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 $X$.
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.
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.
A prime divisor is an integral closed subscheme $Z \subset X$ of codimension $1$.
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
\begin{equation} \label{divisors-equation-Weil-divisor-class} R(X)^* \xrightarrow {\text{div}} \text{Div}(X) \to \text{Cl}(X) \to 0 \end{equation}
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.
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