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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