Throughout my university education, I have studied some theory of Riemann surfaces, focusing particularly on Miranda's Algebraic curves and Riemann surfaces. My current studies however are in the theory of Stein manifolds and Stein spaces. In particular, I am looking at (sheaf-)cohomological methods to solving the Cousin problems.
In the theory of Riemann surfaces, for a meromorphic function $f$ on a Riemann surface $X$, we define the divisor of $f$ to be the function $D$ which maps $$f \longmapsto \sum_{p \in X} \text{ord}_p \cdot p,$$ where $\text{ord}_p$ denotes the order of the root or pole of $f$ at $p$.
In the context of Stein manifolds however, we define a divisor to be an element $d \in H^0(X, \mathscr{D})$, where $\mathscr{D}$ is the quotient sheaf of $\mathscr{M}^{\times}$ by $\mathscr{O}_X^{\times}$, where $\mathscr{M}^{\times}$ is the sheaf of germs of invertible meromorphic functions and $\mathscr{O}_X^{\times}$ is the sheaf of germs of invertible holomorphic functions.
Are these definitions equivalent? If they are, I can't see why.
Note that this does seem like a question that should be posted on Math.StackExchange, but I have found little to no luck asking questions of anything north of the elementary theory of analytic functions of several complex variables on that site.
