Lemma 109.71.3. The canonical map $\mathcal{X}(S) \to \mathop{\mathrm{lim}}\nolimits \mathcal{X}(S_ n)$ is not essentially surjective.

**Sketch of proof.**
Unwinding definitions, it is enough to check that $H^1(X, A_0) \to \mathop{\mathrm{lim}}\nolimits H^1(X_ n, A_0)$ is not surjective. As $X$ is regular and projective, by Lemmas 109.71.2 and 109.71.1 each $A_0$-torsor over $X$ is torsion. In particular, the group $H^1(X, A_0)$ is torsion. It is thus enough to show: (a) the group $H^1(X_0, A_0)$ is non-torsion, and (b) the maps $H^1(X_{n + 1}, A_0) \to H^1(X_ n, A_0)$ are surjective for all $n$.

Ad (a). One constructs a nontorsion $A_0$-torsor $P_0$ on the nodal curve $X_0$ by glueing trivial $A_0$-torsors on each component of $X_0$ using non-torsion points on $A_0$ as the isomorphisms over the nodes. More precisely, let $x \in X_0$ be a node which occurs in a loop consisting of rational curves. Let $X'_0 \to X_0$ be the normalization of $X_0$ in $X_0 \setminus \{ x\} $. Let $x', x'' \in X'_0$ be the two points mapping to $x_0$. Then we take $A_0 \times _{\mathop{\mathrm{Spec}}(k)} X'_0$ and we identify $A_0 \times {x'}$ with $A_0 \times \{ x''\} $ using translation $A_0 \to A_0$ by a nontorsion point $a_0 \in A_0(k)$ (there is such a nontorsion point as $k$ is algebraically closed and not the algebraic closure of a finite field – this is actually not trivial to prove). One can show that the glueing is an algebraic space (in fact one can show it is a scheme) and that it is an nontorsion $A_0$-torsor over $X_0$. The reason that it is nontorsion is that if $[n](P_0)$ has a section, then that section produces a morphism $s : X'_0 \to A_0$ such that $[n](a_0) = s(x') - s(x'')$ in the group law on $A_0(k)$. However, since the irreducible components of the loop are rational to section $s$ is constant on them ( More on Groupoids in Spaces, Lemma 78.11.3). Hence $s(x') = s(x'')$ and we obtain a contradiction.

Ad (b). Deformation theory shows that the obstruction to deforming an $A_0$-torsor $P_ n \to X_ n$ to an $A_0$-torsor $P_{n + 1} \to X_{n + 1}$ lies in $H^2(X_0, \omega )$ for a suitable vector bundle $\omega $ on $X_0$. The latter vanishes as $X_0$ is a curve, proving the claim. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)