The Stacks project

70.7 The Weil divisor class associated to an invertible module

In this section we go through exactly the same progression as in Section 70.6 to define a canonical map $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ on a locally Noetherian integral algebraic space.

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. By Divisors on Spaces, Lemma 69.10.11 there exists a regular meromorphic section $s \in \Gamma (X, \mathcal{K}_ X(\mathcal{L}))$. In fact, by Divisors on Spaces, Lemma 69.10.8 this is the same thing as a nonzero element in $\mathcal{L}_\eta $ where $\eta \in |X|$ is the generic point. The same lemma tells us that if $\mathcal{L} = \mathcal{O}_ X$, then $s$ is the same thing as a nonzero rational function on $X$ (so what we will do below matches the construction in Section 70.6).

Let $Z \subset X$ be a prime divisor and let $\xi \in |Z|$ be the generic point. We are going to define the order of vanishing of $s$ along $Z$. Consider the canonical morphism

\[ c_\xi : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }^ h) \longrightarrow X \]

whose source is the spectrum of the henselian local ring of $X$ as $\xi $ (Decent Spaces, Definition 66.11.7). The pullback $\mathcal{L}_\xi = c_\xi ^*\mathcal{L}$ is an invertible module and hence trivial; choose a generator $s_\xi $ of $\mathcal{L}_\xi $. Since $c_\xi $ is flat, pullbacks of meromorphic functions and (regular) sections are defined for $c_\xi $, see Divisors on Spaces, Definition 69.10.6 and Lemmas 69.10.7 and 69.10.10. Thus we get

\[ c_\xi ^*(s) = f s_\xi \]

for some nonzerodivisor $f \in Q(\mathcal{O}_{X, \xi }^ h)$. Here we are using Divisors, Lemma 31.24.2 to identify the space of meromorphic sections of $\mathcal{L}_\xi \cong \mathcal{O}_{\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \xi }^ h)}$ in terms of the total ring of fractions of $\mathcal{O}_{X, \xi }^ h$. Let us agree to denote this element

\[ s/s_\xi = f \in Q(\mathcal{O}_{X, \xi }^ h) \]

Observe that $f = s/s_\xi $ is replaced by $uf$ where $u \in \mathcal{O}_{X, \xi }^ h$ is a unit if we change our choice of $s_\xi $.

Definition 70.7.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic algebraic space over $S$. 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 $Z \subset X$ with generic point $\xi \in |Z|$ we define the order of vanishing of $s$ along $Z$ as the integer

\[ \text{ord}_{Z, \mathcal{L}}(s) = \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/a \mathcal{O}_{X, \xi }^ h) - \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/b \mathcal{O}_{X, \xi }^ h) \]

where $a, b \in \mathcal{O}_{X, \xi }^ h$ are nonzerodivisors such that the element $s/s_\xi $ of $Q(\mathcal{O}_{X, \xi }^ h)$ constructed above is equal to $a/b$. This is well defined by the above and Algebra, Lemma 10.120.1.

As explained above, a regular meromorphic section $s$ of $\mathcal{O}_ X$ can be written $s = f \cdot 1$ where $f$ is a nonzero rational function on $X$ and we have $\text{ord}_ Z(f) = \text{ord}_{Z, \mathcal{O}_ X}(s)$. As in the case of principal divisors we have the following lemma.

Lemma 70.7.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. 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 subspace $U \subset X$ 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 all correspond to irreducible components of $|X| \setminus |U|$. Hence Lemma 70.6.1. gives the desired result. $\square$

Lemma 70.7.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$ 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 70.7.2 guarantees that the sums are indeed Weil divisors. $\square$

Definition 70.7.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. 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. This is well defined by Lemma 70.7.2.

  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 70.7.3.

As expected this construction is additive in the invertible module.

