The Stacks project

Remark 110.72.2. We can get rid of the $2$-dimensional condition on $W$ in Lemma 110.72.1. Namely, if $W$ is noetherian, regular and integral, then (1) $P(W)\to P(K)$ is bijective for every $G$-torsor $P$ and (2) $H^1(W,G)\to H^1(K,G)$ is injective. A sketch of the proof goes as follows. First (2) clearly follows from (1). For (1), the 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 $W$, so after étale base change we may assume $P=G$ (note that étale covers preserve regularity). Then observe that $G=\underline{\mathrm{Pic}}^0_{G'/W}$ where $G'$ is the dual abelian scheme. The conclusion follows from the fact that $G'$ is a regular scheme, so generically defined invertible sheaves extend. In fact, one can weaken "$W$ is regular" to "smooth $W$-schemes are locally factorial". Laurent Moret-Bailly learned the Picard trick from Raynaud; there should be a reference somewhere in Raynaud's work.