## 31.26 Weil divisors

We will introduce Weil divisors and rational equivalence of Weil divisors for locally Noetherian integral schemes. Since we are not assuming our schemes are quasi-compact we have to be a little careful when defining Weil divisors. We have to allow infinite sums of prime divisors because a rational function may have infinitely many poles for example. For quasi-compact schemes our Weil divisors are finite sums as usual. Here is a basic lemma we will often use to prove collections of closed subschemes are locally finite.

Lemma 31.26.1. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. The collection of irreducible components of $Z$ is locally finite in $X$.

Proof. Let $U \subset X$ be a quasi-compact open subscheme. Then $U$ is a Noetherian scheme, and hence has a Noetherian underlying topological space (Properties, Lemma 28.5.5). Hence every subspace is Noetherian and has finitely many irreducible components (see Topology, Lemma 5.9.2). $\square$

Recall that if $Z$ is an irreducible closed subset of a scheme $X$, then the codimension of $Z$ in $X$ is equal to the dimension of the local ring $\mathcal{O}_{X, \xi }$, where $\xi \in Z$ is the generic point. See Properties, Lemma 28.10.3.

Definition 31.26.2. Let $X$ be a locally Noetherian integral scheme.

1. A prime divisor is an integral closed subscheme $Z \subset X$ of codimension $1$.

2. A Weil divisor is a formal sum $D = \sum n_ Z Z$ where the sum is over prime divisors of $X$ and the collection $\{ Z \mid n_ Z \not= 0\}$ is locally finite (Topology, Definition 5.28.4).

The group of all Weil divisors on $X$ is denoted $\text{Div}(X)$.

Our next task is to define the Weil divisor associated to a rational function. In order to do this we use the order of vanishing of a rational function along a prime divisor which is defined as follows.

Definition 31.26.3. Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. For every prime divisor $Z \subset X$ we define the order of vanishing of $f$ along $Z$ as the integer

$\text{ord}_ Z(f) = \text{ord}_{\mathcal{O}_{X, \xi }}(f)$

where the right hand side is the notion of Algebra, Definition 10.121.2 and $\xi$ is the generic point of $Z$.

Note that for $f, g \in R(X)^*$ we have

$\text{ord}_ Z(fg) = \text{ord}_ Z(f) + \text{ord}_ Z(g).$

Of course it can happen that $\text{ord}_ Z(f) < 0$. In this case we say that $f$ has a pole along $Z$ and that $-\text{ord}_ Z(f) > 0$ is the order of pole of $f$ along $Z$. It is important to note that the condition $\text{ord}_ Z(f) \geq 0$ is not equivalent to the condition $f \in \mathcal{O}_{X, \xi }$ unless the local ring $\mathcal{O}_{X, \xi }$ is a discrete valuation ring.

Lemma 31.26.4. Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. Then the collections

$\{ Z \subset X \mid Z\text{ a prime divisor with generic point }\xi \text{ and }f\text{ not in }\mathcal{O}_{X, \xi }\}$

and

$\{ Z \subset X \mid Z \text{ a prime divisor and }\text{ord}_ Z(f) \not= 0\}$

are locally finite in $X$.

Proof. There exists a nonempty open subscheme $U \subset X$ such that $f$ corresponds to a section of $\Gamma (U, \mathcal{O}_ X^*)$. Hence the prime divisors which can occur in the sets of the lemma are all irreducible components of $X \setminus U$. Hence Lemma 31.26.1 gives the desired result. $\square$

This lemma allows us to make the following definition.

Definition 31.26.5. Let $X$ be a locally Noetherian integral scheme. Let $f \in R(X)^*$. The principal Weil divisor associated to $f$ is the Weil divisor

$\text{div}(f) = \text{div}_ X(f) = \sum \text{ord}_ Z(f) [Z]$

where the sum is over prime divisors and $\text{ord}_ Z(f)$ is as in Definition 31.26.3. This makes sense by Lemma 31.26.4.

Lemma 31.26.6. Let $X$ be a locally Noetherian integral scheme. Let $f, g \in R(X)^*$. Then

$\text{div}_ X(fg) = \text{div}_ X(f) + \text{div}_ X(g)$

as Weil divisors on $X$.

Proof. This is clear from the additivity of the $\text{ord}$ functions. $\square$

We see from the lemma above that the collection of principal Weil divisors form a subgroup of the group of all Weil divisors. This leads to the following definition.

Definition 31.26.7. Let $X$ be a locally Noetherian integral scheme. The Weil divisor class group of $X$ is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: $\text{Cl}(X)$.

By construction we obtain an exact complex

31.26.7.1
$$\label{divisors-equation-Weil-divisor-class} R(X)^* \xrightarrow {\text{div}} \text{Div}(X) \to \text{Cl}(X) \to 0$$

which we can think of as a presentation of $\text{Cl}(X)$. Our next task is to relate the Weil divisor class group to the Picard group.

Comment #4235 by Zhenhua Wu on

In 31.26.4, the argument will be more transparent if we pick $U$ to be the domain of definition of $f$. A more detailed proof is following:

There exists a largest nonempty open subscheme $U\subset X$ such that $f$ corresponds to a section of $\Gamma(U,\mathcal{O} _ X^ * )$ . Hence any prime divisor $Z$ with generic point $\xi$ s.t. $f\notin\mathcal{O} _ {X,\xi}$ must have $\xi\in X\backslash U$. Since $U$ is irreducible, we have $U\cap Z=\emptyset$. Since $Z$ is of codimension 1, it is easy to see it is an irreducible component of $X\backslash U$. Hence Lemma 31.26.1 gives the desired result.

If $\mathop{\mathrm{ord}} _ Z (f)\neq 0$, equivalently $\mathop{\mathrm{ord}}Z(f)<0$ or $\mathop{\mathrm{ord}} _ Z(f^{-1})<0$. Hence we have either $f\notin \mathcal{O} _ {X,\xi}$ or $f^{-1}\notin \mathcal{O} _ {X,\xi}$. Apply previous argument then we are done.

Comment #4414 by on

OK, I actually think this is the same as written, so I am going to leave it as is.

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