Weil divisors on normal Noetherian schemes

algebraic-geometry

I'm looking at Section 7 of Liu's Algebraic Geometry and Arithmetic Curves.

In Section 7, Definition 2.4. defines a Weil divisor as a cycle of codimension 1 on a Noetherian integral scheme. But then Liu discusses Weil divisors on normal Noetherian schemes in Definition 2.7., 2.10., Proposition 2.11, etc., as well as the notion of function field $K(X)$ of a normal Noetherian scheme $X$.

I feel lost as the function field is defined over an integral scheme, and up to my understanding, the closest one to the function field of a normal Noetherian scheme is perhaps the direct sum of the function fields of the finite integral components.

What I'm more interested is whether Weil divisors are well-defined on normal Noetherian schemes and how close it is to be equivalent to invertible sheaves. Could someone provide an explanation on this?

Thank you.

Best Answer

In a normal scheme, irreducible components are connected components: no two irreducible components may meet, as otherwise the local ring at a point in their intersection would not be a domain, contradicting the definition that all local rings in a normal scheme are integrally closed domains. Therefore it makes sense to analyze component-by-component for Weil divisors on normal schemes, and one may just apply the relevant results on each component and conclude from there. In particular, Weil divisors are perfectly well defined on an arbitrary noetherian normal scheme.

As to the comparison between Cartier divisors (line bundles) and Weil divisors (codimension one cycles), the following material is relatively standard. We'll roughly follow Chapter 14 of Vakil's FOAG.

Definition. Let $X$ be a noetherian scheme. Define a map $\operatorname{div}$ from the collection of line bundles $\mathcal{L}$ on $X$ with a rational section $s$ not vanishing on any irreducible component of $X$ to Weil divisors on $X$ as follows:

$$\operatorname{div}(s) = \sum_{Y} \operatorname{val}_Y(s)\cdot [Y] $$

where $Y$ ranges over the codimension one irreducible subschemes of $X$, and $\operatorname{val}_Y$ represents the natural valuation.

This gives a map from the group under tensor product of isomorphism classes of line bundles with rational section to the group of Weil divisors.

Proposition. (Vakil 14.2.1) Let $X$ be a noetherian normal scheme. The map $\operatorname{div}(s):Pic(X)\to Cl(X)$ is injective.

This means for any normal noetherian scheme, any Cartier divisor gives a Weil divisor. We see that the reverse is not always true: the canonical example is the divisor $D$ given by the line $V(x,z)$ inside the cone $V(xy-z^2)\subset \Bbb A^3$ (Vakil exercise 14.2.H). $D$ is not Cartier, because the divisor vanishes to order 2 on the set-theoretic support of the divisor. On the other hand, $2D$ is a Cartier divisor, and this is essentially the only way in which things can go wrong.

Definition. A scheme is called factorial (or locally factorial) if every local ring is a unique factorization domain.

We note that as UFDs are normal domains, this immediately implies that every factorial scheme is in fact normal.

Proposition (Vakil 14.2.10). Let $X$ be a noetherian factorial scheme. Then for any Weil divisor $D$, the sheaf $\mathcal{O}(D)$ is a line bundle, and the map $Pic(X)\to Cl(X)$ is an isomorphism.

In particular, this means on a factorial noetherian scheme, the notions of Weil divisor and Cartier divisor are the same.

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