The Cousin and Poincare Problems

Abstract

In this paper we feproduce the proofs of Cousin and Poincare problems for the structure sheaf A [5] without making explicit nse of flabby sheaf theory. We conciude with a Remark in section 3. For the Solutions of these problems we shall merely make direct appeal to a property inherent to A, i.e., sections defined in A can be extended holomorphically to the entire region of definition. We recall the foUotving Definitions. Let Gc: C", be a region (connected öpen set), and A(G) the ring (C- Algebra) of holomorphic functions on G. Then the set A of ali convergent power series (germs) representing the elements of A(G) is called a restricted sheaf över G. It was proved in [1 ] that A is coherent as soon as G is a region of holomorphy.