The Stacks project

Proof. Let $(U, u) \to (S, s)$ be an étale neighbourhood such that $\mathcal{O}_{Y_ U} = (X_ U \to Y_ U)_*\mathcal{O}_{X_ U}$ and $R^1(X_ U \to Y_ U)_*\mathcal{O}_{X_ U} = 0$ and the same is true after base change by $U' \to U$. Then we replace $S$ by the open image of $U \to S$. Given $S' \to S$ we set $U' = U \times _ S S'$ and we obtain étale coverings $\{ U' \to S'\}$ and $\{ Y_{U'} \to Y_{S'}\}$. Thus the truth of the statement for the base change of $c$ by $S' \to S$ follows from the truth of the statement for the base change of $X_ U \to Y_ U$ by $U' \to U$. In other words, the question is local in the étale topology on $S$. Thus by Lemma 109.4.3 we may assume $X$ and $Y$ are schemes. By More on Morphisms, Lemma 37.72.7 there exists an open subscheme $V \subset Y$ containing $Y_ s$ such that $c_*\mathcal{O}_ X|_ V = \mathcal{O}_ V$ and $R^1c_*\mathcal{O}_ X|_ V = 0$ and such that this remains true after any base change by $S' \to S$. Since $g : Y \to S$ is proper, we can find an open neighbourhood $U \subset S$ of $s$ such that $g^{-1}(U) \subset V$. Then $U$ works. $\square$

Proof. Recall that $\mathcal{O}_{S, s} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ S(U)$ where the colimit is over the system of affine neighbourhoods $U$ of $s$. Thus the category of algebraic spaces of finite presentation over the local ring is the colimit of the categories of algebraic spaces of finite presentation over the affine neighbourhoods of $s$. See Limits of Spaces, Lemma 70.7.1. In this way we reduce to the case where $S$ is the spectrum of a local ring and $s$ is the closed point.

Assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is a local ring and $s$ is the closed point. Write $A = \mathop{\mathrm{colim}}\nolimits A_ j$ with $A_ j$ local Noetherian (say essentially of finite type over $\mathbf{Z}$) and local transition homomorphisms. Set $S_ j = \mathop{\mathrm{Spec}}(A_ j)$ with closed point $s_ j$. We can find a $j$ and families of curves $X_ j \to S_ j$, $Y_{j, i} \to S_ j$, see Lemma 109.5.3 and Limits of Stacks, Lemma 102.3.5. After possibly increasing $j$ we can find morphisms $c_{j, i} : X_ j \to Y_{j, i}$ whose base change to $s$ is $c_ i$, see Limits of Spaces, Lemma 70.7.1. Since $\kappa (s) = \mathop{\mathrm{colim}}\nolimits \kappa (s_ j)$ we can similarly assume there is an isomorphism $Y_{j, 1, s_ j} \cong Y_{j, 2, s_ j}$ compatible with $c_{j, 1, s_ j}$ and $c_{j, 2, s_ j}$. Finally, the assumptions $c_{1, s, *}\mathcal{O}_{X_ s} = \mathcal{O}_{Y_{1, s}}$ and $R^1c_{1, s, *}\mathcal{O}_{X_ s} = 0$ are inherited by $c_{j, 1, s_ j}$ because $\{ s_ j \to s\}$ is an fpqc covering and $c_{1, s}$ is the base of $c_{j, 1, s_ j}$ by this covering (details omitted). In this way we reduce the lemma to the case discussed in the next paragraph.

Assume $S$ is the spectrum of a Noetherian local ring $\Lambda$ and $s$ is the closed point. Consider the scheme theoretic image $Z$ of

The statement of the lemma is equivalent to the assertion that $Z$ maps isomorphically to $Y_1$ and $Y_2$ via the projection morphisms. Since taking the scheme theoretic image of this morphism commutes with flat base change (Morphisms of Spaces, Lemma 67.30.12, we may replace $\Lambda$ by its completion (More on Algebra, Section 15.43).

