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?
Best Answer
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, 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.