- 原书名：Methods of Homological Algebra 2nd edition
- 原出版社： Springer
Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections.
Ⅰ.1 Triangulated Spaces
Ⅰ.2 Simplicial Sets
Ⅰ.3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem
Ⅰ.4 Homology and Cohmology
Ⅰ.6 The Exact Sequence
Ⅱ.Main Notions of the Category Theory
Ⅱ.1 The Language of Categories and Functors
Ⅱ.2 Categories and Structures, Equivalence of Categories
Ⅱ.3 Structures and Categories.Representable Functors
Ⅱ.4 Category Approach to the Construction of Geometrical Objects
Ⅱ.5 Additive and Abelian Categories
Ⅱ.6 Functors in Abelian Categories
Ⅲ.Derived Categories and Derived Functors
Ⅲ.1 Complexes as Generalized Objects
Ⅲ.2 Derived Categories and Localization
Ⅲ.3 Triangles as Generalized Exact Triples
Ⅲ.4 Derived Category as the Localization of Homotopic Category
The authors are extremely grateful to Professor Sergei Merkulov who took upon himself the difficult task of carefully reading the text and incorporating appropriate changes. Without his efforts, most of the errors in the first edition would probably have remained unnoticed. Of course, the authors retain full responsibility for ally mistakes that may still remain in the text. ...
S. Gelfand, Yu. Martin
The history of homological algebra can be divided into three periods. The first one starts in the 1940's with the classical works of Eilenberg and MacLane, D.K. Faddeev, and R. Baer and ends with the appearance in 1956 of the fundamental monograph "Homological Algebra" by Cartan and Eilenberg which has lost none of its significance up to the present day.
A. Grothendieck's long paper "Sur quelques points d'algebre homologique" published in 1957 (its appearance had been delayed three years) marks the starting point of the second period, which was dominated by the influence of Grothendieck and his school of algebraic geometry.
The third period, which extends up to the present time, is marked by the ever-increasing use of derived categories and triangulated categories. The basic technique was developed in tile thesis of Grothendieck's student J.-L. Verdier in 1963, but was slow in spreading beyond tile confines of algebraic geometry. Only in the last fifteen years has tile situation changed. First in the work of M. Sato and his school on microlocal analysis, then in the theory of D-modules and perverse sheaves with applications to representation theory, derived categories started to be used as the most suitable instrument.
We now try to characterize these three periods, although we should apologize to the reader for our subjective evaluation and judgment, and for the incompleteness of the material: of course, many important developments do not fit into our rigid scheme.
The book by Cartan and Eilenberg contains essentially all the constructions of homological algebra that constitute its computational tools, namely standard resolutions and spectral sequences. No less important, it contains an axiomatic definition of derived functors of additive functors on the category of modules over a ring.
It was this idea that determined the contours of the second period. The logic of the internal development of analytic and algebraic geometry led to the formulation of the notion of a sheaf and to the realization of the idea that the natural argument of a homology theory is a pair consisting of a space with a sheaf on it, rather than just a space (or a space and a coefficient group). Here the fundamental contribution of H. Cartan's seminar and J.-P. Serre's paper "Faisceaux algebriques coherents" should be mentioned. Grothendieck's paper of 1957 quoted above stresses the analogy between pairs (space, sheaf of abelian groups on it) and pairs (ring, module over it) from the homological point of view and emphasizes the idea that sheaf cohomology should be defined as the derived functor of global sections.
The break with the axiomatic homology and cohomology theory of Ellenberg and Steenrod is in that now an abelian object (a sheaf), rather than a non-abelian one (a space), serves as a variable argument in a cohomology theory. More precisely, a homology or a cohomology theory with fixed coefficients according to Eilenberg and Steenrod is a graded functor from the category of topological spaces into abelian groups that satisfies certain axioms by which it is uniquely determined. The most important of these axioms are the specification of the homology (or cohomology) of the point, and the exact sequence associated with the "excision axiom". The cohomology theory of a fixed topological space according to Grothendieck is a graded functor from the category of sheaves of abelian groups on this space into abelian groups, also satisfying a number of axioms by which it is uniquely determined. The most important of these are the specification of zero-dimensional cohomology as global sections and the exact sequence associated with a short exact sequence of sheaves.
