The Stacks project

42.24 The divisor associated to an invertible sheaf

The following definition is the analogue of Divisors, Definition 31.27.4 in our current setup.

Definition 42.24.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. 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 $s$ is the $(n - 1)$-cycle

    \[ \text{div}_\mathcal {L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z] \]

    defined in Divisors, Definition 31.27.4. This makes sense because Weil divisors have $\delta $-dimension $n - 1$ by Lemma 42.16.1.

  2. We define Weil divisor associated to $\mathcal{L}$ as

    \[ c_1(\mathcal{L}) \cap [X] = \text{class of }\text{div}_\mathcal {L}(s) \in \mathop{\mathrm{CH}}\nolimits _{n - 1}(X) \]

    where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Divisors, Lemma 31.27.3.

Let $X$ and $S$ be as in Definition 42.24.1 above. Set $n = \dim _\delta (X)$. It is clear from the definitions that $Cl(X) = \mathop{\mathrm{CH}}\nolimits _{n - 1}(X)$ where $Cl(X)$ is the Weil divisor class group of $X$ as defined in Divisors, Definition 31.26.7. The map

\[ \mathop{\mathrm{Pic}}\nolimits (X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{n - 1}(X), \quad \mathcal{L} \longmapsto c_1(\mathcal{L}) \cap [X] \]

is the same as the map $\mathop{\mathrm{Pic}}\nolimits (X) \to Cl(X)$ constructed in Divisors, Equation (31.27.5.1) for arbitrary locally Noetherian integral schemes. In particular, this map is a homomorphism of abelian groups, it is injective if $X$ is a normal scheme, and an isomorphism if all local rings of $X$ are UFDs. See Divisors, Lemmas 31.27.6 and 31.27.7. There are some cases where it is easy to compute the Weil divisor associated to an invertible sheaf.

Lemma 42.24.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a nonzero global section. Then

\[ \text{div}_\mathcal {L}(s) = [Z(s)]_{n - 1} \]

in $Z_{n - 1}(X)$ and

\[ c_1(\mathcal{L}) \cap [X] = [Z(s)]_{n - 1} \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 1}(X)$.

Proof. Let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $n - 1$. Let $\xi \in Z$ be its generic point. Choose a generator $s_\xi \in \mathcal{L}_\xi $. Write $s = fs_\xi $ for some $f \in \mathcal{O}_{X, \xi }$. By definition of $Z(s)$, see Divisors, Definition 31.14.8 we see that $Z(s)$ is cut out by a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ such that $\mathcal{I}_\xi = (f)$. Hence $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{Z(s), \xi }) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{X, \xi }/(f)) = \text{ord}_{\mathcal{O}_{X, x}}(f)$ as desired. $\square$

The following lemma will be superseded by the more general Lemma 42.26.2.

Lemma 42.24.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \dim _\delta (Y)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Y$-module. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Then

\[ f^*(c_1(\mathcal{L}) \cap [Y]) = c_1(f^*\mathcal{L}) \cap [X] \]

in $\mathop{\mathrm{CH}}\nolimits _{n + r - 1}(X)$.

Proof. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. We will show that actually $f^*\text{div}_\mathcal {L}(s) = \text{div}_{f^*\mathcal{L}}(f^*s)$ and hence the lemma holds. To see this let $\xi \in Y$ be a point and let $s_\xi \in \mathcal{L}_\xi $ be a generator. Write $s = gs_\xi $ with $g \in R(Y)^*$. Then there is an open neighbourhood $V \subset Y$ of $\xi $ such that $s_\xi \in \mathcal{L}(V)$ and such that $s_\xi $ generates $\mathcal{L}|_ V$. Hence we see that

\[ \text{div}_\mathcal {L}(s)|_ V = \text{div}_ Y(g)|_ V. \]

In exactly the same way, since $f^*s_\xi $ generates $f^*\mathcal{L}$ over $f^{-1}(V)$ and since $f^*s = g f^*s_\xi $ we also have

\[ \text{div}_\mathcal {L}(f^*s)|_{f^{-1}(V)} = \text{div}_ X(g)|_{f^{-1}(V)}. \]

Thus the desired equality of cycles over $f^{-1}(V)$ follows from the corresponding result for pullbacks of principal divisors, see Lemma 42.17.2. $\square$


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