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Introduction to Algebraic Geometry and Commutative Algebra

Dilip P. Patil and Uwe Storch
World Scientific
Publication Date: 
Number of Pages: 
IISc Lecture Notes Series
[Reviewed by
John Perry
, on

This book aims to introduce the reader to contemporary algebraic geometry and its necessary foundations in commutative algebra. It is self-contained, and touches on major themes that do not appear in many introductory books, such as schemes and the Riemann-Roch Theorem. Based on lectures given at a course at the Indian Institute of Science, they have been arranged into a textbook suitable for self-study or for a class of graduate students. Interesting exercises are sprinkled throughout the text, and the logical development is at a reasonable pace. The content itself is, therefore, of high quality.

Readers should be warned, however, that the book's formatting makes it hard either to read or to use as a reference book. Definitions are not separated from the main body of the text, which is not inherently objectionable, but the terms being defined are hard to pick out unless you are reading along. The choice used to highlight the term is to put spaces between its letters; this means (1) that it is very hard to find when you need to look it up again, and (2) the reader's eyes are greatly annoyed in the course of reading. The following passage on page 2, which defines 5 terms, is a good illustration. It is painful on the eyes on the first pass through the text, and the defined terms are nearly impossible to pick out from a page full of text if you are looking back to find a forgotten definition:

The image of ε is the smallest A-subalgebra of B containing { xi | i ∈ I } and is denoted by A[xi | i ∈ I]. We call it the A - s u b a l g e b r a   g e n e r a t e d by the family xi, i in I. We say that B is an A - a l g e b r a   g e n e r a t e d by the family xi, i ∈ I, if B = A[xi | i ∈ I]. Further, we say that B is a f i n i t e l y   g e n e r a t e d   A - a l g e b r a or an A - a l g e b r a   o f   f i n i t e   t y p e   if there exists a finite family x1, …, xn of elements of B such that B = A[x1, … xn]. A ring homomorphism φ : A → B is called a h o m o m o r p h i s m   o f   f i n i t e   t y p e   if B is an A-algebra of finite type with respect to φ.

That issue aside, this is an excellent text.

John Perry is Associate Professor of Mathematics at the University of Southern Mississippi.

  • Finitely Generated Algebras
  • The K-Spectrum and the Zariski Topology
  • Prime Spectra and Dimension
  • Schemes
  • Projective Schemes
  • Regular, Normal and Smooth Points
  • Riemann–Roch Theorem