Timeline for vanishing of étale cohomology of affine surface

8 events

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Jun 14, 2014 at 14:42	history	edited	user19475	CC BY-SA 3.0	added 743 characters in body
Jun 14, 2014 at 13:39	comment	added	user19475		Thank you. What I mean is: I am looking for cases of $U$ and $\mathscr{A}/U$ such that $H^2(U,\mathscr{A}[\ell^n]^\vee(2)) = 0$.
Jun 14, 2014 at 13:34	comment	added	Martin Bright		Certainly there are some. Take $U$ to be $\mathbb{A}^2$ and $\mathscr{A}$ a constant Abelian variety over $U$. Then you have $\mathrm{Pic}(U)=0$ and $\mathrm{Br}(U)=0$, so $H^2(U,\mu_\ell)$ vanishes. Perhaps you could clarify what you mean by "vanishing results".
Jun 14, 2014 at 12:18	history	edited	user19475	CC BY-SA 3.0	added 88 characters in body
Jun 14, 2014 at 10:04	history	edited	user19475	CC BY-SA 3.0	added 92 characters in body
Jun 14, 2014 at 10:03	comment	added	user19475		My question was if there are $U$ such that this holds true, not if it is true for all $U$.
Jun 14, 2014 at 9:41	comment	added	Martin Bright		This seems unlikely. E.g. if $\mathcal{A}$ is a relative elliptic curve with full $\ell$-torsion, then $\mathcal{A}[\ell]$ is isomorphic to $(\mu_\ell)^2$ (ignoring twists since the field is algebraically closed) and $H^2(U,\mu_\ell)$ very much depends on the geometry of $U$.
Jun 14, 2014 at 8:25	history	asked	user19475	CC BY-SA 3.0