Customer Reviews

Algebra V: Homological Algebra (Encyclopaedia of Mathematical Sciences)

5 star:		(1)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.