I am trying to follow Griffiths & Harris, "Principles of Algebraic Geometry", page 131, where they explain how from a given divisor one obtains an element of the quotient sheaf $\mathcal M^*/ \mathcal O^*$. In their notation, $M$ is a complex manifold, $\mathcal M^*$ is the sheaf of not-identically-$0$ meromorphic functions, and $\mathcal O^*$ is the sheaf of nowhere zero holomorphic functions.
Now a divisor on $M$ is a locally finite sum of the form $$ \sum_i a_i V_i $$ where $V_i$ are irreducible analytic hypersurfaces of $M$ and $a_i \in \mathbb Z$. We take an open cover $\{ U_\alpha \}$ of $M$ such that on each $U_\alpha$, each $V_i$ is represented as the zero set of a holomorphic function $g_{i\alpha} \in \mathcal O(U_\alpha)$.
The first step to seeing $D$ as a section in $\mathcal M^*/ \mathcal O^*$ is considering the local sections $$ \prod_i g_{i\alpha}^{a_i} \in \mathcal M^*(U_\alpha) $$ This is all fine. But then they claim that it is easy to see that we obtain a global section in $\mathcal M^*/\mathcal O^*$, which I am not seeing.
As far as I can tell, this amounts to saying that if we have other representing functions for $V_i$, say $g_{\alpha, i}'$, then considering the function $f_\alpha'$ that we get from the $g_{\alpha, i}'$, we have that the quotient $f_\alpha/f_\alpha'$ is a nowhere vanishing holomorphic functions. Why would this be? I feel like this comes down to something which I don't know about the functions locally defining hypersurfaces.
So essentially, my question I think boils down to the simpler case where $D = V$ with $V$ an irreducible analytic hypersurface. Then if $V$ is $f_\alpha = 0$ on $U_\alpha$ and $f_\beta = 0$ on $U_\beta$ with $U_\alpha \cap U_\beta \neq \emptyset$, why is it that $f_\alpha/f_\beta \in \mathcal O^*$? Why is it that the quotient even makes sense, given that $f_\beta$ vanishes on $V$?