Sheaves and More Cohomology

of a Laurent series. If Op is the local ring of holomorphic functions around p. Take Mp the field of meromorphic functions around p, a principle part is just an element of the quotient groupMp/Op. The Mittag-Leffler question is, given a discrete set {pn} of points in S and a principle part at pn for each n, does there exist a meromorphic function f on S, holomorphic outside {pn}, whose principle part at each pn is the one specified? The question is clearly trivial locally, and so the problem is one of passage from local to global data. Here are two approaches, both of which lead to cohomology theories.