Algebraic geometry. 1 : from algebraic varieties to schemes / Kenji Ueno

Auteur: Ueno, Kenji (1945-) - AuteurType de document: MonographieCollection: Translations of mathematical monographs ; 185Langue: anglaisPays: Etats UnisÉditeur: Providence : American Mathematical Society, 1999Description: 1 vol. (XIX-154 p.) ; 22 cm ISBN: 0821808621 ; br. Note: Contient des exercicesRésumé: This book is the first one by the author in a series of three on algebraic geometry. Here the author provides a basic foundation of Grothendieck's theory of schemes, which is developed in detail in the second book. He starts by introducing algebraic sets and (affine and projective) algebraic varieties over an algebraically closed field. By pointing out the insufficiency of this approach when working over the rationals or the integers, the author comes quite naturally to the spectrum of a ring and the notion of schemes. The theory of schemes is then also considered from a categorical point of view. This treatise may serve as a first introduction for any student interested in algebraic geometry in the style of Grothendieck. It provides basic concepts and definitions, even introducing such notions as localizations, tensor products and inductive and projective limits. The material is illustrated by examples and figures, and some exercises provide the option to verify one's progress. The book does not yet lead, however, to any deep results in algebraic geometry, and the reader who wants to work in this field has to continue his or her studies far beyond the scope of this treatise. The second and the third parts of this book (which are not yet available in English) will presumably serve this purpose. (MathSciNet).Bibliographie: Index. Sujets MSC: 14-01 Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.)
14A15 Algebraic geometry -- Foundations -- Schemes and morphisms
En-ligne: Zentralblatt | MathSciNet | AMS
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Contient des exercices

Index

This book is the first one by the author in a series of three on algebraic geometry. Here the author provides a basic foundation of Grothendieck's theory of schemes, which is developed in detail in the second book. He starts by introducing algebraic sets and (affine and projective) algebraic varieties over an algebraically closed field. By pointing out the insufficiency of this approach when working over the rationals or the integers, the author comes quite naturally to the spectrum of a ring and the notion of schemes. The theory of schemes is then also considered from a categorical point of view.
This treatise may serve as a first introduction for any student interested in algebraic geometry in the style of Grothendieck. It provides basic concepts and definitions, even introducing such notions as localizations, tensor products and inductive and projective limits. The material is illustrated by examples and figures, and some exercises provide the option to verify one's progress. The book does not yet lead, however, to any deep results in algebraic geometry, and the reader who wants to work in this field has to continue his or her studies far beyond the scope of this treatise. The second and the third parts of this book (which are not yet available in English) will presumably serve this purpose. (MathSciNet)

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