Synopsis
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. <P>The text covers the important topics of sheaf theory, including types of sheaves and the fundamental operations on them, such as ... <P>coherent and quasicoherent sheaves. <BR>proper and projective morphisms. <BR>direct and inverse images. <BR>Cech cohomology. <P>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. As a result, he is able to make algebraic geometry very accessible to a wide audience of non-specialists.
Review
"The author includes in each chapter many examples, problems and exercises illustrating the topics discussed ... The book is aimed at graduate and upper-level undergraduate students who want to learn modern algebraic geometry, but it is also very useful for other mathematicians, for example, as a very good textbook for lecture courses ... Overall, this book is an excellent instructional exposition for an introduction to algebraic geometry." ---- Mathematical Reviews
"This is a concept that has gained in importance in recent years, and the book under review offers an insight into why this should be so ... good material ... It is a clean treatment ... this is a place to dig into and get facts and examples about a number of important areas in algebraic geometry ... This physically compact book thus provides a good pocket guide to a subject of increasing importance." ---- Bulletin of the LMS
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