The Stacks project

Proof. Taking base change, this immediately reduces to the following problem: given a valuation ring $R$ with fraction field $K$ and an algebraic space $X$ proper and smooth over $R$ and an invertible $\mathcal{O}_{X_ K}$-module $\mathcal{L}_ K$, show there exists an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ whose generic fibre is $\mathcal{L}_ K$. Observe that $X_ K$ is Noetherian, separated, and regular (use Morphisms of Spaces, Lemma 67.28.6 and Spaces over Fields, Lemma 72.16.1). Thus we can write $\mathcal{L}_ K$ as the difference in the Picard group of $\mathcal{O}_{X_ K}(D_ K)$ and $\mathcal{O}_{X_ K}(D'_ K)$ for two effective Cartier divisors $D_ K, D'_ K$ in $X_ K$, see Divisors on Spaces, Lemma 71.8.4. Finally, we know that $D_ K$ and $D'_ K$ are restrictions of effective Cartier divisors $D, D' \subset X$, see Divisors on Spaces, Lemma 71.8.5. $\square$