Sections of smooth algebras of relative dimension one

Let B be an A-algebra together with an A-algebra retraction σ : B →ₐ[A] A, the algebraic form of a section of Spec B → Spec A. If B is standard smooth of relative dimension one over A, the ideal ker σ of the section is, near the section, generated by a single nonzerodivisor: there are t ∈ ker σ, a nonzerodivisor of B, and r ∈ B with σ r = 1 and r • ker σ ≤ (t), so that ker σ becomes the principal ideal (t) after inverting r. Geometrically, a section of a smooth relative curve is an effective Cartier divisor.

The proof uses an étale coordinate: a standard smooth algebra of relative dimension one is étale over a polynomial ring P = A[X₀] in one variable (Algebra.IsStandardSmoothOfRelativeDimension.exists_etale_mvPolynomial). The retraction σ restricts to the evaluation of P at a = σ X₀, whose kernel is generated by the nonzerodivisor u = X₀ - a, and its image t in B is a nonzerodivisor of B by flatness. Since B is formally unramified over P, Algebra.FormallyUnramified.exists_smul_ker_le_span_singleton provides r.

Main results

• Algebra.IsStandardSmoothOfRelativeDimension.exists_isSMulRegular_smul_ker_le_span_singleton: for a standard smooth algebra of relative dimension one, a nonzerodivisor t ∈ ker σ and an r with σ r = 1 and r • ker σ ≤ (t).

References

• The Stacks Project, Divisors, sections Regular immersions and Relative effective Cartier divisors: a section of a smooth morphism of relative dimension one is a relative effective Cartier divisor.

theorem Algebra.IsStandardSmoothOfRelativeDimension.exists_isSMulRegular_smul_ker_le_span_singleton {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [IsStandardSmoothOfRelativeDimension 1 A B] (σ : B →ₐ[A] A) : ∃ t ∈ RingHom.ker σ, IsSMulRegular B t ∧ ∃ (r : B), σ r = 1 ∧ r • RingHom.ker σ ≤ Ideal.span {t}

The local equation of a section of a smooth relative curve. Let B be a standard smooth A-algebra of relative dimension one and σ : B →ₐ[A] A a retraction. Then ker σ contains a nonzerodivisor t of B, and some r ∈ B with σ r = 1 satisfies r • ker σ ≤ (t); so after inverting r, an element which is a unit along the section, ker σ is generated by t.