Smoothness of invertible elements 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? 
 A: Suppose $S$ is noetherian (otherwise suppose $X\to S$ is finitely presented). It is known that the functor
$$ T \mapsto \Gamma(X_T, O_{X_T})$$ 
is represented by an $S$-scheme $V$ which is the Spec of the symetric algebra of some finitely generated module $M$ over $R=O(S)$ (see Bosch-Lütkebohmert-Raynaud: Néron models, §8.1, Corollary 8), and your functor is represented by an open subscheme $V^*$ of $V$ (op. cit., 8.1, Lemma 10). By construction, the fibers of $V\to S$ are vector spaces and $V^*$ is fiberwise dense in $V$. 
If $V^*\to S$ is smooth, then $V$ is smooth because $V\to S$ is a group scheme. The converse is obviously true because $V^*$ is open in $V$. So $V^*\to S$ is smooth if and only if $V\to S$ is smooth.
Now $V\to S$ is smooth if and only if $X\to S$ is cohomologically flat in dimension $0$ (op. cit., 8.1, Cor.8), i.e. $\Gamma(X_T, O_{X_T})=\Gamma(X, O_X)\otimes_R O(T)$ for any affine $T$ over $S$. This is also equivalent to 
$$\Gamma(X_s, O_{X_s})=\Gamma(X, O_X)\otimes_R k(s), \quad \forall s\in S.$$ 
This is true if the fibers of $X\to S$ are geometrically reduced (EGA III, 7.8.6). In particular, if the fibers of $X\to S$ are nodal curves, then your functor is smooth. 
