31.27 The Weil divisor class associated to an invertible module
In this section we go through exactly the same progression as in Section 31.26 to define a canonical map $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ on a locally Noetherian integral scheme.
Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\xi \in X$ be a point. If $s_\xi , s'_\xi \in \mathcal{L}_\xi $ generate $\mathcal{L}_\xi $ as $\mathcal{O}_{X, \xi }$-module, then there exists a unit $u \in \mathcal{O}_{X, \xi }^*$ such that $s_\xi = u s'_\xi $. The stalk of the sheaf of meromorphic sections $\mathcal{K}_ X(\mathcal{L})$ of $\mathcal{L}$ at $x$ is equal to $\mathcal{K}_{X, x} \otimes _{\mathcal{O}_{X, x}} \mathcal{L}_ x$. Thus the image of any meromorphic section $s$ of $\mathcal{L}$ in the stalk at $x$ can be written as $s = fs_\xi $ with $f \in \mathcal{K}_{X, x}$. Below we will abbreviate this by saying $f = s/s_\xi $. Also, if $X$ is integral we have $\mathcal{K}_{X, x} = R(X)$ is equal to the function field of $X$, so $s/s_\xi \in R(X)$. If $s$ is a regular meromorphic section, then actually $s/s_\xi \in R(X)^*$. On an integral scheme a regular meromorphic section is the same thing as a nonzero meromorphic section. Finally, we see that $s/s_\xi $ is independent of the choice of $s_\xi $ up to multiplication by a unit of the local ring $\mathcal{O}_{X, x}$. Putting everything together we see the following definition makes sense.
Definition 31.27.1. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{K}_ X(\mathcal{L}))$ be a regular meromorphic section of $\mathcal{L}$. For every prime divisor $Z \subset X$ we define the order of vanishing of $s$ along $Z$ as the integer
\[ \text{ord}_{Z, \mathcal{L}}(s) = \text{ord}_{\mathcal{O}_{X, \xi }}(s/s_\xi ) \]
where the right hand side is the notion of Algebra, Definition 10.121.2, $\xi \in Z$ is the generic point, and $s_\xi \in \mathcal{L}_\xi $ is a generator.
As in the case of principal divisors we have the following lemma.
Lemma 31.27.2. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \mathcal{K}_ X(\mathcal{L})$ be a regular (i.e., nonzero) meromorphic section of $\mathcal{L}$. Then the sets
\[ \{ Z \subset X \mid Z \text{ a prime divisor with generic point }\xi \text{ and }s\text{ not in }\mathcal{L}_\xi \} \]
and
\[ \{ Z \subset X \mid Z \text{ is a prime divisor and } \text{ord}_{Z, \mathcal{L}}(s) \not= 0\} \]
are locally finite in $X$.
Proof.
There exists a nonempty open subscheme $U \subset X$ such that $s$ corresponds to a section of $\Gamma (U, \mathcal{L})$ which generates $\mathcal{L}$ over $U$. 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.
$\square$
Lemma 31.27.3. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s, s' \in \mathcal{K}_ X(\mathcal{L})$ be nonzero meromorphic sections of $\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have
\[ \sum \text{ord}_{Z, \mathcal{L}}(s)[Z] = \sum \text{ord}_{Z, \mathcal{L}}(s')[Z] + \text{div}(f) \]
as Weil divisors.
Proof.
This is clear from the definitions. Note that Lemma 31.27.2 guarantees that the sums are indeed Weil divisors.
$\square$
Definition 31.27.4. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.
For any nonzero meromorphic section $s$ of $\mathcal{L}$ we define the Weil divisor associated to $s$ as
\[ \text{div}_\mathcal {L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z] \in \text{Div}(X) \]
where the sum is over prime divisors.
We define Weil divisor class associated to $\mathcal{L}$ as the image of $\text{div}_\mathcal {L}(s)$ in $\text{Cl}(X)$ where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Lemma 31.27.3.
As expected this construction is additive in the invertible module.
Lemma 31.27.5. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$, $\mathcal{N}$ be invertible $\mathcal{O}_ X$-modules. Let $s$, resp. $t$ be a nonzero meromorphic section of $\mathcal{L}$, resp. $\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes \mathcal{N}$, and
\[ \text{div}_{\mathcal{L} \otimes \mathcal{N}}(st) = \text{div}_\mathcal {L}(s) + \text{div}_\mathcal {N}(t) \]
in $\text{Div}(X)$. In particular, the Weil divisor class of $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}$ is the sum of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$.
Proof.
