Homomorphisms of rings of germs of analytic functions

Abstract:

Let S and S’ be complex analytic manifolds with S Stein. Let $X \subset S$ and $X’ \subset S’$ be compact sets with X holomorphically convex. Denote by $\mathcal {O}(X)$ (respectively $\mathcal {O}(X’)$) the ring of germs on X (respectively X’) of functions analytic near X (respectively X’). It is shown that each nonzero homomorphism of $\mathcal {O}(X)$ into $\mathcal {O}(X’)$ is given by composition with an analytic map defined in a neighborhood of X’ and taking values in S.