Additivity of the Abel map

For relative effective Cartier divisors, addition multiplies ideal sheaves. The ideal sheaf of the sum is the tensor product of the individual ideal sheaves, so the Abel map sends the sum to the product of their classes in the relative Picard presheaf. This identifies the group law that the Abel maps from symmetric powers must respect.

The construction follows Kleiman, The Picard scheme, Section 9.3, and the Stacks Project, Divisors, Section "Effective Cartier divisors".

theorem TauCeti.AlgebraicGeometry.abelMap_app_mul {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) (T : (CategoryTheory.Over S)ᵒᵖ) (D E : (relativeEffectiveCartierSubfunctor f).toFunctor.obj T) : (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := D * E } = (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := D } * (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := E }

The Abel map sends a sum of relative effective Cartier divisors to the tensor product of their line-bundle classes in the relative Picard presheaf.

noncomputable def TauCeti.AlgebraicGeometry.abelMapMonoidHom {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) (T : (CategoryTheory.Over S)ᵒᵖ) : (relativeEffectiveCartierSubfunctor f).toFunctor.obj T →* ↑((relativePicardPresheaf f).obj T)

At every base change, the Abel map is a monoid homomorphism from effective Cartier divisors under addition to the relative Picard group under tensor product.

theorem TauCeti.AlgebraicGeometry.abelMapMonoidHom_apply {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) (T : (CategoryTheory.Over S)ᵒᵖ) (D : (relativeEffectiveCartierSubfunctor f).toFunctor.obj T) : (abelMapMonoidHom f T) D = (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := D }

The monoid homomorphism agrees pointwise with the Abel map.