Example 42.68.11. Consider the local ring $R = \mathbf{Z}_ p$. Set $M = \mathbf{Z}_ p/(p^2) \oplus \mathbf{Z}_ p/(p^3)$. Let $u : M \to M$ be the map given by the matrix
\[ u = \left( \begin{matrix} a
& b
\\ pc
& d
\end{matrix} \right) \]
where $a, b, c, d \in \mathbf{Z}_ p$, and $a, d \in \mathbf{Z}_ p^*$. In this case $\det _\kappa (u)$ equals multiplication by $a^2d^3 \bmod p \in \mathbf{F}_ p^*$. This can easily be seen by consider the effect of $u$ on the symbol $[p^2e, pe, pf, e, f]$ where $e = (0 , 1) \in M$ and $f = (1, 0) \in M$.
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