Lemma 107.23.3. 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 107.5.3 and Limits of Stacks, Lemma 100.3.5. 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 70.9.3). 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 15.49.8 and Proposition 15.49.12. 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 37.31.5). 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 91.10.6. (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 97.15.9. By More on Morphisms of Spaces, Lemma 74.43.3 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 100.3.5) and using Limits of Spaces, Lemma 68.7.1 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E7C. Beware of the difference between the letter 'O' and the digit '0'.