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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?

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1 Answer 1

up vote 2 down vote accepted

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.

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There are 2 questions: 1)the smoothness is weaker than the cohomological flatness in dimension zero. Namely we only need lifting over nilpotent extensions, do you know when this happens? 2) If I change $\mathcal{O}_X^{*}$ with a locally(\'etale topology) constant sheaf of finitely generated abelian group $G$ on $X$ do you know under which conditions on the family the functor $T\rightarrow f_{T,\'et,*}(G_T)$ is formally smooth? –  unky Nov 6 '12 at 18:48
    
There are 2 questions: 1)the smoothness is weaker than the cohomological flatness in dimension zero. Namely we only need lifting over nilpotent extensions, do you know when this happens? 2) If I change $\mathcal{O}_X^{*}$ with a locally(\'etaletopology)constant sheaf of finitely generated abelian group $G$ on $X$ do you know under which conditions on the family the functor $T\rightarrow f_{T,\'et,}(G_T)$ is formally smooth ? –  unky Nov 6 '12 at 18:53
    
@unky:the lifting property for nilpotent extension is equivalent to the smoothness for scheme of finite type. I explained that the smoothness is equivalent to coh. flatness in dimension $0$. Anyway you said you are satisfied with semi-stable curve over DVR. They are coh. flat. in dimension $0$. For (2), please post a different message, preferably at mathoverflow rather than here. –  user18119 Nov 6 '12 at 20:53

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