I am reading section 14, A Finiteness Theorem of Otto Forster's book Lectures on Riemann Surfaces, and come across a problem on Theorem 14.15 on page 117. In the proof Forster introduces a function

$$F=\det(f\delta_{\nu\mu}-c_{\nu\mu})_{\nu\mu} $$

which is holomorphic, where $f$ is holomorphic, but I don't know why it follows that $F\xi_\nu\mid_Y=0$. I am wondering whether we should replace $F$ with the matrix $(f\delta_{\nu\mu}-c_{\nu\mu})$, but since the proof relies heavily on this claim, I get puzzled. Is there something wrong or am I misunderstanding some stuff? How should I understand this theorem?