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In Le group de Brauer II, Grothendieck states

Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] Alors les groupes $H^q(X,\underline{G_m})$ sont des groupes de torsion pour $q\geq 2$.

Which I translate as saying (in modern terminology) that for any Noetherian scheme whose local rings have strict Henselisations that are factorial, we have only torsion elements in the given cohomology groups (I gather this is étale cohomology). I believe the proof (not stated explicitly) is via a Leray spectral sequence argument and that the étale cohomology of $Spec(k)$ is torsion, but don't hold me to this.

What are some simple schemes for which the condition on the local rings holds? In particular I have a smooth affine variety $V$ over $\mathbb{C}$, for which I am considering $H^2$ with values in $\underline{G_m}$, and would like to know if this is a torsion group. I know that the corresponding analytic sheaf cohomology group $H^2(V,\mathcal{O}^\ast)$ is torsion-free.

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The henselization of a regular local ring is regular, and hence by Auslander-Buchsbaum it is a UFD. So any nonsingular variety satisfies the hypotheses.

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  • $\begingroup$ Thanks - I knew it would be something obvious to those in the know. The result is slightly unsatisfying, though, since the hints at a proof are not enlightening for me. $\endgroup$ Jun 12 '14 at 2:20

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