# Local Tate duality for F-vector space

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?

• You have explained the notation $A'$ and $(\ )^\vee$, but not $V^*(1)$ and $(\ )^*$. Please kindly explain! Commented Apr 22, 2023 at 12:34
• Probably, when expalining, you will see yourself that (2) follows from (1)... Commented Apr 22, 2023 at 12:40
• $()^*=Hom_{\mathbb F}(\cdot,\mathbb F)$ is the $\mathbb F$-linear dual and $V^*(1)=V^*\otimes_{\mathbb Z_p}\mathbb Z_p(1)$ means the Tate twist of $V^*$, where $\mathbb Z_p(1)$ is the p-adic cyclotomic character of $G_K$. Commented Apr 22, 2023 at 14:13
• Does it help to note that if $\mathbb{L} / \mathbb{F}$ is an extension of finite fields, then the functors on $\mathbb{L}$-vector spaces given by $Hom_{\mathbb{L}}(-, \mathbb{L})$ and $Hom_{\mathbb{F}}(-, \mathbb{F})$ are canonically the same? Commented Apr 22, 2023 at 21:31
• @MikhailBorovoi The natural transformation $Hom_L(-, L) \to Hom_F(-, F)$ is composing with trace. The transformation $Hom_F(-, F) \to Hom_L(-, L)$ is the composite of base-extension $Hom_F(V, F) \to Hom_L(V \otimes_F L, L)$ with the embedding $V \into V \otimes_F L$. Commented May 4, 2023 at 18:31

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.
• I cannot treat the case of $q=p^l$ when $l>1$ because I don't understand your definition of $V^*(1)$. Try to unravle the definition and to construct a $G_K$-equivariant bilinear pairing $$V\times V^*(1) \to \mu_q.$$ Commented Apr 22, 2023 at 17:33
• Thanks for your answer. In the definition of $V^*(1)$, we view $V^*=Hom_{\mathbb F}(V,\mathbb F)$ as a $\mathbb Z_p$-module by letting $\mathbb Z_p$ acts on $\mathbb F$ via $\mathbb Z_p\to \mathbb F_p\to \mathbb F$. In case it helps, this statement of local Tate duality appears on p.32 of link, though his notation is slightly different from mine. Commented Apr 22, 2023 at 21:04