From: Torsten Ekedahl <teke@tofslan.matematik.su.se>
Subject: Re: References requested.
Date: 24 Jan 2001 06:50:02 +0100
Newsgroups: sci.math,sci.math.research
Summary: Functional equation on complex plane solved cohomological

anderbrasil@alternex.com.br (Anderson Brasil) writes:

> Consider the following problem. Given an entire function g(z), find
> another entire function f(z) such that f(z+1)-f(z)=g(z) for every z in
> the complex domain. 
> 
> Has someone ever studied that problem? Is it related to any part of
> mathematics?
> 

The relation I can think of is the following:

Suppose one wants to computer the cohomology of the structure sheaf of
the complex manifold C^*. Of course, this is a Stein manifold so all
higher cohomology vanishes. On the other hand, we may use the regular
covering space map exp: C --> C^* and use the Hochschild-Serre
spectral sequence to compute. The group of deck transformations is Z
so that the E_2 term of it is H^i(Z,H^j(C,O)), where O is the
structure sheaf. As C also is Stein H^j(C,O)=0 when j > 0 and so we
get an equality

H^i(Z,E) = H^i(C^*,O),

where E is the space of entire functions. Now, for any Z-module E
H^1(Z,E) is the cokernel of the map e |--> se - e, where s is the
generator of Z (1 that is). Hence H^1(C^*,E) is seen as the
obstruction for solving the equation f(z+1)-f(z)=g(z), i. e., the
equation is solvable iff the image of g in it is zero. Now, as C^* is
Stein H^1(C^*,O)=0 so the equation is always solvable.