**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2001; 184 pp; Softcover

MSC: Primary 14;

**Print ISBN: 978-0-8218-1357-7**

Product Code: MMONO/197

Product Code: MMONO/197

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

**Electronic ISBN: 978-1-4704-4623-9
Product Code: MMONO/197.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

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# Algebraic Geometry 2: Sheaves and Cohomology

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*Kenji Ueno*

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.
- Čech 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. As a result, he is able to make algebraic geometry very accessible to a wide audience of non-specialists.

The book contains numerous problems and exercises with solutions. It would be an excellent text for the second part of a course in algebraic geometry.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.

#### Reviews & Endorsements

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