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Let $X$ be a the spectrum of a regular local ring. What is known about the vanishing of the Zariski cohomology group $$ H^n(\mathbb{A}^k_X,\mathbb{G}_m) $$ for $n,k\geq 0$?

If $X$ has dimension $d$ and if $n>k+d$ so that $\mathbb{A}^k_X$ has dimension smaller than $n$, then the group is zero but can one say more?

user8463524
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  • You know that $H^1(\mathbb{A}^k_X,\mathbb{G}_m)=0$, since $\mathbb{A}_X^k$ is a UFD (since $X$ is a UFD). – Alex Youcis Aug 19 '14 at 02:26
  • Thank you Alex, yes this is the Picard group of $\mathbb{A}^k_X$. It is probably not true that $H^n(\mathbb{A}^k_X,\mathbb{G}_m)\cong H^n(X,\mathbb{G}_m)$ because $X$ is regular, is it? – user8463524 Aug 19 '14 at 09:12
  • Honestly, I don't know. To be frank, I don't even have a good handle on what $H^2(X,\mathbb{G}_m)$ means. For the etale topology it is (in tame cases) the Brauer group, but for the Zariski site, I'm not sure. And, for $H^n(X,\mathbb{G}_m)$ in general, I have no idea. Are you aware of a nice way to think about it? – Alex Youcis Aug 19 '14 at 09:38

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