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The Geometry of Schemes (Graduate Texts in Mathematics) (Englisch) Taschenbuch – 24. Oktober 2007

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Produktinformation

  • Taschenbuch: 312 Seiten
  • Verlag: Springer; Auflage: 1st ed. 2000. Corr. 2nd printing 2001 (24. Oktober 2007)
  • Sprache: Englisch
  • ISBN-10: 0387986375
  • ISBN-13: 978-0387986371
  • Größe und/oder Gewicht: 15,2 x 1,8 x 22,9 cm
  • Durchschnittliche Kundenbewertung: 4.5 von 5 Sternen  Alle Rezensionen anzeigen (2 Kundenrezensionen)
  • Amazon Bestseller-Rang: Nr. 148.967 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
  • Komplettes Inhaltsverzeichnis ansehen

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Produktbeschreibungen

Pressestimmen

"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

Synopsis

The theory of schemes is the foundation for algebraic geometry proposed and elaborated by Alexander Grothendieck and his coworkers. It has allowed major progress in classical areas of algebraic geometry such as invariant theory and the moduli of curves. It integrates algebraic number theory with algebraic geometry, fulfilling the dreams of earlier generations of number theorists. This integration has led to proofs of some of the major conjectures in number theory (Deligne's proof of the Weil Conjectures, Faltings proof of the Mordell Conjecture). This book is intended to bridge the chasm between a first course in classical algebraic geometry and a technical treatise on schemes. It focuses on examples, and strives to show "what is going on" behind the definitions. There are many exercises to test and extend the reader's understanding. The prerequisites are modest: a little commutative algebra and an acquaintance with algebraic varieties, roughly at the level of a one-semester course. The book aims to show schemes in relation to other geometric ideas, such as the theory of manifolds. Some familiarity with these ideas is helpful, though not required.

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Einleitungssatz
The theory of schemes is the foundation for algebraic geometry formulated by Alexandre Grothendieck and his many coworkers. Lesen Sie die erste Seite
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0 von 1 Kunden fanden die folgende Rezension hilfreich Von davide am 22. Juni 2014
Format: Taschenbuch Verifizierter Kauf
A great book by great authors. Always near my desk!

The edition is good. Not wonderful, but it is not too expensive.
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0 von 6 Kunden fanden die folgende Rezension hilfreich Von Ein Kunde am 11. Mai 2000
Format: Taschenbuch
Very good book for scheme theoritical approach to Algebraic Geometry
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Die hilfreichsten Kundenrezensionen auf Amazon.com (beta)

Amazon.com: 6 Rezensionen
38 von 38 Kunden fanden die folgende Rezension hilfreich
Crystal clear overview of a traditionally abstract subject 13. Juni 2001
Von Dr. Lee D. Carlson - Veröffentlicht auf Amazon.com
Format: Taschenbuch
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. When departing from the assumption of a field that is not finitely generated, extra points will have to be added to classical varieties. The fact that only one closed point appears is compared to the case of complex manifolds, via the concept of a germ. This is a very helpful comparison, and one that further solidifies the understanding of a scheme in the mind of the reader. The authors give the reader a short peek at the etale topology in one of the examples. Examples are then given where the field is not algebraically closed, generalizing classical number theory, and non-reduced schemes, where nilpotents are present. The chapter ends with examples of arithmetic schemes where the spectra of rings are finitely generated over the integers.
Projective schemes are the subject of Chapter 3, and are defined in terms of graded algebras and invariants of projective schemes embedded in projective space are discussed. The Grasmannian scheme is discussed in detail as an example of a projective scheme. Interestingly, Bezout's theorem, very familiar from elementary algebraic geometry, is generalized here to projective schemes.
Constructions from classical algebraic geometry are generalized to schemes in Chapter 4. The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety. The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes. The notion of blowing up is also generalized to the scheme setting, with the authors motivating the discussion by blowing up the plane. The discussion of blow-ups along non-reduced subschemes of a scheme and blow-ups of arithmetic schemes is fascinating and the presentation is crystal clear. Fano varieties are also generalized to Fano schemes in the chapter. Most of the information about these schemes are contained in the exercises, and some of these need to be worked out for a thorough understanding.
The next chapter is more categorical in nature, and deals with generalizations of the classical Sylvester construction of resultants and discriminants to the scheme setting.
In the last chapter the authors return to the functor of points, and motivate the discussion by asking for a parametrization of families of schemes. The authors show, interestingly, that using the functor of points one can more easily compute geometric information about a scheme than using its equations. They illustrate this for the Zariski tangent space. Then after an overview of Hilbert schemes they close the book by introducing the reader to moduli spaces and a hint of algebraic stacks. No end in sight for this beautiful subject..........
34 von 38 Kunden fanden die folgende Rezension hilfreich
A very good start 24. September 2000
Von Colin McLarty - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
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.
8 von 10 Kunden fanden die folgende Rezension hilfreich
Good book! 30. Juni 2009
Von A. Denkert - Veröffentlicht auf Amazon.com
Format: Taschenbuch
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.
6 von 18 Kunden fanden die folgende Rezension hilfreich
good for a diffrent point of few. 11. Januar 2007
Von A. Glang - Veröffentlicht auf Amazon.com
Format: Taschenbuch
I like the book in a way he explains the connection between alg. geom. and com. algebra. So, if you're quiet good in on of those both theories (this is ness. for this book), then it is a good book to learn more about the other side. To be good means you had at least one good course.

It's more or less a student book (4 year or further on) to get a better few to the connection of alg geom with com algebra.
5 von 18 Kunden fanden die folgende Rezension hilfreich
Supplement 9. Februar 2008
Von Ronald Reagan - Veröffentlicht auf Amazon.com
Format: Taschenbuch
This book is a strategic step in my campaign to be able to read EGA. Namely, I bought "The Geometry of Schemes" in order to get a better intuition for schemes (which, sadly, Hartshorne failed to provide). So far so good. There are pictures and the Eisenbud clarity I so like. I still don't get schemes, but since I haven't really read too much of the text that is to be expected.
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