Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2) [Paperback]

Kenji Ueno (Author)

Book Description

ISBN-10: 0821813579 | ISBN-13: 978-0821813577 | Publication Date: August 28, 2001

Modern algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes was explained in Algebraic Geometry 1: From Algebraic Varieties to Schemes, (see Volume 185 in the same series, Translations of Mathematical Monographs). In the present book, Ueno turns to the theory of sheaves and their cohomology. Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local holomorphic functions on a complex manifold or the local sections of a vector bundle. To study schemes, it is useful to study the sheaves defined on them, especially the coherent and quasicoherent sheaves. The primary tool in understanding sheaves is cohomology. For example, in studying ampleness, it is frequently useful to translate a property of sheaves into a statement about its cohomology. <P>The text covers the important topics of sheaf theory, including types of sheaves and the fundamental operations on them, such as ... <P>coherent and quasicoherent sheaves. <BR>proper and projective morphisms. <BR>direct and inverse images. <BR>Cech cohomology. <P>For the mathematician unfamiliar with the language of schemes and sheaves, algebraic geometry can seem distant. However, Ueno makes the topic seem natural through his concise style and his insightful explanations. He explains why things are done this way and supplements his explanations with illuminating examples. As a result, he is able to make algebraic geometry very accessible to a wide audience of non-specialists.

Editorial Reviews

Review

"The author includes in each chapter many examples, problems and exercises illustrating the topics discussed ... The book is aimed at graduate and upper-level undergraduate students who want to learn modern algebraic geometry, but it is also very useful for other mathematicians, for example, as a very good textbook for lecture courses ... Overall, this book is an excellent instructional exposition for an introduction to algebraic geometry." ---- Mathematical Reviews

"This is a concept that has gained in importance in recent years, and the book under review offers an insight into why this should be so ... good material ... It is a clean treatment ... this is a place to dig into and get facts and examples about a number of important areas in algebraic geometry ... This physically compact book thus provides a good pocket guide to a subject of increasing importance." ---- Bulletin of the LMS

Product Details

• Paperback: 184 pages
• Publisher: American Mathematical Society (August 28, 2001)
• Language: English
• ISBN-10: 0821813579
• ISBN-13: 978-0821813577
• Product Dimensions: 8.3 x 5.5 x 0.5 inches
• Shipping Weight: 7.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #2,058,539 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

12 of 13 people found the following review helpful:

4.0 out of 5 stars A good book, but not Hartshorne, October 5, 2005

By 

Colin McLarty (Chardon, OH USA) - See all my reviews

This review is from: Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2) (Paperback)

This is a good book on important ideas. But it competes with Hartshorne ALGEBRAIC GEOMETRY and that is a tough challenge. It has roughly the same prerequisites as Hartshorne and covers much the same ideas. The three volumes together are actually a bit longer than Hartshorne. I had hoped this would be a lighter, more easily surveyable book than Hartshorne's. The subject involves a huge amount of material, an overall survey showing how the parts fit together could be very helpful, and the IWANAMI SERIES has some terrific, brief, easy to read, overviews of such subjects--which give proof techniques but refer elsewhere for the details of some longer proofs.

But it turns out that Ueno differs from Hartshorne in the other direction: He gives more explicit nuts and bolts of the basic constructions. Overall it is easier to get an overview from Hartshorne. Ueno does also give a lot of "insider information" on how to look at things. It is a good book. The annotated bibliography is very interesting. But I have to say Hartshorne is better.

If you get stuck on an exercise in Hartshorne this book might help. If you are working through Hartshorne on your own, you will find this alternative exposition useful as a companion. You might like the more extensive elementary treatment of representable functors, or sheaves, or Abelian categories--but you could get those from references in Hartshorne as well.

Someday some textbook will supercede Hartshorne. Even Rome fell after enough centuries. But here is my prediction, for what it is worth: That successor textbook will not be more elementary than Hartshorne. It will take advantage of progress since Hartshorne wrote (almost 30 years ago now) to make the same material quicker and simpler. It will include number theory examples and will treat coherent cohomology as a special case of etale cohomology---as Hartshorne himself does briefly in his appendices. It will be written by someone who has mastered every aspect of the mathematics and exposition of Hartshorne's book and of Milne's ETALE COHOMOLOGY, and like both of those books it will draw heavily on Grothendieck's brilliant, original, but thorny Elements de Geometrie Algebrique. Of course some people have that level of mastery, notably Deligne, Hartshorne, and Milne who have all written great exposition. But they can't do everything and no one has yet boiled this down to a textbook successor to Hartshorne. If you write this successor *please* let me know as I am dying to read it.