Homomorphisms of rings of germs of analytic functions

Abstract

Let S and S’ be complex analytic manifolds with S Stein. Let X ⊂ S X \subset S and X ′ ⊂ S ′ X’ \subset S’ be compact sets with X holomorphically convex. Denote by O ( X ) \mathcal {O}(X) (respectively O ( X ′ ) \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 O ( X ) \mathcal {O}(X) into O ( X ′ ) \mathcal {O}(X’) is given by composition with an analytic map defined in a neighborhood of X’ and taking values in S.