5
$\begingroup$

Let $S$ be the spectrum of a discrete valuation ring and $f:X\rightarrow S$ be a relative projective curve with generic fiber smooth and special fiber semistable. How much differ the sheaf $R^2f_{\operatorname{et},*}\mathbb{G}_m$ (computed in the étale topology) from $R^2f_{\operatorname{Zar},*}\mathbb{G}_m$ (computed in the Zariski topology, this is zero)? If they are not the same is there a condition on $f$ to ensure that they coincide?

$\endgroup$
1
  • $\begingroup$ +1 for the austere efficiency of the title. $\endgroup$ Jul 2, 2013 at 7:04

1 Answer 1

4
$\begingroup$

For a smooth morphism, see Grothendieck, Dix exposés, Groupe de Brauer III, Corollaire 3.2.

$\endgroup$
1
  • 2
    $\begingroup$ Actually the result cited (stating that the étale one vanishes) is more general than suggested: it doesn't require that the morphism be smooth, just that $X$ be regular. Also it's worth pointing out that in characteristic zero this follows easily from the Kummer sequence and proper base change, and doesn't need $X$ to be regular; the meat of the theorem proved by Artin is the vanishing of the $p$-torsion in characteristic $p$. $\endgroup$ Jul 2, 2013 at 9:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.