consider a proper, flat family of schemes $X\rightarrow S$, with $S$ affine. I would like to know under which condition on the family the functor $Sch/S \rightarrow Ab$
$ T \rightarrow \Gamma(X_T,\mathcal{O}_T^{*}) $
is smooth.
Namely if $Spec(B)\rightarrow Spec(A)$ is a nilpotent immersion between artinian $S$-rings then the restriction
$ \Gamma(X_{Spec(A)},\mathcal{O}_{X_{Spec(A)}}^{*})\rightarrow\Gamma(X_{Spec(B)},\mathcal{O}_{X_{Spec(B)}}^{*}) $ is surjective. Is this at least true for a family of curves over a discrete valuation ring, with generic fiber smooth and special fiber nodal?