Lemma 109.23.3. Let $f : X \to S$ be a family of curves. Let $s \in S$ be a point. Let $h_0 : X_ s \to Y_0$ be a morphism to a proper scheme $Y_0$ over $\kappa (s)$ such that $h_{0, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_0}$ and $R^1h_{0, *}\mathcal{O}_{X_ s} = 0$. Then there exist an elementary étale neighbourhood $(U, u) \to (S, s)$, a family of curves $Y \to U$, and a morphism $h : X_ U \to Y$ over $U$ whose fibre in $u$ is isomorphic to $h_0$.
Proof. We first do some reductions; we urge the reader to skip ahead. The question is local on $S$, hence we may assume $S$ is affine. Write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a cofiltered limit of affine schemes $S_ i$ of finite type over $\mathbf{Z}$. For some $i$ we can find a family of curves $X_ i \to S_ i$ whose base change is $X \to S$. This follows from Lemma 109.5.3 and Limits of Stacks, Lemma 102.3.5. Let $s_ i \in S_ i$ be the image of $s$. Observe that $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ i)$ and that $X_ s$ is a scheme (Spaces over Fields, Lemma 72.9.3). After increasing $i$ we may assume there exists a morphism $h_{i, 0} : X_{i, s_ i} \to Y_ i$ of finite type schemes over $\kappa (s_ i)$ whose base change to $\kappa (s)$ is $h_0$, see Limits, Lemma 32.10.1. After increasing $i$ we may assume $Y_ i$ is proper over $\kappa (s_ i)$, see Limits, Lemma 32.13.1. Let $g_{i, 0} : Y_0 \to Y_{i, 0}$ be the projection. Observe that this is a faithfully flat morphism as the base change of $\mathop{\mathrm{Spec}}(\kappa (s)) \to \mathop{\mathrm{Spec}}(\kappa (s_ i))$. By flat base change we have
\[ h_{0, *}\mathcal{O}_{X_ s} = g_{i, 0}^*h_{i, 0, *}\mathcal{O}_{X_{i, s_ i}} \quad \text{and}\quad R^1h_{0, *}\mathcal{O}_{X_ s} = g_{i, 0}^*Rh_{i, 0, *}\mathcal{O}_{X_{i, s_ i}} \]
see Cohomology of Schemes, Lemma 30.5.2. By faithful flatness we see that $X_ i \to S_ i$, $s_ i \in S_ i$, and $X_{i, s_ i} \to Y_ i$ satisfies all the assumptions of the lemma. This reduces us to the case discussed in the next paragraph.
Assume $S$ is affine of finite type over $\mathbf{Z}$. Let $\mathcal{O}_{S, s}^ h$ be the henselization of the local ring of $S$ at $s$. Observe that $\mathcal{O}_{S, s}^ h$ is a G-ring by More on Algebra, Lemma 15.50.8 and Proposition 15.50.12. Suppose we can construct a family of curves $Y' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ and a morphism
\[ h' : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h) \longrightarrow Y' \]
over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h)$ whose base change to the closed point is $h_0$. This will be enough. Namely, first we use that
\[ \mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(U, u)} \mathcal{O}_ U(U) \]
where the colimit is over the filtered category of elementary étale neighbourhoods (More on Morphisms, Lemma 37.35.5). Next, we use again that given $Y'$ we can descend it to $Y \to U$ for some $U$ (see references given above). Then we use Limits, Lemma 32.10.1 to descend $h'$ to some $h$. This reduces us to the case discussed in the next paragraph.
Assume $S = \mathop{\mathrm{Spec}}(\Lambda )$ where $(\Lambda , \mathfrak m, \kappa )$ is a henselian Noetherian local G-ring and $s$ is the closed point of $S$. Recall that the map
\[ \mathcal{D}\! \mathit{ef}_{X_ s \to Y_0} \to \mathcal{D}\! \mathit{ef}_{X_ s} \]
is an equivalence, see Deformation Problems, Lemma 93.10.6. (This is the only important step in the proof; everything else is technique.) Denote $\Lambda ^\wedge $ the $\mathfrak m$-adic completion. The pullbacks $X_ n$ of $X$ to $\Lambda /\mathfrak m^{n + 1}$ define a formal object $\xi $ of $\mathcal{D}\! \mathit{ef}_{X_ s}$ over $\Lambda ^\wedge $. From the equivalence we obtain a formal object $\xi '$ of $\mathcal{D}\! \mathit{ef}_{X_ s \to Y_0}$ over $\Lambda ^\wedge $. Thus we obtain a huge commutative diagram
\[ \xymatrix{ \ldots \ar[r] & X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & X_ s \ar[d] \\ \ldots \ar[r] & Y_ n \ar[r] \ar[d] & Y_{n - 1} \ar[r] \ar[d] & \ldots \ar[r] & Y_0 \ar[d] \\ \ldots \ar[r] & \mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^{n + 1}) \ar[r] & \mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^ n) \ar[r] & \ldots \ar[r] & \mathop{\mathrm{Spec}}(\kappa ) } \]
The formal object $(Y_ n)$ comes from a family of curves $Y' \to \mathop{\mathrm{Spec}}(\Lambda ^\wedge )$ by Quot, Lemma 99.15.9. By More on Morphisms of Spaces, Lemma 76.43.3 we get a morphism $h' : X_{\Lambda ^\wedge } \to Y'$ inducing the given morphisms $X_ n \to Y_ n$ for all $n$ and in particular the given morphism $X_ s \to Y_0$.
To finish we do a standard algebraization/approximation argument. First, we observe that we can find a finitely generated $\Lambda $-subalgebra $\Lambda \subset A \subset \Lambda ^\wedge $, a family of curves $Y'' \to \mathop{\mathrm{Spec}}(A)$ and a morphism $h'' : X_ A \to Y''$ over $A$ whose base change to $\Lambda ^\wedge $ is $h'$. This is true because $\Lambda ^\wedge $ is the filtered colimit of these rings $A$ and we can argue as before using that $\mathcal{C}\! \mathit{urves}$ is locally of finite presentation (which gives us $Y''$ over $A$ by Limits of Stacks, Lemma 102.3.5) and using Limits of Spaces, Lemma 70.7.1 to descend $h'$ to some $h''$. Then we can apply the approximation property for G-rings (in the form of Smoothing Ring Maps, Theorem 16.13.1) to find a map $A \to \Lambda $ which induces the same map $A \to \kappa $ as we obtain from $A \to \Lambda ^\wedge $. Base changing $h''$ to $\Lambda $ the proof is complete. $\square$
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