Complex-Analytic Spaces and Elements

Abstract

A ringed space is a pair $(X,\mathbb {O})$, where X is a topological space (called the underlying topological space) and $\mathbb O$ is a sheaf of commutative rings with multiplicative unity (called the structure sheaf). Given two ringed spaces $(X,\mathbb {O})$ and $(X',\mathbb {O}')$, a morphism of ringed spaces between them is a pair (f, f^∗), where f is a continuous map from X to X′ and f^∗ is an f-homomorphism from $\mathbb {O}'$ to $\mathbb {O}$, i.e., a collection of ring homomorphisms (mapping unity to unity) ${f^*}_{U'} : \mathbb {O}'(U') \to \mathbb {O} (f^{-1}(U') )$, one for each open subset U′ of X′, such that for every $U_1^{\prime } \supset U_2^{\prime }$ the diagram