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My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ for every $s \in S$, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is formally smooth.

The proof proceeds by taking an affine test scheme $Z$ and a closed subscheme $Z_0 \subset Z$ defined by an ideal of square zero and proving that the map $$R^1(f\times_S Z)_* \mathscr{O}^*_{X\times_S Z} \rightarrow R^1(f \times_S Z_0)_* \mathscr{O}^*_{X\times_S Z_0}$$ is a surjection of sheaves (on the etale or the fppf site). My question is: how is this enough for the claimed formal smoothness? It seems to me that what one wants is the surjectivity of $$\mathrm{Pic}_{X/S}(Z) = H^0(Z, R^1(f\times_S Z)_* \mathscr{O}^*_{X\times_S Z}) \rightarrow H^0(Z_0, R^1(f \times_S Z_0)_* \mathscr{O}^*_{X\times_S Z_0}) = \mathrm{Pic}_{X/S}(Z_0),$$ which doesn't seem to follow formally because a surjection of sheaves need not be a surjection on global sections.

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    $\begingroup$ If the kernel of the surjection of sheaves is a quasi-coherent $\mathcal{O}_X$-module, then the vanishing of higher cohomology of quasi-coherent sheaves on affine schemes implies surjectivity of the necessary map of global sections. Note that the two sheaves you have written are just Abelian sheaves, not even $\mathcal{O}_X$-modules. But infinitesimal deformation theory identifies the kernel sheaf with a quasi-coherent $\mathcal{O}_X$-module. $\endgroup$ – Jason Starr Jan 5 '15 at 20:11
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I'll write $p = f \times_S Z$, and ${\cal O}^\times$ (resp. $\bar{{\cal O}}^\times$) for the units in the structure sheaf of $X \times_S Z$ (resp. $X \times_S Z_0$, viewed as a sheaf on the same space).

First, one shows that the map $p_*{\cal O}^\times \to p_* \bar{{\cal O}}^\times$ is a surjective map of sheaves on $Z$. For example, one could check this in the case where $Z$ is contained inside an affine coordinate chart of $S$.

From the short exact sequence of sheaves $0 \to 1 + {\cal I} \to {\cal O}^\times \to \bar{{\cal O}}^\times \to 0$ on $X \times_S Z$ we have the long exact sequence $$ 0 \to R^0 p_* {\cal I} \to R^0 p_* {\cal O}^\times \to R^0 p_* \bar{{\cal O}}^\times \to R^1 p_* {\cal I} \to R^1 p_* {\cal O}^\times \to R^1 p_* \bar{{\cal O}}^\times \to \cdots $$ from sheafifying the functorial long exact sequence of $p^{-1} U$ in cohomology. The surjectivity from above and from your statement allows us to extract from this a short exact sequence $$ 0 \to R^1 p_* {\cal I} \to R^1 p_* {\cal O}^\times \to R^1 p_* \bar{{\cal O}}^\times \to 0. $$ Therefore, the obstructions to surjectivity of $H^0(Z; R^1 p_* {\cal O}^\times) \to H^0(Z; R^1 p_* \bar{{\cal O}}^\times)$ live in the sheaf cohomology group $H^1(Z; R^1 p_* {\cal I})$. As Jason Starr points out, this is cohomology of an affine scheme with coefficients in a quasicoherent sheaf, hence vanishes.

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  • $\begingroup$ No problem; sorry for the redundancy, but I'm glad the two of you sorted it out. $\endgroup$ – Tyler Lawson Jan 5 '15 at 20:41

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