_4} [Fintype ι] [Finite σ] (g : σ → P) [Module.Finite (P ⧸ Ideal.span (Set.range g)) Ω[P ⧸ Ideal.span (Set.range g)⁄K]] (b : Module.Basis ι P Ω[P⁄K]) (k : ℕ) : fittingIdeal (P ⧸ Ideal.span (Set.range g)) Ω[P ⧸ Ideal.span (Set.range g)⁄K] k = Ideal.map (Ideal.Quotient.mk (Ideal.span (Set.range g))) (Ideal.span (Set.range fun (rc : (Fin (Fintype.card ι - k) → ι) × (Fin (Fintype.card ι - k) → σ)) => ((Matrix.of fun (i : ι) (j : σ) => (b.coord i) ((KaehlerDifferential.D K P) (g j))).submatrix rc.1 rc.2).det))

The Fitting ideals of the differentials of a quotient. If Ω[P⁄K] has a basis b indexed by ι, the k-th Fitting ideal of the differentials of P ⧸ (g) is generated by the images of the (|ι| - k) × (|ι| - k) minors of the Jacobian matrix (∂ᵢ gⱼ), whose entries are the coordinates of the dgⱼ in b. Rows and columns of a minor may be chosen independently.

theorem TauCeti.fittingIdeal_kaehlerDifferential_quotient_span_singleton {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] {ι : Type u_3} [Fintype ι] [Nonempty ι] (f : P) [Module.Finite (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K]] (b : Module.Basis ι P Ω[P⁄K]) : fittingIdeal (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K] (Fintype.card ι - 1) = Ideal.map (Ideal.Quotient.mk (Ideal.span {f})) (Ideal.span (Set.range fun (i : ι) => (b.coord i) ((KaehlerDifferential.D K P) f)))

The Fitting ideal of the differentials of a hypersurface. If Ω[P⁄K] has a basis b indexed by ι, the (|ι| - 1)-st Fitting ideal of the differentials of P ⧸ (f) is generated by the images of the coordinates of df in b, the partial derivatives of f.

theorem TauCeti.comap_fittingIdeal_kaehlerDifferential_quotient_span_singleton {K : Type u_1} {P : Type u_2} [CommRing K] [CommRing P] [Algebra K P] {ι : Type u_3} [Fintype ι] [Nonempty ι] (f : P) [Module.Finite (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K]] (b : Module.Basis ι P Ω[P⁄K]) : Ideal.comap (Ideal.Quotient.mk (Ideal.span {f})) (fittingIdeal (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K] (Fintype.card ι - 1)) = Ideal.span (insert f (Set.range fun (i : ι) => (b.coord i) ((KaehlerDifferential.D K P) f)))

The Jacobian ideal of a hypersurface. If Ω[P⁄K] has a basis b indexed by ι, the preimage in P of the (|ι| - 1)-st Fitting ideal of the differentials of P ⧸ (f) is generated by f and the coordinates of df in b, the partial derivatives of f.