 equality follows from the definition of $\Phi^\bullet(M^\vee)$ and $\underline{\Psi}^{\bullet}(M)$, respectively. The third equality uses the property that $\psi_M$ is left adjoint to $\varphi_{M^\vee}$ and $\varphi_M$ is left adjoint to $\psi_{M^\vee}$. Similarly,
	\begin{align*}
	\mathcal{H}^1(\Phi^\bullet(M^\vee))^\vee &= ((M^\vee)/({\varphi_{M^\vee}-id})M^\vee)^\vee\\& = ((M^\vee)^\vee)^{(\varphi_{M^\vee}=id)^\vee}\\ &= M^{\psi_M=id}\\ & = \mathcal{H}^0(\underline{\Psi}^{\bullet}(M)).
	\end{align*}
	Hence $$\mathcal{H}^i(\Phi^\bullet(M^\vee))^{\vee} \cong \mathcal{H}^{1-i}(\underline{\Psi}^{\bullet}(M))$$ for all $i$. In other words, the pairing given by (\ref{perfect}) is perfect.
\end{proof} 
}\fi
\begin{remark}\label{Remark-duality}
Note that the functors $\mathcal{H}^i(\Phi^\bullet(-))$ and $\mathcal{H}^i(\underline{\Psi}^\bullet(-))$ commute with direct limits. Therefore the pairing exists for any $M\in \varinjlim {\bf Mod}^{\varphi_q,\Gamma_{LT},\acute{e}t,tor}_{/\mathcal{O}_{\mathcal{E}}}$. 
\end{remark}
\iffalse{
	This induces a perfect pairing of $\mathcal{O}_K$-modules
	\begin{equation*}
	\mathcal{H}^i(\Phi^\bullet(M^\vee))\times \mathcal{H}^{1-i}(\underline{\Psi}^{\bullet}(M))\rightarrow K/\mathcal{O}_K.
	\end{equation*}
	In other words, we have a canonical isomorphism
	\begin{equation}\label{duality}
	\mathcal{H}^i(\Phi^\bullet(M^\vee))^{\vee} \cong \mathcal{H}^{1-i}(\underline{\Psi}^{\bullet}(M)).
	\end{equation}
}\fi
	Let $V \in {\bf Rep}_{\mathcal{O}_K}(G_K)$, and we define the Iwasawa cohomology over $K_{\infty}$ by
	\begin{equation*}
	H^i_{Iw}(K_{\infty}/K,V):= \varprojlim H^i(L,V),
	\end{equation*}
	where $L$ varies over the finite Galois extensions of $K$ contained in $K_{\infty}$, and the projective limit is taken with respect to the cohomological co-restriction maps.	
We now recall the following theorem of \cite[Theorem 5.13]{SV}, which gives a description of the Iwasawa cohomology groups in terms of $\underline{\Psi}^\bullet$ complex. 
	\begin{theorem}\label{Iwasawa cohomology}
Let $V \in {\bf Rep}_{\mathcal{O}_K-tor}^{dis}(G_K) $. Then the complex 
		\begin{equation*}
		\underline{\Psi}^{\bullet}(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))): 0\rightarrow \mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))\xrightarrow{\psi-id}\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))\rightarrow 0,
		\end{equation*} 
		where $\psi= \psi_{\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))}$, and $\chi_{cyc}$ is the cyclotomic character, computes the Iwasawa cohomology groups $H^i_{Iw}({K_{\infty}/K},V)$ for all $i\geq 1$.
