Algebra V: Homological Algebra (Encyclopaedia of Mathematical Sciences) [Hardcover]

A. J. Kostrikin (Author), I. R. Shafarevich (Editor)

Book Description

Publication Date: July 1994 | ISBN-10: 0387533737 | ISBN-13: 978-0387533735

This volume of the Encyclopaedia presents a modern approach to homological algebra, which is based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geomtry and algebraic topology.

Product Details

• Hardcover: 222 pages
• Publisher: Springer-Verlag (July 1994)
• Language: English
• ISBN-10: 0387533737
• ISBN-13: 978-0387533735
• Product Dimensions: 9.5 x 6.5 x 0.8 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #8,094,337 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

5 of 5 people found the following review helpful:

5.0 out of 5 stars Homology Category, March 5, 2001

By 

Weiping Li (Stillwater, OK USA) - See all my reviews

This review is from: Algebra V: Homological Algebra (Encyclopaedia of Mathematical Sciences) (Hardcover)

This is an excellent book for whom are familiar with the previous homological algebra techniques (as Cartan and Eilenberg's Homological algebra as first generation, Hilton and Stammbach's A course in Homological algebra as second generation, this book serves as third generation of derived category. This is not written for graduate students who are trying to learn this material as the book is in part of encyclopaedia.

The book explains well why, how and what the derived functors and derived category are introduced. Many concrete exmaples in algebra and geometry are listed in Chapter 3, in particular, Chapter 6 (Mixed Hodge structure), chapter 7(perverse sheaf) and chapter 8(D-modules) are served very good samples on the Derived categories and their equivalence which interprete many deep and fundamental theorems in these areas from category point of view.

Chapter 4(Derived categories and derived functors) and Chapter 5 (Triangulated categories) are the core of the homology category theory. The authors used the localization of homotopy category to construct the universal derived category which is familiar construction for category people under the name of fraction category. In fact many results can be obtained from calculus of fractions (see chapter 19 of the book by H. Schubert "Categories").

It would be better to include the rational homotopy model of Quillen and Sullivan's results and many other more explicit examples from infinitesimally topological calculations. There are few typos and undefined notations. Overall this is a very good reference book indeed.