The development of this idea led to a very far-reaching generalization of basic notions of algebraic geometry - Grotheudieck topologies and topoi. Tile essence of this generalization is that since tile cohomological properties of a Space are completely determined by the category of sheaves over it, it is these categories that should be the primary objects of study 'in topology, rather than topological spaces themselves. After a suitable axiomatization of tile properties of such categories we arrive at the notion of a topos. The development of these abstract ideas was motivated by a very concrete problem the famous conjectures of A. Well on the number of solutions of congruences modulo a prime. The very statements of these conjectures include the assumption about the existence of a certain cohomology theory of algebraic varieties in characteristic p ] 0, which would allow us to apply to this situation the Lefsehetz fixed point formula; a cohomology theory of this type was provided by the cohomology of the etale topos constructed by A. Grothendieck and developed by his students.
The main product of the homological algebra of this period was the computation and properties of various derived functors RPF, where F is the functor of global sections, of direct image, of tensor product and so on. These derived functors arise as the cohomology of complexes of the form F(F), where I' are resolutions consisting of injective, projective, fiat, or some other objects suitably adapted to F. The choice of a resolution is highly non-unique, but RPF does not depend on this choice.
In the course of time it came to be understood that one should study all complexes, rather that just resolutions I' (and complexes obtained by applying functors to these resolutions, but modulo a quite complicated equivalence relation, which identifies certain complexes having the same cohomology. ..
The final version of this equivalence relation seems still not to be completely understood. However, a working definition which has proved its worth was formulated in Verdier's thesis of 1963. The categories of complexes obtained in this way are called derived categories, and axiomatization of their properties leads to the notion of triangularted categories.
It seems to us that the main feature of the third period of homologieal algebra is the development of a special kind of "thinking in terms of complexes" as opposed to the "thinking in terms of objects and their cohomologieal invariants" that was typical for the first two periods. Perhaps this appears most vividly in the theory of perverse sheaves; it was shown that the cohomological properties of topological manifolds extend to a substantial degree to spaces with singularities, if we take as coefficients not sheaves but special complexes of sheaves (as objects of the corresponding derived category). The conormal complexes of Grothendieck and Illusie and the dualizing complexes of Grothendieck and Verdier can be considered as earlier constructions of the same kind.
This book is intended as an introductory textbook on the technique of derived categories. Up to now, as far as we know, a mathematician willing to learn this subject has had to turn either to tile two original sources, the abstract of Verdier's thesis and the notes of Hartshorne's seminar, or to tile oral tradition, in those mathematical centers where it still has been maintained.
Thus the central part of the book is Chaps. III-IV, and the reader with even a slight acquaintance with abelian categories and functors can start directly from Chap. III.
Chapter II is directed to the reader who has hardly had anything to do with categories, and we have tried to make clear tile intuitive meaning of standard categorical constructions, and to give examples of "thinking in categories". The main practical aim of this chapter is an introduction to abelian categories.
Finally, Chaps. I and V resulted from our attempt (which had cost us a lot of trouble) to separate off homological algebra from algebraic topology, without burning the bridge between them. Triangulated spaces and simplicial sets are perhaps the most direct methods of describing topology in terms of algebra, and we decided to start the book with an introduction to simplicial methods. On the other hand, algebraic topology is unthinkable without homotopy theory, and the book ends with a treatment of the foundations of homotopic algebra in Chap. V.
We worked on this book with the disquieting feeling that the development of homological algebra is currently in a state of flux, and that the basic definitions and constructions of the theory of triangulated categories, despite their widespread use, are of only preliminary nature (this applies even more to homotopic algebra). There is no doubt that similar thoughts have occurred to the founders of the theory, and to everyone who has seriously worked with it; the absence of a monographic exposition is one of the symptoms.
Nevertheless, this period has already lasted twenty years; papers whose main results cannot even be stated in the old language are multiplying; the need for a textbook is growing. We therefore present this book to the benevolent judgment of the reader.
The plan of the book evolved gradually over several years when the authors were running seminars in the Mathematics Department of Moscow University, and were in contact with members of the "Homological Algebra Fan Club". A.A. Beilinson, M.M. Kapranov, V.V. Schechtman, whose papers and explanations provided us with live examples of thinking in complexes.