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?



$R^2f_{\operatorname{et},*}\mathbb{G}_m$ vs $R^2f_{\operatorname{Zar},*}\mathbb{G}_m$
For a smooth morphism, see Grothendieck, Dix exposés, Groupe de Brauer III, Corollaire 3.2.


