Algebraic Geometry 1: From Algebraic Varieties to SchemesShare this page
This is the first of three volumes on algebraic geometry. The second volume, Algebraic Geometry 2: Sheaves and Cohomology, is available from the AMS as Volume 197 in the
Translations of Mathematical Monographs series.
Early in the 20th century, algebraic geometry underwent a significant overhaul, as mathematicians, notably Zariski, introduced a much stronger emphasis on algebra and rigor into the subject. This was followed by another fundamental change in the 1960s with Grothendieck's introduction of schemes. Today, most algebraic geometers are well-versed in the language of schemes, but many newcomers are still initially hesitant about them. Ueno's book provides an inviting introduction to the theory, which should overcome any such impediment to learning this rich subject.
The book begins with a description of the standard theory of algebraic varieties. Then, sheaves are introduced and studied, using as few prerequisites as possible. Once sheaf theory has been well understood, the next step is to see that an affine scheme can be defined in terms of a sheaf over the prime spectrum of a ring. By studying algebraic varieties over a field, Ueno demonstrates how the notion of schemes is necessary in algebraic geometry.
This first volume gives a definition of schemes and describes some of their elementary properties. It is then possible, with only a little additional work, to discover their usefulness. Further properties of schemes will be discussed in the second volume.
Ueno's book is a self-contained introduction to this important circle of ideas, assuming only a knowledge of basic notions from abstract algebra (such as prime ideals). It is suitable as a text for an introductory course on algebraic geometry.
Table of Contents
Table of Contents
Algebraic Geometry 1: From Algebraic Varieties to Schemes
Undergraduates and first-year graduate students seeking an introduction to algebraic geometry.
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.
-- Mathematical Reviews
This masterly written text, now also available to the international mathematical audience, tells its own tale and represents a highly welcome addition to the great standard textbooks on algebraic geometry.
-- Zentralblatt MATH