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Algebraic Geometry 2

Algebraic Geometry 2 - 99 edition

ISBN13: 978-0821813577

Cover of Algebraic Geometry 2 99 (ISBN 978-0821813577)
ISBN13: 978-0821813577
ISBN10: 0821813579
Cover type:
Edition/Copyright: 99
Publisher: American Mathematical Society
Published: 1999
International: No

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Algebraic Geometry 2 - 99 edition

ISBN13: 978-0821813577

Kenji Ueno

ISBN13: 978-0821813577
ISBN10: 0821813579
Cover type:
Edition/Copyright: 99
Publisher: American Mathematical Society

Published: 1999
International: No
Summary

Modern algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes was explained in Algebraic Geometry 1: From Algebraic Varieties to Schemes (see Volume 185 in the same series, Translations of Mathematical Monographs). In the present book, Ueno turns to the theory of sheaves and their cohomology. Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local holomorphic functions on a complex manifold or the local sections of a vector bundle. To study schemes, it is useful to study the sheaves defined on them, especially the coherent and quasicoherent sheaves. The primary tool in understanding sheaves is cohomology. For example, in studying ampleness, it is frequently useful to translate a property of sheaves into a statement about its cohomology. The text covers the important topics of sheaf theory, including types of sheaves and the fundamental operations on them, such as * coherent and quasicoherent sheaves. * proper and projective morphisms. * direct and inverse images. * Cech cohomology. For the mathematician unfamiliar with the language of schemes and sheaves, algebraic geometry can seem distant. However, Ueno makes the topic seem natural through his concise style and his insightful explanations. He explains why things are done this way and supplements his explanations with illuminating examples.

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