Let $s$, resp. $t$ be a nonzero meromorphic section of $\mathcal{L}$, resp. $\mathcal{N}$. Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes \mathcal{N}$. Let $Z \subset X$ be a prime divisor. Let $\xi \in Z$ be its generic point. Choose generators $s_\xi \in \mathcal{L}_\xi $, and $t_\xi \in \mathcal{N}_\xi $. Then $s_\xi t_\xi $ is a generator for $(\mathcal{L} \otimes \mathcal{N})_\xi $. So $st/(s_\xi t_\xi ) = (s/s_\xi )(t/t_\xi )$. Hence we see that
\[ \text{div}_{\mathcal{L} \otimes \mathcal{N}, Z}(st) = \text{div}_{\mathcal{L}, Z}(s) + \text{div}_{\mathcal{N}, Z}(t) \]
by the additivity of the $\text{ord}_ Z$ function.
$\square$
In this way we obtain a homomorphism of abelian groups
31.27.5.1
\begin{equation} \label{divisors-equation-c1} \mathop{\mathrm{Pic}}\nolimits (X) \longrightarrow \text{Cl}(X) \end{equation}
which assigns to an invertible module its Weil divisor class.
Lemma 31.27.6. Let $X$ be a locally Noetherian integral scheme. If $X$ is normal, then the map (31.27.5.1) $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ is injective.
Proof.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module whose associated Weil divisor class is trivial. Let $s$ be a regular meromorphic section of $\mathcal{L}$. The assumption means that $\text{div}_\mathcal {L}(s) = \text{div}(f)$ for some $f \in R(X)^*$. Then we see that $t = f^{-1}s$ is a regular meromorphic section of $\mathcal{L}$ with $\text{div}_\mathcal {L}(t) = 0$, see Lemma 31.27.3. We will show that $t$ defines a trivialization of $\mathcal{L}$ which finishes the proof of the lemma. In order to prove this we may work locally on $X$. Hence we may assume that $X = \mathop{\mathrm{Spec}}(A)$ is affine and that $\mathcal{L}$ is trivial. Then $A$ is a Noetherian normal domain and $t$ is an element of its fraction field such that $\text{ord}_{A_\mathfrak p}(t) = 0$ for all height $1$ primes $\mathfrak p$ of $A$. Our goal is to show that $t$ is a unit of $A$. Since $A_\mathfrak p$ is a discrete valuation ring for height one primes of $A$ (Algebra, Lemma 10.157.4), the condition signifies that $t \in A_\mathfrak p^*$ for all primes $\mathfrak p$ of height $1$. This implies $t \in A$ and $t^{-1} \in A$ by Algebra, Lemma 10.157.6 and the proof is complete.
$\square$
Lemma 31.27.7. Let $X$ be a locally Noetherian integral scheme. Consider the map (31.27.5.1) $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$. The following are equivalent
the local rings of $X$ are UFDs, and
$X$ is normal and $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ is surjective.
In this case $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ is an isomorphism.
Proof.
If (1) holds, then $X$ is normal by Algebra, Lemma 10.120.11. Hence the map (31.27.5.1) is injective by Lemma 31.27.6. Moreover, every prime divisor $D \subset X$ is an effective Cartier divisor by Lemma 31.15.7. In this case the canonical section $1_ D$ of $\mathcal{O}_ X(D)$ (Definition 31.14.1) vanishes exactly along $D$ and we see that the class of $D$ is the image of $\mathcal{O}_ X(D)$ under the map (31.27.5.1). Thus the map is surjective as well.
Assume (2) holds. Pick a prime divisor $D \subset X$. Since (31.27.5.1) is surjective there exists an invertible sheaf $\mathcal{L}$, a regular meromorphic section $s$, and $f \in R(X)^*$ such that $\text{div}_\mathcal {L}(s) + \text{div}(f) = [D]$. In other words, $\text{div}_\mathcal {L}(fs) = [D]$. Let $x \in X$ and let $A = \mathcal{O}_{X, x}$. Thus $A$ is a Noetherian local normal domain with fraction field $K = R(X)$. Every height $1$ prime of $A$ corresponds to a prime divisor on $X$ and every invertible $\mathcal{O}_ X$-module restricts to the trivial invertible module on $\mathop{\mathrm{Spec}}(A)$. It follows that for every height $1$ prime $\mathfrak p \subset A$ there exists an element $f \in K$ such that $\text{ord}_{A_\mathfrak p}(f) = 1$ and $\text{ord}_{A_{\mathfrak p'}}(f) = 0$ for every other height one prime $\mathfrak p'$. Then $f \in A$ by Algebra, Lemma 10.157.6. Arguing in the same fashion we see that every element $g \in \mathfrak p$ is of the form $g = af$ for some $a \in A$. Thus we see that every height one prime ideal of $A$ is principal and $A$ is a UFD by Algebra, Lemma 10.120.6.
$\square$
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