The Stacks project

Example 42.68.19. Let $k$ be a field. Consider the ring $R = k[T]/(T^2)$ of dual numbers over $k$. Denote $t$ the class of $T$ in $R$. Let $M = R$ and $\varphi = ut$, $\psi = vt$ with $u, v \in k^*$. In this case $\det _ k(M)$ has generator $e = [t, 1]$. We identify $I_\varphi = K_\varphi = I_\psi = K_\psi = (t)$. Then $\gamma _\varphi (t \otimes t) = u^{-1}[t, 1]$ (since $u^{-1} \in M$ is a lift of $t \in I_\varphi $) and $\gamma _\psi (t \otimes t) = v^{-1}[t, 1]$ (same reason). Hence we see that $\det _ k(M, \varphi , \psi ) = -u/v \in k^*$.