Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?



Galois cohomology of the field of Laurent series
No to everything. More generally, consider any complete discrete valuation ring $R$ with uniformizer denoted $t$ and separably closed residue field $k$, and let $K = {\rm{Frac}}(R)$. We allow ${\rm{char}}(K) = 0$, since that case has some interest (e.g., $R$ could be the completion of the maximal unramified extension of the henselized localization of the 2-dimensional regular local ring $\mathbf{Z}\_p[\![u,t]\!]$ at the height-1 prime $(u)$; this $R$ has generic characteristic 0, uniformizer $t$, and residue field $\mathbf{F}\_p(\!(u)\!)\_{\rm{sep}}$.)


Since it is well-known that $K$ has cohomological dimension $\le 1$ when $k$ is algebraically closed, and that fields of cohomological dimension $\le 1$ have vanishing degree-1 Galois cohomology for tori, we just have to show that whenever $k$ is imperfect there is a $K$-torus $T$ for which ${\rm{H}}^1(K,T) \ne 1$.


If $K'/K$ is finite separable then the norm map from $K'$ to $K$ provides an exact sequence of $K$-tori $$1 \rightarrow T \rightarrow {\rm{R}}\_{K'/K}({\rm{GL}}\_1) \rightarrow {\rm{GL}}\_1 \rightarrow 1,$$ so by Hilbert 90 we see that ${\rm{H}}^1(K,T)=1$ if and only if the norm ${K'}^{\times} \rightarrow K^{\times}$ is surjective. 


Let $p = {\rm{char}}(k) > 0$ and choose $a \in R^{\times}$ lifting
$a\_0 \in k-k^p$. The monic polynomial $f = x^p-tx-a \in K[x]$ is irreducible by Gauss' Lemma since it is monic in $R[x]$ with reduction $x^p - a\_0 \in k[x]$ that is irreducible. Thus, it is separable if ${\rm{char}}(K)=0$, and it is also separable if ${\rm{char}}(K) = p$ by direct differentiation. Hence, $K' := K[x]/(f)$ is a degree-$p$ separable extension of $K$ and it has valuation ring $R' = R[x]/(f)$ with uniformizer $t$ since this $R'$ is a 1-dimensional noetherian local domain such that $R'/tR' = k[x]/(x^p-a\_0)$ is a field. In particular, $K'$ has uniformizer $t$ and residue field $k(a\_0^{1/p})$. 


(Beware that in the equicharacteristic case, if we fix an isomorphism $R \simeq k[\![t]\!]$
as we may do by the Cohen structure theorem then the residue field of $R'$ does *not* lift into $R'$ as a $k$-algebra, since $a\_0$ is not a $p$th power in $K$ and so cannot be one in $K'$ either, as $K'$ is separable over $K$. That is, although $R' \simeq k(a\_0^{1/p})[\![t]\!]$ abstractly as rings -- or even as $\mathbf{F}\_p[\![t]\!]$-algebras -- by the Cohen structure theorem, there is no such isomorphism as $k$-algebras or even just as $\mathbf{F}\_p(a\_0)$-algebras.)


Since ${K'}^{\times} = {R'}^{\times} \times t^{\mathbf{Z}}$ and the norm of $t$ is $t^p$, it is clear that the norm map caries ${K'}^{\times}$ into $R^{\times} \times t^{p\mathbf{Z}}$, so the norm is not surjective. (The norm map on integral units is also generally not surjective, since modulo 1-units the induced map $k(a\_0^{1/p})^{\times} \rightarrow k^{\times}$ is readily checked to be the norm map relative to the field extension $k \rightarrow k(a\_0^{1/p})$, namely the $p$-power map, and its image $k^p(a\_0)^{\times}$ is generally quite thin inside $k^{\times}$, though for $k =\mathbf{F}\_p(\!(u)\!)\_{\rm{sep}}$ such thinness doesn't occur. But digging one step into the 1-units reveals failure of surjectivity for any $k$ due to the residue field extension being purely inseparable.)