	\end{theorem}
	\begin{proof}
		Since $V \in {\bf Rep}_{\mathcal{O}_K-tor}^{dis}(G_K)$, i.e., $V$ is a discrete $\pi$-primary representation of $G_K$, we have an isomorphism 
		\begin{equation*}
		H^i_{Iw}({K_{\infty}/K},V)\cong H^{2-i}(\operatorname{Gal}(\bar{K}/K_{\infty}),V^{\vee}(\chi_{cyc}))^{\vee},
		\end{equation*}
		which is induced from the Local Tate duality. For more details see \cite[Remark 5.11]{SV}. Moreover,
	\begin{align*}
	H^{2-i}(\operatorname{Gal}(\bar{K}/K_{\infty}),V^{\vee}(\chi_{cyc}))^{\vee} &= \mathcal{H}^{2-i}(\Phi^{\bullet}(\mathbb{D}_{LT}(V^\vee(\chi_{cyc}))))^{\vee}\\ &=\mathcal{H}^{2-i}(\Phi^\bullet(\mathbb{D}_{LT}(V^\vee)(\chi_{cyc})))^\vee \\&= \mathcal{H}^{2-i}(\Phi^\bullet(\mathbb{D}_{LT}(V)^\vee(\chi_{LT}^{-1}\chi_{cyc})))^\vee\\& = \mathcal{H}^{2-i}(\Phi^\bullet(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))^\vee))^\vee\\&\cong \mathcal{H}^{i-1}(\underline{\Psi}^{\bullet}(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT})))).
	\end{align*} 
	Here the first equality follows from Proposition \ref{H_K-cohomology}. The second and third equality uses Remark 4.6 and Remark 5.6 of \cite{SV}, respectively, while the last isomorphism comes from Remark \ref{Remark-duality}. Hence
	\begin{equation*}
	H^i_{Iw}({K_{\infty}/K},V)\cong \mathcal{H}^{i-1}(\underline{\Psi}^{\bullet}(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT})))).
	\end{equation*}
	This proves the theorem.
	\end{proof}
	\begin{corollary}\label{Iwasawa cor.}
		For any $V \in {\bf Rep}_{\mathcal{O}_K}(G_K)$, we have $H^i_{Iw}({K_{\infty}/K},V)\cong \mathcal{H}^{i-1}(\underline{\Psi}^{\bullet}(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))))$ for $i\geq 1$.
	\end{corollary}
	\begin{proof}
		Since the transition maps are surjective in the projective system $(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V/\pi^nV(\chi_{cyc}^{-1}\chi_{LT}))))_n$ of co-chain complexes of abelian groups, the first hyper-cohomology spectral sequence degenerates at $E_2$. Moreover, $\varprojlim\limits_n{}^{1} \mathcal{H}^i(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V/\pi^nV(\chi_{cyc}^{-1}\chi_{LT}))))=0$. Then the second hyper-cohomology spectral sequence 
		\begin{equation*}
		\varprojlim\limits_n{}^{i} \mathcal{H}^j(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V/\pi^nV(\chi_{cyc}^{-1}\chi_{LT})))) \Rightarrow \mathcal{H}^{i+j}(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT})))),
		\end{equation*} 
		also degenerates at $E_2$. Therefore $\varprojlim\limits_n \mathcal{H}^i(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V/\pi^nV(\chi_{cyc}^{-1}\chi_{LT}))))= \mathcal{H}^i(\underline{\Psi}^\bullet(\mathbb{D}_{LT}(V(\chi_{cyc}^{-1}\chi_{LT}))))$. Moreover, it follows from Lemma \ref{commutes inverse} that the functor $H^i_{Iw}({K_{\infty}/K},-)$ commutes with the inverse limits. Now the result follows from Theorem \ref{Iwasawa cohomology} by taking the inverse limits.
	\end{proof}
	Next, we generalize most of our results to the case of any complete local Noetherian ring whose residue field is finite extension of $\mathbb{F}_p$.
	
\section{An Equivalence of Categories over the Coefficient Ring}\label{sec6}
\subsection{Background on Coefficient Rings}\label{sec5}
In this section, we recall some basic results on coefficient rings. Recall that a \emph{coefficient ring} $R$ is a complete Noetherian local ring with finite residue field $k_R$ of characteristic $p$, i.e., $k_R$ is a finite extension of $\mathbb{F}_p$. Then $R$ has