Lemma 70.7.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. 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 $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}$ 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 $ with notation as described earlier in this section. Then $s_\xi \otimes t_\xi $ is a generator for $(\mathcal{L} \otimes \mathcal{N})_\xi $. So $st/(s_\xi t_\xi ) = (s/s_\xi )(t/t_\xi )$ in $Q(\mathcal{O}_{X, \xi }^ h)$. Applying the additivity of Algebra, Lemma 10.120.1 we conclude that

\[ \text{div}_{\mathcal{L} \otimes \mathcal{N}, Z}(st) = \text{div}_{\mathcal{L}, Z}(s) + \text{div}_{\mathcal{N}, Z}(t) \]

Some details omitted. $\square$

Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. By the constructions and lemmas above we obtain a homomorphism of abelian groups

70.7.5.1
\begin{equation} \label{spaces-over-fields-equation-c1} \mathop{\mathrm{Pic}}\nolimits (X) \longrightarrow \text{Cl}(X) \end{equation}

which assigns to an invertible module its Weil divisor class.

Lemma 70.7.6. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. If $X$ is normal, then the map (70.7.5.1) $\mathop{\mathrm{Pic}}\nolimits (X) \to \text{Cl}(X)$ 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 70.7.3. We claim that $t$ defines a trivialization of $\mathcal{L}$. The claim finishes the proof of the lemma. Our proof of the claim is a bit awkward as we don't yet have a lot of theory at our dispposal; we suggest the reader skip the proof.

We may check our claim ├ętale locally. Let $U \in X_{\acute{e}tale}$ be affine such that $\mathcal{L}|_ U$ is trivial. Say $s_ U \in \Gamma (U, \mathcal{L}|_ U)$ is a trivialization. By Properties, Lemma 28.7.5 we may also assume $U$ is integral. Write $U = \mathop{\mathrm{Spec}}(A)$ as the spectrum of a normal Noetherian domain $A$ with fraction field $K$. We may write $t|_ U = f s_ U$ for some element $f$ of $K$, see Divisors on Spaces, Lemma 69.10.4 for example. Let $\mathfrak p \subset A$ be a height one prime corresponding to a codimension $1$ point $u \in U$ which maps to a codimension $1$ point $\xi \in |X|$. Choose a trivialization $s_\xi $ of $\mathcal{L}_\xi $ as in the beginning of this section. Choose a geometric point $\overline{u}$ of $U$ lying over $u$. Then

\[ (\mathcal{O}_{X, \xi }^ h)^{sh} = \mathcal{O}_{X, \overline{u}} = \mathcal{O}_{U, u}^{sh} = (A_\mathfrak p)^{sh} \]

see Decent Spaces, Lemmas 66.11.9 and Properties of Spaces, Lemma 64.22.1. The normality of $X$ shows that all of these are discrete valuation rings. The trivializations $s_ U$ and $s_\xi $ differ by a unit as sections of $\mathcal{L}$ pulled back to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{u}})$. Write $t = f_\xi s_\xi $ with $f_\xi \in Q(\mathcal{O}_{X, \xi }^ h)$. We conclude that $f_\xi $ and $f$ differ by a unit in $Q(\mathcal{O}_{X, \overline{u}})$. If $Z \subset X$ denotes the prime divisor corresponding to $\xi $ (Lemma 70.4.7), then $0 = \text{ord}_{Z, \mathcal{L}}(t) = \text{ord}_{\mathcal{O}_{X, \xi }^ h}(f_\xi )$ and since $\mathcal{O}_{X, \xi }^ h$ is a discrete valuation ring we see that $f_\xi $ is a unit. Thus $f$ is a unit in $\mathcal{O}_{X, \overline{u}}$ and hence in particular $f \in A_\mathfrak p$. This implies $f \in A$ by Algebra, Lemma 10.155.6. We conclude that $t \in \Gamma (X, \mathcal{L})$. Repeating the argument with $t^{-1}$ viewed as a meromorphic section of $\mathcal{L}^{\otimes -1}$ finishes the proof. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ENV. Beware of the difference between the letter 'O' and the digit '0'.