In Scholze's paper The Langlands–Kottwitz approach for the modular curve (published version) about $\operatorname{GL}_2$ when he wants to find the relation between the semisimple trace and the usual trace he claims that if $I = I_K\subset G_K$ is the inertia group of a local number field $K$ with residue field $k$ and $V$ is a representation of $G_k$ such that $I_k$ acts through a finite cyclic quotient then $R_I(V)$, $R_{I}:D(\operatorname{Rep}_{\overline{\mathbb Q_l}}(G_K))\to D(\operatorname{Rep}_{\overline{\mathbb Q_l}}(G_k))$ is represented by $$\dotsb\to 0\to V^I\to V^I(1)\to \ 0 \to\dotsb.$$
I think essentially we can reduce the problem to $\mathbb F_l$ coefficients and we can assume $I=\langle s\rangle$ is a finite cyclic group and $G_K=I\rtimes \hat{\mathbb{Z}}$ but I can't compute the derived functor in this case. I think that if $w:D(\operatorname{Rep}_{\mathbb F_l}(\hat{\mathbb{Z}}))\to D(\mathbb F_l)$ is the forgetful functor then $R_I\circ w$ is represented by $$0\to V[I]\xrightarrow{s-1} V[I] \xrightarrow{N}V[I]\to \dotsb$$ but then $R_I$ can't be represented by the complex in the first paragraph.
Any help is appreciated.
