The Stacks project

Lemma 109.71.1. Let $W$ be a two dimensional regular integral Noetherian scheme with function field $K$. Let $G \to W$ be an abelian scheme. Then the map $H^1_{fppf}(W, G) \to H^1_{fppf}(\mathop{\mathrm{Spec}}(K), G)$ is injective.

Sketch of proof. Let $P \to W$ be an fppf $G$-torsor which is trivial in the generic point. Then we have a morphism $\mathop{\mathrm{Spec}}(K) \to P$ over $W$ and we can take its scheme theoretic image $Z \subset P$. Since $P \to W$ is proper (as a torsor for a proper group algebraic space over $W$) we see that $Z \to W$ is a proper birational morphism. By Spaces over Fields, Lemma 71.3.2 the morphism $Z \to W$ is finite away from finitely many closed points of $W$. By (insert future reference on resolving indeterminacies of morphisms by blowing quadratic transformations for surfaces) the irreducible components of the geometric fibres of $Z \to W$ are rational curves. By More on Groupoids in Spaces, Lemma 78.11.3 there are no nonconstant morphisms from rational curves to group schemes or torsors over such. Hence $Z \to W$ is finite, whence $Z$ is a scheme and $Z \to W$ is an isomorphism by Morphisms, Lemma 29.54.8. In other words, the torsor $P$ is trivial. $\square$


Comments (2)

Comment #727 by Kestutis Cesnavicius on

Why is this a consequence of the cited result of Raynaud as claimed?

Comment #730 by on

You are absolutely right, this reference is probably not correct. Bhargav, who wrote the section this result is in, had a different result which is a special case of the result of Raynaud, but then I edited but just carried along the reference blindly. Argh! I have removed it and also fixed the other two snafus you pointed out. Many thanks! You can find the changes in this commit.


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