User:TakuyaMurata/Weil divisor

Weil divisors

Let X be an integral locally Noetherian scheme. A prime divisor or irreducible divisor on X is an integral closed subscheme Z of codimension 1 in X. A Weil divisor on X is a formal sum over the prime divisors Z of X,

$\sum _{Z}n_{Z}Z,$

where the collection $\{Z\colon n_{Z}\neq 0\}$ is locally finite. If X is quasi-compact, local finiteness is equivalent to $\{Z\colon n_{Z}\neq 0\}$ being finite. The group of all Weil divisors is denoted Div(X). A Weil divisor D is effective if all the coefficients are non-negative. One writes D ≥ D′ if the difference D − D′ is effective.

For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points. A divisor on Spec Z is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in Q. A similar characterization is true for divisors on $\operatorname {Spec} {\mathcal {O}}_{K}$, where K is a number field.

If Z ⊂ X is a prime divisor, then the local ring O_X,Z has Krull dimension one. If f ∈ O_X,Z is non-zero, then the order of vanishing of f along Z, written ord_Z(f), is the length of O_X,Z / (f). This length is finite, and it is additive with respect to multiplication, that is, ord_Z(fg) = ord_Z(f) + ord_Z(g). If k(X) is the field of rational functions on X, then any non-zero f ∈ k(X) may be written as a quotient g / h, where g and h are in O_X,Z, and the order of vanishing of f is defined to be ord_Z(g) - ord_Z(h). With this definition, the order of vanishing is a function ord_Z : k(X)^* → Z. If X is normal, then the local ring O_X,Z is a discrete valuation ring, and the function ord_Z is the corresponding valuation. For a non-zero rational function f on X, the principal Weil divisor associated to f is defined to be the Weil divisor

$\operatorname {div} f=\sum _{Z}\operatorname {ord} _{Z}(f)Z.$

It can be shown that this sum is locally finite and hence that it indeed defines a Weil divisor. The principal Weil divisor associated to f is also notated (f). If f is a regular function, then its principal Weil divisor is effective, but in general this is not true. The additivity of the order of vanishing function implies that

$\operatorname {div} fg=\operatorname {div} f+\operatorname {div} g.$

Consequently div is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors.

Let X be a normal integral Noetherian scheme. Every Weil divisor D determines a coherent sheaf O_X(D) on X whose local sections have poles at most those specified by D. Concretely it may be defined as subsheaf of the sheaf of rational functions

$\Gamma (U,{\mathcal {O}}_{X}(D))=\{f\in k(X)\colon f=0{\text{ or }}\operatorname {div} (f)+D\geq 0{\text{ on }}U\}.$

That is, a nonzero rational function f is a section of O_X(D) over U if and only if for any prime divisor Z intersecting U,

$\operatorname {ord} _{Z}(f)\geq -n_{z}$

where n_Z is the coefficient of Z in D. If D is principal, so D is the divisor of a rational function g, then there is an isomorphism O(D) → O_X via $f\mapsto fg$ (since $\operatorname {div} (fg)$ is an effective divisor and so fg is regular thanks to the normality of X.) Conversely, if O(D) is isomorphic to O_X as an O_X-module, then D is principal (roughly because a fractional idea in an integral domain is principal if and only if it is free and this fact remains valid in the sheaf-theoretic context.)

Note: even though O_X(D) may be defined as the subsheaf of the sheaf of rational functions, in practice, one usually does not see its sections as rational functions (this is the implemention detail). For example, if D is effective, the constant function 1 is a section of O_X(D); then the unique section of O_X(D) corresponds to 1 is called the canonical section and is denoted by s_D.

Assume that X is a normal integral separated scheme of finite type over a field. Let D be a Weil divisor. Then O(D) is a rank one reflexive sheaf, and since O(D) is defined as a subsheaf of M_X, it is a fractional ideal sheaf (see below). Conversely, every rank one reflexive sheaf corresponds to a Weil divisor: The sheaf can be restricted to the regular locus, where it becomes free and so corresponds to a Cartier divisor (again, see below), and because the singular locus has codimension at least two, the closure of the Cartier divisor is a Weil divisor.

Note: Each Cartier divisor D determines the isomorphism class of a line bundle say L(D) (see #Cartier divisors). If s_D is a nonzero rational section of L(D) such that $\operatorname {div} (s_{D})=D$, where $\operatorname {div} (s_{D})$ is defined just like for rational functions, then there is an isomorphism (depending on a choice of s_D):

$O_{X}(D){\overset {\sim }{\to }}L(D),\,f\mapsto fs_{D}$.

Because of this isomorphism, one often uses O_X(D) and L(D) interchangably. If D is effective, then the constant function "1" corresponds to s_D (see #Global sections of line bundles and linear systems.)