There is a descent spectral sequence computing $\pi\_\*L\_{K(n)}S^0$ with $E\_2$-term
$$E\_2^{s,t}\cong H^s\_c(\mathbb{G}\_n,(E\_n)\_t)$$
It is mentioned in [Barthel-Beaudry](https://arxiv.org/pdf/1901.09004.pdf) (in the description of Figure 3.30) that if $p>2$ and $2(p-1)>n^2$, then there are only nonzero entries when $t$ is a multiple of $2(p-1)$.



> 
> **Question.** Why does it have this property?
> 
> 
> 


I’m trying to figure out the reason for this, but I could not read directly from the group cohomology, nor can I find a reference for that.



On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere
This sparsity holds even without the assumptions that $p>2$ and $2(p-1)>n^2$. (Those are used to get a horizontal vanishing line). As in the comments, it comes down to the fact that you can use $\widehat{E(n)}\simeq L\_{K(n)}E(n)$ in place of $E\_n$, and $\widehat{E(n)}$ has homotopy groups concentrated in degrees that are a multiple of $2(p-1)$.


One way to get sparsity directly from the group cohomology is to use that there's a copy of $\mathbb{Z}\_p^\times$ sitting inside the center of $\mathbb{G}\_n$, and further sitting inside this is the subgroup $\mathbb{F}\_p^\times\subset\mathbb{Z}\_p^\times$ of roots of unity (the Teichmüller lifts). Because the order of $\mathbb{F}\_p^\times$ is invertible in $\pi\_\ast E\_n$, you get an isomorphism
$$
H^\ast\_c(\mathbb{G}\_n;\pi\_\ast E\_n)\cong H^\ast\_c(\mathbb{G}\_n/\mathbb{F}\_p^\times;(\pi\_\ast E\_n)^{\mathbb{F}\_p^\times}).
$$
An element $\ell\in\mathbb{Z}\_p^\times$ corresponds to multiplication by $\ell$ on the formal group of $E\_n$, so acts on $\pi\_{2t}E\_n$ as multiplication by $\ell^t$. This implies
$$
(\pi\_t E\_n)^{\mathbb{F}\_p^\times} = 
\begin{cases}
\pi\_t E\_n &2(p-1)\mid n\\0&\text{otherwise},
\end{cases}
$$
and thus $H^\ast\_c(\mathbb{G}\_n;\pi\_t E\_n) = 0$ unless $2(p-1)\mid n$.


