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.
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.
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. 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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