A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q\_p$, $A$ be a finite $G\_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect pairing $H^i(G\_K,A)\times H^{2-i}(G\_K,A')\to \mathbb Q/\mathbb Z$, where $A'=Hom(A,\overline K^\times)$. In particular, there is a canonical isomorphism
$$H^i(G\_K,A)\cong H^{2-i}(G\_K,A')^\vee,$$ where $^\vee=Hom(\cdot,\mathbb Q/\mathbb Z).$


However, another version is used when studying the global/local deformation ring: Let $K$ be a finite extension of $\mathbb Q\_p$, $\mathbb F$ be its residue field, and $V$ be a finite dimensional $\mathbb F$-vector space with a continuous $G\_K$ action. Let $V^\*$ be the dual representation. Then for $0\le i\le 2$, there is a canonical isomorphism
$$H^i(G\_K,V) \cong H^{2-i}(G\_K,V^\*(1))^\*. $$


Can I ask whether it is possible to derive the second version from the first one? If not, is there a reference for the proof of the second version?



Local Tate duality for F-vector space
EDIT: I treat the general case.
Write ${\Bbb F}={\Bbb F}\_q$ where $q=p^l$ for some natural $l$.
Then ${{\Bbb F}}\_q\supseteq {{\Bbb F}}\_p={\Bbb Z}/p{\Bbb Z}$.


Using [a comment of @DavidLoeffler](https://mathoverflow.net/questions/445288/local-tate-duality-for-f-vector-space?noredirect=1#comment1152371_445288),
we obtain that
for an ${\Bbb F}$-vector space $W$, we can identify
\begin{multline\*}
W^\*={\rm Hom}\_{{\Bbb F}}(W, {\Bbb F})\cong {\rm Hom}\_{{{\Bbb F}\_p}}(W, {\Bbb F}\_p)
={\rm Hom}(W, {\Bbb Z}/p{\Bbb Z})\\
\cong{\rm Hom}(W, \tfrac1p{\Bbb Z}/{\Bbb Z})
={\rm Hom}(W, {\Bbb Q}/{\Bbb Z})=W^\vee.
\end{multline\*}


By definition, our $V$ is a finite $G\_K$-module and
$$ V^\*(1)\cong{\rm Hom}\_{{\Bbb F}}(V,{{\Bbb F}})\otimes\_{{{{\Bbb F}\_p}}}\mu\_p$$
where $\mu\_p$ denotes the group of roots of unity of degree dividing $p$ in $\overline K^\times$.
We can identify
\begin{multline\*}
V^\*(1)\cong{\rm Hom}\_{{\Bbb F}}(V,{{\Bbb F}})\otimes\_{{{{\Bbb F}\_p}}}\mu\_p\cong {\rm Hom}(V,{\Bbb Z}/p{\Bbb Z})\otimes\_{{{\Bbb F}\_p}} \mu\_p\\
\cong {\rm Hom}(V,\mu\_p)={\rm Hom}(V, \overline K^\times)=V'.
\end{multline\*}


We conclude that
$$H^{2-i}(G\_K,V^\*(1))^\*\cong H^{2-i}(G\_K, V')^\vee. $$
Now we see that the second assertion of the question is a special case of the first one.


