Algebraic Geometry 3: Further Study of Schemes (Translations of Mathematical Monographs)

Algebraic geometry plays an important role in several branches of science and technology. This is the last of three volumes by Kenji Ueno algebraic geometry. This, in together with Algebraic Geometry 1 and Algebraic Geometry 2, makes an excellent textbook for a course in algebraic geometry. In this volume, the author goes beyond introductory notions and presents the theory of schemes and sheaves with the goal of studying the properties necessary for the full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. Ueno also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. Cover Title page Contents Contents to Volumes 1 and 2 Chapter 7. Fundamental Properties of Scheme Theory 7.1. Algebraic Schemes and Algebraic Varieties a. Maximal Spectrum b. Algebraic Varieties c. Algebraic Schemes 7.2. Dimension a. Krull Dimension b. Dimension of a Scheme c. Function Fields of Algebraic Varieties and Their Dimensions d. Normal Schemes and Regular Schemes e. Normalization Morphism f. Weil Divisors and Cartier Divisors 7.3. Flat Morphisms and Proper Morphisms a. Flat Morphisms b. Flat Families c. Chow's Lemma and the Cohomology of Proper Morphisms 7.4. Regular Schemes and Smooth Morphisms a. Kähler Differentials b. The Sheaf of Relative Differential Forms c. Regular Schemes and Non-Singular Algebraic Varietie d. Smooth Morphisms 7.5. Completion and Zariski's Main Theorem a. Completion b. Formal Schemes and Zariski's Main Theorem Summary Exercises Chapter 8. Algebraic Curves and Jacobian Varieties 8.1. Algebraic Curves a. Riemann-Roch Theorem b. Algebraic Curves and Algebraic Function Fields c. Frobenius Morphisms and Étale Morphisms 8.2. Jacobian Varieties a. Elliptic Curves b. Group Schemes c. Jacobian Varieties Summary Exercises Chapter 9. Algebraic Geometry and Analytic Geometry 9.1. Analytic Geometry 9.2. Kodaira's Vanishing Theorem Summary Exercise Overview and References Solutions to Problems Chapter 7 Chapter 8 Solutions to Exercises Chapter 7 Chapter 8 Chapter 9 Fundamental properties of scheme theory Index Copyr/reprinting page Back Cover

This is the third part of the textbook on algebraic geometry by Kenji Ueno (the first two parts were published by the AMS as Volumes 185 and 197 of this series). Here the author presents the theory of schemes and sheaves beyond introductory notions, with the goal of studying properties of schemes and coherent sheaves necessary for full development of modern algebraic geometry. The main topics discussed in the book include dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion and Zariski's main theorem. The author also presents the theory of algebraic curves and their Jacobians and the relation between algebraic and analytic geometry, including Kodaira's Vanishing Theorem. The book contains numerous exercises and problems with solutions, which makes it (together with two previous parts) appropriate for a graduate course on algebraic geometry or for self-study.

Algebraic geometry plays an important role in several branches of science and technology. This book discusses dimension theory, flat and proper morphisms, regular schemes, smooth morphisms, completion, and Zariski's main theorem. It also presents the theory of algebraic curves and their Jacobians. 1. From algebraic varieties to schemes 2. Sheaves and cohomology 3. Further study of schemes.