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Tag 02SE

30.25. The Weil divisor class associated to an invertible module

In this section we go through exactly the same progression as in Section 30.24 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 30.25.1. 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 10.120.2, $\xi \in Z$ 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 30.25.2. 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 30.24.1. gives the desired result. $\square$

Lemma 30.25.3. 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 30.25.2 guarantees that the sums are indeed Weil divisors. $\square$

Definition 30.25.4. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.

  1. 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.
  2. 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 30.25.3.

As expected this construction is additive in the invertible module.

Lemma 30.25.5. 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 \begin{equation} \tag{30.25.5.1} \mathop{\mathrm{Pic}}\nolimits(X) \longrightarrow \text{Cl}(X) \end{equation} which assigns to an invertible module its Weil divisor class.

Lemma 30.25.6. Let $X$ be a locally Noetherian integral scheme. If $X$ is normal, then the map (30.25.5.1) $\mathop{\mathrm{Pic}}\nolimits(X) \to \text{Cl}(X)$ 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 30.25.3. 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 10.151.4), 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 10.151.6 and the proof is complete. $\square$

Lemma 30.25.7. Let $X$ be a locally Noetherian integral scheme. Consider the map (30.25.5.1) $\mathop{\mathrm{Pic}}\nolimits(X) \to \text{Cl}(X)$. The following are equivalent

  1. the local rings of $X$ are UFDs, and
  2. $X$ 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 10.119.11. Hence the map (30.25.5.1) is injective by Lemma 30.25.6. Moreover, every prime divisor $D \subset X$ is an effective Cartier divisor by Lemma 30.15.7. In this case the canonical section $1_D$ of $\mathcal{O}_X(D)$ (Definition 30.14.1) vanishes exactly along $D$ and we see that the class of $D$ is the image of $\mathcal{O}_X(D)$ under the map (30.25.5.1). Thus the map is surjective as well.

Assume (2) holds. Pick a prime divisor $D \subset X$. Since (30.25.5.1) 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 10.151.6. 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 10.119.6. $\square$

    The code snippet corresponding to this tag is a part of the file divisors.tex and is located in lines 6186–6439 (see updates for more information).

    \section{The Weil divisor class associated to an invertible module}
    \label{section-c1}
    
    \noindent
    In this section we go through exactly the same progression as in
    Section \ref{section-Weil-divisors} to define a canonical map
    $\Pic(X) \to \text{Cl}(X)$
    on a locally Noetherian integral scheme.
    
    \medskip\noindent
    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.
    
    \begin{definition}
    \label{definition-order-vanishing-meromorphic}
    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
    {\it 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 \ref{algebra-definition-ord},
    $\xi \in Z$ is the generic point,
    and $s_\xi \in \mathcal{L}_\xi$ is a generator.
    \end{definition}
    
    \noindent
    As in the case of principal divisors we have the following lemma.
    
    \begin{lemma}
    \label{lemma-divisor-meromorphic-locally-finite}
    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 $X$.
    \end{lemma}
    
    \begin{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 \ref{lemma-components-locally-finite}.
    gives the desired result.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-divisor-meromorphic-well-defined}
    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.
    \end{lemma}
    
    \begin{proof}
    This is clear from the definitions.
    Note that Lemma \ref{lemma-divisor-meromorphic-locally-finite}
    guarantees that the sums are indeed Weil divisors.
    \end{proof}
    
    \begin{definition}
    \label{definition-divisor-invertible-sheaf}
    Let $X$ be a locally Noetherian integral scheme.
    Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
    \begin{enumerate}
    \item For any nonzero meromorphic section $s$ of $\mathcal{L}$
    we define the {\it 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.
    \item We define {\it 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 \ref{lemma-divisor-meromorphic-well-defined}.
    \end{enumerate}
    \end{definition}
    
    \noindent
    As expected this construction is additive in the invertible module.
    
    \begin{lemma}
    \label{lemma-c1-additive}
    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 $\mathcal{N}$.
    \end{lemma}
    
    \begin{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.
    \end{proof}
    
    \noindent
    In this way we obtain a homomorphism of abelian groups
    \begin{equation}
    \label{equation-c1}
    \Pic(X) \longrightarrow \text{Cl}(X)
    \end{equation}
    which assigns to an invertible module its Weil divisor class.
    
    \begin{lemma}
    \label{lemma-normal-c1-injective}
    Let $X$ be a locally Noetherian integral scheme. If $X$ is normal,
    then the map (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$
    is injective.
    \end{lemma}
    
    \begin{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 \ref{lemma-divisor-meromorphic-well-defined}.
    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 $X = \Spec(A)$ 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 \ref{algebra-lemma-criterion-normal}), 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
    \ref{algebra-lemma-normal-domain-intersection-localizations-height-1}
    and the proof is complete.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-local-rings-UFD-c1-bijective}
    Let $X$ be a locally Noetherian integral scheme. Consider the map
    (\ref{equation-c1}) $\Pic(X) \to \text{Cl}(X)$.
    The following are equivalent
    \begin{enumerate}
    \item the local rings of $X$ are UFDs, and
    \item $X$ is normal and $\Pic(X) \to \text{Cl}(X)$
    is surjective.
    \end{enumerate}
    In this case $\Pic(X) \to \text{Cl}(X)$ is an isomorphism.
    \end{lemma}
    
    \begin{proof}
    If (1) holds, then $X$ is normal by
    Algebra, Lemma \ref{algebra-lemma-UFD-normal}.
    Hence the map (\ref{equation-c1}) is injective by
    Lemma \ref{lemma-normal-c1-injective}. Moreover,
    every prime divisor $D \subset X$ is an effective
    Cartier divisor by Lemma \ref{lemma-weil-divisor-is-cartier-UFD}.
    In this case the canonical section $1_D$ of $\mathcal{O}_X(D)$
    (Definition \ref{definition-invertible-sheaf-effective-Cartier-divisor})
    vanishes exactly along $D$ and we see that the class of $D$ is the
    image of $\mathcal{O}_X(D)$ under the map (\ref{equation-c1}).
    Thus the map is surjective as well.
    
    \medskip\noindent
    Assume (2) holds. Pick a prime divisor $D \subset X$.
    Since (\ref{equation-c1}) 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 $\Spec(A)$. 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
    \ref{algebra-lemma-normal-domain-intersection-localizations-height-1}.
    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 \ref{algebra-lemma-characterize-UFD-height-1}.
    \end{proof}

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