## Homological Algebra: Homological Algebra.A.I. Kostrikin, Igorʹ Rostislavovich Shafarevich, I.R. Shafarevich This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, 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 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 geometry and algebraic topology. |

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### Contents

Complexes and Cohomology | 8 |

2 Standard Complexes in Algebra and in Geometry | 9 |

3 Spectral Sequence | 17 |

Bibliographic Hints | 21 |

The Language of Categories | 22 |

2 Additive and Abelian Categories | 35 |

3 Functors in Abelian Categories | 42 |

4 Classical Derived Functors | 47 |

Bibliographic Hints | 139 |

Mixed Hodge Structures | 140 |

1 The Category of Hodge Structures | 142 |

2 Mixed Hodge Structures on Cohomology with Constant Coefficients | 145 |

3 Hodge Structures on Homotopic Invariants | 148 |

4 HodgeDeligne Complexes | 153 |

5 HodgeDeligne Complexes for Singular and Simplicial Varieties | 155 |

6 HodgeBeilinson Complexes and Derived Categories of Hodge Structures | 157 |

Homology Groups in Algebra and in Geometry | 52 |

2 Obstructions Torsors Characteristic Classes | 56 |

3 Cyclic CoHomology | 60 |

4 NonCommutative Differential Geometry | 67 |

5 CoHomology of Discrete Groups | 71 |

6 Generalities on Lie Algebra Cohomology | 76 |

7 Continuous Cohomology of Lie Groups | 77 |

8 Cohomology of InfiniteDimensional Lie Algebras | 81 |

Bibliographic Hints | 85 |

Derived Categories and Derived Functors | 86 |

2 Derived Category as the Localization of Homotopic Category | 92 |

3 Structure of the Derived Category | 97 |

4 Derived Functors | 102 |

5 Sheaf Cohomology | 110 |

Bibliographic Hints | 120 |

Triangulated Categories | 121 |

2 Examples | 128 |

3 Cores | 133 |

7 Variations of Hodge Structures | 159 |

Perverse Sheaves | 163 |

2 Glueing | 168 |

Bibliographic Hints | 172 |

PModules | 173 |

1 The Weyl Algebra | 175 |

2 Algebraic PModules | 182 |

3 Inverse Image | 188 |

4 Direct Image | 190 |

5 Holonomic Modules | 195 |

6 Regular Connections | 202 |

7 PModules with Regular Singularities | 205 |

8 Equivalence of Categories RiemannHilbert Correspondence | 208 |

Bibliographic Hints | 210 |

211 | |

217 | |

219 | |

### Common terms and phrases

abelian category abelian groups acyclic complex algebraic varieties An-module arbitrary axioms canonically isomorphic category of sheaves Chap cochain cocycle coefficients cohomology coincides cokernel commutative compact consider construction corresponding cyclic define definition Deligne Denote derived category derived functors diagram differential dimension direct image distinguished triangle easily verify elements equivalence of categories exact functor exact sequence exact triple example exists filtration finite finite-dimensional formula full subcategory functor F G-module Geometry graded Hence holonomic holonomic modules homological algebra homomorphisms homotopic induced injective resolution inverse image kernel left adjoint left exact Lemma Let F Lie algebra linear locally free manifold mixed Hodge structures morphism of functors notion perverse sheaves phisms projective proof Proposition prove quasi-isomorphism regular singularities resp right exact ring SAbx Sect sheaves of abelian smooth spectral sequence t-structure tensor product Theorem theory topological space triangulated category unique variation of Hodge vector fields zero