Let $k$ be a field. I am interested in the notion of the higher Brauer group defined as follows: For X a smooth scheme over $k$, $Br^r(X):=H^{2r+1}_{et}(X, \mathbb{Z}(r))$, an etale motivic cohomology group. I would like to get a vanishing of this group for $X=Spec~ k$, in terms of the etale cohomological dimension of $k$.

My crude attempt at this is as follows: Fix a prime $l$. The $l$-primary torsion subgroup of Br^r(k) is given by $H^{2r+1}_{et}(Spec~k, \mathbb{Z}_{l}(r))$. Now $H^{2r+1}_{et}(Spec~k, \mathbb{Z}_{l}(r))\simeq H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))$.

Now depending on the prime $l$, for $l\neq char~k$,

$\mathbb{Q}_l/\mathbb{Z}_{l}\simeq \lim_n\mu_{l^n}^{\otimes r}$. For $l=char~k$, it is isomorphic to $\lim_n \nu_n(r)[-r]$. This is in B. Kahn's paper Definition 2.7.

Using these if $l\neq char~k$: $H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))=0$ for $2r>cd(k)$,

and for $l= char~k$: $H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))=H^{r}_{et}(Spec~k,\lim_n \nu_n(r))=0$

for $r>cd(k)$.

I am unsure about the last steps as the sheaves involved are non-torsion. Any comments are welcome!

Let $k$ be a field. I am interested in the notion of the higher Brauer group defined as follows: For X a smooth scheme over $k$, $Br^r(X):=H^{2r+1}_{et}(X, \mathbb{Z}(r))$, an etale motivic cohomology group. I would like to get a vanishing of this group for $X=Spec~ k$, in terms of the etale cohomological dimension of $k$.

My crude attempt at this is as follows: Fix a prime $l$. The $l$-primary torsion subgroup of $Br^r(k) $ is given by $H^{2r+1}_{et}(Spec~k, \mathbb{Z}_{l}(r))$. Now $H^{2r+1}_{et}(Spec~k, \mathbb{Z}_{l}(r))\simeq H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))$.

Now depending on the prime $l$, for $l\neq char~k$,

$\mathbb{Q}_l/\mathbb{Z}_{l}(r)\simeq \lim_n\mu_{l^n}^{\otimes r}$. For $l=char~k$, it is isomorphic to $\lim_n \nu_n(r)[-r]$. This is in B. Kahn's paper Definition 2.7.

Using these if $l\neq char~k$: $H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))=0$ for $2r>cd(k)$,

and for $l= char~k$: $H^{2r}_{et}(Spec~k,\mathbb{Q}_l/\mathbb{Z}_{l}(r))=H^{r}_{et}(Spec~k,\lim_n \nu_n(r))=0$

for $r>cd(k)$.

I am unsure about the last steps as the sheaves involved are non-torsion. Any comments are welcome!