The degree of a sum of relative effective Cartier divisors #
The degree of a relative effective Cartier divisor D on X over S, finite over S, is the
rank of D ⟶ S. This file proves that the degree is additive: deg (D + E) = deg D + deg E,
where the sum D + E is cut out by the product of the ideal sheaves
(Scheme.IdealSheafData.IsRelativeEffectiveCartier.finrank_mul_subschemeι_comp). Consequently the
sum of divisors of degrees d and e on the base changes of a proper relative curve has degree
d + e (TauCeti.AlgebraicGeometry.mul_mem_relativeEffectiveCartierDegreeSubfunctor_obj), so
addition of divisors restricts to the fixed-degree functors Div^d_{X/S}. Together with the empty
divisor having degree zero, this gives the degrees of multiples and of finite sums of divisors
(TauCeti.AlgebraicGeometry.pow_mem_relativeEffectiveCartierDegreeSubfunctor_obj and
TauCeti.AlgebraicGeometry.prod_mem_relativeEffectiveCartierDegreeSubfunctor_obj).
More generally, the rank of any finite sum of relative effective Cartier divisors whose total
divisor is finite over the base is the sum of their ranks
(TauCeti.AlgebraicGeometry.finrank_prod_subschemeι_comp).
Since ranks of finite flat morphisms and relative effective Cartier divisors are both compatible
with base change, the statement reduces to the fibres, that is, to a scheme Y over a field K
(Scheme.IdealSheafData.IsEffectiveCartier.finrank_mul_subschemeι_comp_of_field). There the
closed subscheme Z = V(I * J) is finite over K, so it is affine with a finite-dimensional ring
of global sections B, and V(I) and V(J) are cut out of Z by ideals 𝔞 and 𝔟 of B. Near
every point of Z, on an affine chart U of Y where I = (α) for a nonzerodivisor α, the
sections of Z are Γ(Y, U) ⧸ (α) J(U), the ideal 𝔞 is generated by α, and the annihilator of
α is 𝔟. The local criterion TauCeti.finrank_quotient_add_finrank_quotient_of_forall then
gives dim B = dim (B ⧸ 𝔞) + dim (B ⧸ 𝔟). Only I needs to be effective Cartier over a field;
the second ideal sheaf J is arbitrary.
References #
- The Stacks Project, Divisors, Section Effective Cartier divisors: the sum of effective Cartier divisors.
- S. Kleiman, The Picard scheme, in Fundamental Algebraic Geometry: Grothendieck's FGA Explained, Section 9.3.
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §1.1.
Additivity of degree over a field. Let I be an effective Cartier divisor and J any
ideal sheaf on a scheme Y over a field K, such that the closed subscheme V(I * J) is finite
over K. Then the degree of V(I * J) over K is the sum of the degrees of V(I) and V(J).
The degree of a sum of relative effective Cartier divisors is the sum of the degrees. If
I and J are relative effective Cartier divisors on X over S and the closed subscheme of
I * J, the divisor D + E, is finite over S, then its rank over S is the sum of the ranks of
the closed subschemes of I and J.
Ranks add for a finite sum of relative effective Cartier divisors whose total divisor is finite over the base.
Degrees add under addition of divisors. On the base changes of a proper relative curve,
the sum of relative effective Cartier divisors of degrees d and e has degree d + e.
The empty divisor has degree zero. On the base changes of a proper relative curve, the
identity of the monoid of relative effective Cartier divisors lies in Div⁰_{X/S}.
Degrees multiply under repeated addition. On the base changes of a proper relative curve,
n times a relative effective Cartier divisor of degree d has degree n * d.
Degrees add over finite sums of divisors. On the base changes of a proper relative curve,
if D i has degree d i for every i ∈ s, then the sum of the D i, written multiplicatively as
∏ i ∈ s, D i, has degree ∑ i ∈ s, d i.