\begin{align*} [a]_{f,f}(\alpha) \begin{cases} =0, &a\in \mfr^n_K;\\ \in \Lambda_{f,n-i}\setminus\Lambda_{f,n-i-1}, &a\in \mfr^i_K\setminus \mfr^{i+1}_K, 0\le i\le n-1, \end{cases}\\ \forall a\in \ocal_K,\alpha\in \Lambda_{f,n}\setminus \Lambda_{f,n-1}. \end{align*}

\begin{equation*} \begin{cases} \Phi_L(\vp)\overset{\operatorname{def}}{=}\vp(1+I): \operatorname{End}_A((A/I)_L)\to (A/I)^\mathrm{rev},\\ \Phi_R(\vp)\overset{\operatorname{def}}{=}\vp(1+I): \operatorname{End}_A((A/I)_R)\to A/I, \end{cases} \end{equation*}

\begin{equation*} \begin{cases} \operatorname{Im} \vp_{\pi,n} &= \{\tau\in \operatorname{Aut}(L_{\pi,n}/K): \tau|_L\in \sigma^\mathbb{Z}\},\\ \ker \vp_{\pi,n} &=\begin{cases} (1+\mfr^n_K)\times \operatorname{Nm}_{K_m/K}(\pi)^\mathbb{Z}, &L = K_m;\\ 1+\mfr^n_K, &L = \Khat. \end{cases} \end{cases} \end{equation*}

\label{prod}为方便起见, 即使$A$无单位元, 对$a\in A$和$\bm{k}\in\mathbb{N}^n$, 我们同样使用记号$a \displaystyle{\prod^{n}_{i=1}} X^{k_i}_i$, 此时它只是形式记号.