$ \mathbb{Q}_p $-admissible p-adic representaion

Question

In Serin Hong's notes on $p$-adic hodge theory, he claimed that every $p$-adic representation is $\mathbb{Q}_p$-admissible, as $D_{\mathbb{Q}_p}$ is the identity functor.But by the definition of the $D_{\mathbb{Q}_p}$, we have $$ D_{\mathbb{Q}_p}(V)=(V\otimes_{\mathbb{Q}_p}\mathbb{Q}_p)^{G_K}=V^{G_K}. $$ If $D_{\mathbb{Q}_p}(V)$ is the identity functor, then $V^{G_K}=V$, but this is obviously wrong. Is the statement he claimed incorrect？

Answer 1

I believe what Hong meant is the following: Let $K=\mathbb Q_p$ and let $B=\overline{\mathbb{Q}}_p$ with the usual $\Gamma_K$-action. Then we have $E:=B^{\Gamma_K}=\mathbb Q_p$, and hence there is a functor $D_{B}\colon\mathrm{Rep}_{\mathbb Q_p}(\Gamma_{\mathbb Q_p})\to\mathrm{Vec}_{\mathbb Q_p}$ given by $V\mapsto(V\otimes_{\mathbb Q_p}\overline{\mathbb Q}_p)^{\Gamma_{\mathbb Q_p}}$.

Now the claim is that $D_B$ is the forgetful functor, i.e., there is a natural isomorphism $V\simeq(V\otimes_{\mathbb Q_p}\overline{\mathbb Q}_p)^{\Gamma_{\mathbb Q_p}}$. This is just the statement of Galois descent! (In other words, $D_B$ factors as $\mathrm{Rep}_{\mathbb Q_p}(\Gamma_{\mathbb Q_p})\xrightarrow{-\otimes\overline{\mathbb Q}_p}\mathrm{Rep}_{\overline{\mathbb Q}_p}(\Gamma_{\mathbb Q_p})\simeq\mathrm{Vec}_{\mathbb Q_p}$)