Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j\_\*\mathbb G\_m$ ?


It should vanish, shouldn't it? In this case, it would be enough to show that its stalks vanish. Then, if $p \in \mathbb P^1$ we have


$(R^1j\_\*\mathbb G\_m)\_p = \varprojlim H^1(j^{-1}V,\mathbb G\_m) =H^1(\varprojlim j^{-1}V,\mathbb G\_m)$


where the limit is taken over etale (fppf?) neighborhoods $V$ of point $p$. I could not proceed from this point, please help.



Compute higher direct image for Gm under open embedding
As $R^1 j\_\*\mathbb{G}\_m$ is the sheafification of $(V\to \mathbb{P}^1 \text{ etale})\mapsto H^1(V\_U, \mathbb{G}\_m)$ (here $V\_U = V\times\_{\mathbb{P}^1} U$), it suffices to prove that given an etale $V\to \mathbb{P}^1$, an element $\zeta\in H^1(V\_U, \mathbb{G}\_m)$, and a geometric point $\bar x$ of $V$, there exists an etale neighborhood $V'$ of $\bar x$ in $V$ such that the image of $\zeta$ in $H^1(V'\_U, \mathbb{G}\_m)$ is zero.


Note that for a scheme $X$, we have $H^1(X, \mathbb{G}\_m) = Pic(X)$ (this is still true in the etale topology by descent). So we have to prove that given a line bundle $L$ on $V\_U$ and a geometric point $\bar x$ of $V$, there is an etale neighborhood $V'$ of $\bar x$ in $V$ such that the restriction of $L$ to $V'\_U$ is trivial. If $x\in V\_U$, this is clear, as $L$ is locally trivial. In any case, $L$ extends to a line bundle on $V$ (write $L=\mathcal{O}\_{V\_U}(D)$ for some divisor $D$, then the extension can be taken as $\tilde L = \mathcal{O}\_V(D)$), hence is trivial in a neighborhood of $\bar x$. 