Assume $S$ is the spectrum of a complete Noetherian local ring $\Lambda$. Observe that $X$, $Y_1$, $Y_2$ are schemes in this case (More on Morphisms of Spaces, Lemma 76.43.6). Denote $X_ n$, $Y_{1, n}$, $Y_{2, n}$ the base changes of $X$, $Y_1$, $Y_2$ to $\mathop{\mathrm{Spec}}(\Lambda /\mathfrak m^{n + 1})$. Recall that the arrow

is an equivalence, see Deformation Problems, Lemma 93.10.6. Thus there is an isomorphism of formal objects $(X_ n \to Y_{1, n}) \cong (X_ n \to Y_{2, n})$ of $\mathcal{D}\! \mathit{ef}_{X_ s \to Y_{1, s}}$. Finally, by Grothendieck's algebraization theorem (Cohomology of Schemes, Lemma 30.28.3) this produces an isomorphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$. $\square$

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

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

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

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

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

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. $\square$

Proof. Let $s \in S$. Let $c_0 : X_ s \to Y_0$ be the contraction of Algebraic Curves, Section 53.24 (more precisely Algebraic Curves, Lemma 53.24.2). By Lemma 109.23.3 there exists an elementary étale neighbourhood $(U, u)$ and a morphism $c : X_ U \to Y$ of families of curves over $U$ which recovers $c_0$ as the fibre at $u$. Since $\omega _{Y_0}$ is ample, after possibly shrinking $U$, we see that $Y \to U$ is a stable family of genus $g$ by the openness inherent in Lemmas 109.22.3 and 109.22.5. After possibly shrinking $U$ once more, assertion (1) of the lemma for $c : X_ U \to Y$ follows from Lemma 109.23.1. Moreover, part (2) holds by the uniqueness in Algebraic Curves, Lemma 53.24.2. We conclude that a morphism $c$ as in the lemma exists étale locally on $S$. More precisely, there exists an étale covering $\{ U_ i \to S\}$ and morphisms $c_ i : X_{U_ i} \to Y_ i$ over $U_ i$ where $Y_ i \to U_ i$ is a stable family of curves having properties (1) and (2) stated in the lemma.

To finish the proof it suffices to prove uniqueness of $c : X \to Y$ (up to unique isomorphism). Namely, once this is done, then we obtain isomorphisms

satisfying the cocycle condition (by uniqueness) over $U_ i \times U_ j \times U_ k$. Since $\overline{\mathcal{M}_ g}$ is an algebraic stack, we have effectiveness of descent data and we obtain $Y \to S$. The morphisms $c_ i$ descend to a morphism $c : X \to Y$ over $S$. Finally, properties (1) and (2) for $c$ are immediate from properties (1) and (2) for $c_ i$.

Finally, if $c_1 : X \to Y_ i$, $i = 1, 2$ are two morphisms towards stably families of curves over $S$ satisfying (1) and (2), then we obtain a morphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$ at least locally on $S$ by Lemma 109.23.2. We omit the verification that these morphisms are unique (hint: this follows from the fact that the scheme theoretic image of $c_1$ is $Y_1$). Hence these locally given morphisms glue and the proof is complete. $\square$

Proof. To see this is true, it suffices to check that the construction of Lemma 109.23.4 is compatible with base change (and isomorphisms but that's immediate), see the (abuse of) language for algebraic stacks introduced in Properties of Stacks, Section 100.2. To see this it suffices to check properties (1) and (2) of Lemma 109.23.4 are stable under base change. This is immediately clear for (1). For (2) this follows either from the fact that the contractions of Algebraic Curves, Lemmas 53.22.6 and 53.23.6 are stable under ground field extensions, or because the conditions characterizing the morphisms on fibres in Algebraic Curves, Lemma 53.24.2 are preserved under ground field extensions. $\square$