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.
$\square$

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