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
Comments (0)