Example 42.68.21. Let R = k be a field. Let M = k^{\oplus a} \oplus k^{\oplus b} be l = a + b dimensional. Let \varphi and \psi be the following diagonal matrices
\varphi = \text{diag}(u_1, \ldots , u_ a, 0, \ldots , 0), \quad \psi = \text{diag}(0, \ldots , 0, v_1, \ldots , v_ b)
with u_ i, v_ j \in k^*. In this case we have
\det \nolimits _ k(M, \varphi , \psi ) = \frac{u_1 \ldots u_ a}{v_1 \ldots v_ b}.
This can be seen by a direct computation or by computing in case l = 1 and using the additivity of Lemma 42.68.18.
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