Holomorphic convexity of compact sets in analytic spaces and the structure of algebras of holomorphic germs

Abstract

Let ( X , O X ) (X,{\mathcal {O}_X}) be a reduced analytic space and let K be a compact, holomorphically convex subset of X. It is shown that analogs of Cartan’s Theorems A and B are valid for coherent analytic sheaves on K. This result is applied to the study of the algebra of germs on K of functions holomorphic near K. In particular, characterizations of finitely generated ideals, prime ideals and homomorphisms are obtained.