Extension of line bundle on regular deformation

Question

Let $R$ be a DVR and $\pi : S \rightarrow \text{Spec}(R)$ a regular smoothing of a nodal curve $C$ (with regular components). Given a line bundle $L$ on the (regular) generic fiber $\pi^{-1}(\eta)=S_\eta$ it is possible to extend $L$ to a line bundle $\mathcal{L}$ on $S$.

Can some one give me a reference where it is proven and why it is important that $S$ is regular?

Answer 1

First, extend $L$ to any coherent sheaf $F$ on $S$, then consider its reflexive hull $F^{\vee\vee}$, and use the fact that any reflexive sheaf of rank $1$ on a regular variety is invertible.