Editing Cousin problems (section)

==First Cousin problem==
The '''first Cousin problem''' or '''additive Cousin problem''' assumes that each difference

:<math>f_i-f_j</math>

is a [[holomorphic function]], where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that

:<math>f-f_i</math>

is ''holomorphic'' on ''U<sub>i</sub>''; in other words, that ''f'' shares the [[mathematical singularity|singular]] behaviour of the given local function. The given condition on the <math>f_i-f_j</math> is evidently ''necessary'' for this; so the problem amounts to asking if it is sufficient. The case of one variable is the [[Mittag-Leffler theorem]] on prescribing poles, when ''M'' is an open subset of the [[complex plane]]. [[Riemann surface]] theory shows that some restriction on ''M'' will be required. The problem can always be solved on a [[Stein manifold]].

The first Cousin problem may be understood in terms of [[sheaf cohomology]] as follows. Let '''K''' be the [[sheaf (mathematics)|sheaf]] of meromorphic functions and '''O''' the sheaf of holomorphic functions on ''M''. A global section <math>f</math> of '''K''' passes to a global section <math>\phi(f)</math> of the quotient sheaf '''K'''/'''O'''. The converse question is the first Cousin problem: given a global section of '''K'''/'''O''', is there a global section of '''K''' from which it arises? The problem is thus to characterize the image of the map

:<math>H^0(M,\mathbf{K}) \, \xrightarrow{\phi} \, H^0(M,\mathbf{K}/\mathbf{O}).</math>

By the [[long exact sequence in homology|long exact cohomology sequence]],

:<math>H^0(M,\mathbf{K}) \,\xrightarrow{\phi}\, H^0(M,\mathbf{K}/\mathbf{O})\to H^1(M,\mathbf{O})</math>

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group ''H''<sup>1</sup>(''M'','''O''') vanishes.  In particular, by [[Cartan's theorems A and B|Cartan's theorem B]], the Cousin problem is always solvable if ''M'' is a Stein manifold.