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The Stacks project

Lemma 110.72.1. 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 72.3.2 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 79.11.3 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 (4)

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.

Comment #9903 by Doug Liu on

Is the notion "abelian scheme" defined in Stacks Project?

Comment #9904 by Laurent Moret-Bailly on

You can actually get rid of the 2-dimensional condition on , just assuming is noetherian, regular and integral. Then one concludes that: (1) is bijective for every -torsor ; (2) is injective.

Proof: (2) clearly follows from (1). For (1), injectivity is trivial, and surjectivity (as in the current proof) is about proving that the schematic closure of a rational section is a section. This question is fppf-local on , so after étale base change we may assume (note that "étale" allows to preserve regularity). Then just observe that where is the dual abelian scheme. The conclusion follows from the fact that is a regular scheme, so generically defined invertible sheaves extend.

Remarks: One cas weaken "regular" to "smooth -schemes are locally factorial". I learned the Picard trick from Raynaud; there should be a reference somewhere in his work.

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