Let $G$ be a profinite group, let $K/\mathbb{Q}_\ell$ be a finite extension, let $\mathcal{O}_K$ be its ring of integers, let $\lambda \subset \mathcal{O}_K$ be its maximal ideal, and let $k := \mathcal{O}_K/\lambda$ be its residue field. Let
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\begin{align*}
\rho: G &\to GL_n(K)
\end{align*}
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be a continuous representation. Suppose $L_1 \subset K^n$ and $L_2 \subset K^n$ are $\mathcal{O}_K$-lattices that are stable under $\rho(G)$. For $i \in \{1,2\}$, let
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\begin{align*}
\bar{\rho}_i: G &\to GL_k(L_i/\lambda L_i)
\end{align*}
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be the residual representation induced by the action of $G$ on $L_i/\lambda L_i$.
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Assume that there is a [dense subset](/page/Dense%20Subset) $D \subset G$ such that, for every $g \in D$, the characteristic polynomials of $\bar{\rho}_1(g)$ and $\bar{\rho}_2(g)$ in $k[T]$ are equal. Then the semisimplifications of the residual representations are isomorphic: