Equivalence of the term “Divisor”

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.

These two definitions are not the same and if you think about it, it is not at all surprising. In the first one you are looking at the definition of a divisor of a meromorphic function while in the second one the more general definition of a divisor.

So, perhaps you are thinking of comparing the definition of a divisor as an integral linear combination of points (or more generally of codimension 1 subspaces). This is matching your first description, but it allows divisors that do not come from meromorphic functions. And this is essentially what's in the second definition. It defines a divisor to be locally what you have in your first definition, but allows for the possibility that it does not come from a single meromorphic function.

More generally, the first definition gives you a Weil divisor, the second gives you a Cartier divisor. They are not equivalent in general, but they are on manifolds. More precisely, every Cartier divisor (i.e., one that is locally defined by a single equation) gives you a Weil divisor (i.e., an integral linear combination of codimension 1 subspaces) by looking at its zero set with multiplicities (essentially as in your first definition). A Weil divisor gives you a Cartier divisor if every codimension 1 subspace is defined locally by a single equation. This is true on manifolds, so you have nothing to worry about. Also, when you are looking at divisors of meromorphic functions, for those the two notions are automatically the same.