Locally finite morphism implies globaly finite

Question

I'm trying to understand the following theorem from Shafarevich's Basic Algebraic Geometry I:

I understand why we can assume the system $D(g_{\alpha})$ to be finite, so that $(g_{\alpha_1},...,g_{\alpha_k})=(1)$. But how does he know that $g_{\alpha}^{n_\alpha}$ generates the unit ideal?