Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation.


In his paper on $p$-divisible groups, Tate proves that if $K$ is a $p$-adic field, then the continuous cohomology groups $H^{i}(G, C)$ are one-dimensional over $K$ when $i = 0, 1$ and vanish otherwise.


Are these continuous cohomology groups known when $K \simeq \mathbb{F}\_{q}((t))$ is non-Archimedean of equal characteristic?



Galois cohomology of separable closure
Yes, they are known: They vanish in degrees $i>0$, and for $i=0$ one gets the completed perfection of $K$.


Indeed, let $K'$ be the completed perfection of $K$. Then $G\_K=G\_{K'}$ as both perfection and completion do not change the etale site. But now $K'$ is already perfectoid, so the same techniques of almost mathematics that Tate uses as an intermediate step in his computation apply to prove that $H^i(G\_{K'},C)=0$ for $i>0$, and $H^0(G\_{K'},C)=K'$.


**[Edit:** I realize that it may be worth mentioning that after completion, $\overline{K^{\mathrm{sep}}}$ is perfect.**]**


