In the second edition of our book, many of the typos, errors, and imprecise statements that found their way into the first edition, are corrected.

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 any mistakes that may still remain in the text.

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.Simplicial Sets

Ⅰ.1 Triangulated Spaces

Ⅰ.2 Simplicial Sets

Ⅰ.3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem

Ⅰ.4 Homology and Cohmology

Ⅰ.5 Sheaves

Ⅰ.6 The Exact Sequence

Ⅰ.7 Complexes

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

Ⅲ.5 The Structure of the Derived Category

Ⅲ.6 Derived Functors

Ⅲ.7 Derived Functor of the Composition.Spectral Sequence

Ⅲ.8 Sheaf Cohomology

Ⅳ.Triangulated Categories

Ⅳ.1 Triangulated Categories

Ⅳ.2 Derived Categories Are Triangulated

Ⅳ.3 An Example: The Triangulated Category of A-Modules

Ⅳ.4 Cores

Ⅴ.Introduction to Homotopic Algebra

Ⅴ.1 Closed Model Categories

Ⅴ.2 Homotopic Characterization of Weak Equivalences.

Ⅴ.3 DG-Algebras as a Closed Model Category

Ⅴ.4 Minimal Algebras

Ⅴ.5 Equivalence of Homotopy Categories

References

Index