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The Geometry of Schemes (Graduate Texts in Mathematics) 1st ed. 2000. Corr. 2nd printing 2001 Edition

4.6 out of 5 stars 7 customer reviews
ISBN-13: 978-0387986371
ISBN-10: 0387986375
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Editorial Reviews

Review

"A great subject and expert authors!"
Nieuw Archief voor Wiskunde,June 2001

"Both Eisenbud and Harris are experienced and compelling educators of modern mathematics. This book is strongly recommended to anyone who would like to know what schemes are all about."
Newsletter of the New Zealand Mathematical Society, No. 82, August 2001

About the Author

David Eisenbud is a Professor in the Department of Mathematics at the University of California, Berkeley.

"Benedict Gross" is the Leverett Professor of Mathematics and Dean of Harvard College.

"Joe Harris" is the Higgins Professor of Mathematics and Chair of the Mathematics Department at Harvard.

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Product Details

  • Series: Graduate Texts in Mathematics (Book 197)
  • Paperback: 300 pages
  • Publisher: Springer; 1st ed. 2000. Corr. 2nd printing 2001 edition (October 24, 2007)
  • Language: English
  • ISBN-10: 0387986375
  • ISBN-13: 978-0387986371
  • Product Dimensions: 6 x 0.7 x 9 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #962,456 in Books (See Top 100 in Books)

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Top Customer Reviews

By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on June 12, 2001
Format: Paperback
The theory of schemes is usually thought to be highly abstract and esoteric, and one that makes the study of algebraic geometry even more difficult. The authors definitely dispel this notion in this book, which could have been called "A Concrete Introduction to Schemes", because of the clarity with which the concepts are introduced and explained. After studying this book, one will understand and appreciate the power of schemes in algebraic geometry. The authors do an even better job than they did in their earlier and short work "Schemes: The Language of Modern Algebraic Geometry", which is now out of print.
In chapter 1, the main definitions are given and the basic concepts behind schemes outlined. That schemes are more complicated than varieties is readily apparent even in this beginning chapter, where they are thought of as corresponding to the spectrum of a commutative ring with identity. Very elementary exercises are given to help the reader gain confidence in the constructions involved. They authors do have to discuss some sheaf theory, but they show its relevance nicely in this chapter. They also discuss the notion of a fibered product as a generalization of the idea of a preimage of a set under the application of a function and relate it to the construction of the functor of points. The role of the functor of points as reducing schemes to a kind of set theory is brought out beautifully here.
The next chapter gives many examples of schemes, with the first examples being reduced schemes over algebraically closed fields, these being essentially the ordinary varieties of classical algebraic geometry. The authors then give examples of schemes, the local schemes, which are more general than varieties.
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This book is clear, well written, and has a nice balance of generalities and examples. If you know the basics of rings and modules, this book will show you what schemes are and why they are useful for several different problems: for example, number theory, or studying singularities. I find it a helpful companion to Hartshorne's ALGEBRAIC GEOMETRY. But this book does not get to cohomology, and so cannot actually get to the working methods in the subject. For that, you need Hartshorne.
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If you're interested in learning the basics of Algebraic Geometry and Hartshorne seems too daunting, try this book instead! The authors take their time developing the material and supplement it with exercises and examples, so the student gets an intuition and a feeling for Algebraic Geometry.
Definitely a good addition to a mathematician's library.
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