Complex-Analytic Spaces and Elements

José Manuel Aroca, Heisuke Hironaka and José Luis Vicente
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José Manuel Aroca: Universidad de Valladolid, Catedrático de Geometría y Topología
Heisuke Hironaka: Harvard University, Professor Emeritus
José Luis Vicente: Universidad de Sevilla, Catedrático de Álgebra

Chapter Chapter 1 in Complex Analytic Desingularization, 2018, pp 1-42 from Springer

Abstract: Abstract A ringed space ringed space is a pair ( X , O ) $$(X,\mathbb {O})$$ , where X is a topological space (called the underlying topological space) and O $$\mathbb O$$ is a sheaf of commutative rings with multiplicative unity (called the structure sheaf). Given two ringed spaces ( X , O ) $$(X,\mathbb {O})$$ and ( X ′ , O ′ ) $$(X',\mathbb {O}')$$ , a morphism of ringed spaces 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 O ′ $$\mathbb {O}'$$ to O $$\mathbb {O}$$ , i.e., a collection of ring homomorphisms (mapping unity to unity) f ∗ U ′ : O ′ ( U ′ ) → O ( f − 1 ( U ′ ) ) $${f^*}_{U'} : \mathbb {O}'(U') \to \mathbb {O} (f^{-1}(U') )$$ , one for each open subset U′ of X′, such that for every U 1 ′ ⊃ U 2 ′ $$U_1^{\prime } \supset U_2^{\prime }$$ the diagram

Date: 2018
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DOI: 10.1007/978-4-431-49822-3_1

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