## An Introduction to Homological AlgebraThe landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. |

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### Contents

II | 1 |

III | 5 |

IV | 10 |

V | 15 |

VI | 18 |

VII | 25 |

VIII | 30 |

IX | 33 |

XLIV | 198 |

XLV | 203 |

XLVI | 206 |

XLVII | 216 |

XLIX | 219 |

L | 223 |

LI | 238 |

LII | 242 |

X | 38 |

XI | 43 |

XII | 49 |

XIII | 51 |

XIV | 58 |

XV | 66 |

XVI | 68 |

XVII | 73 |

XVIII | 76 |

XIX | 80 |

XX | 87 |

XXI | 91 |

XXII | 95 |

XXIII | 99 |

XXIV | 104 |

XXV | 111 |

XXVI | 115 |

XXVII | 120 |

XXVIII | 122 |

XXIX | 127 |

XXX | 131 |

XXXI | 135 |

XXXII | 141 |

XXXIII | 145 |

XXXIV | 150 |

XXXV | 153 |

XXXVI | 160 |

XXXVII | 167 |

XXXVIII | 171 |

XXXIX | 174 |

XL | 177 |

XLI | 182 |

XLII | 189 |

XLIII | 195 |

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### Common terms and phrases

abelian category abelian group additive adjoint Applying associated bounded called central extension chain complex chain homotopy Chapter cochain cohomology commutative composition cone consider construction converges Corollary corresponding cyclic define Definition denote derived functors diagram differential dimension direct double complex element equivalence exact functor Example Exercise exists fact factor field filtration finite flat formula given gives graded Hence homology homomorphism homotopy ideal identity implies induces injective isomorphism k-algebra Lemma Lie algebra lifting localization module morphism multiplication natural Note object obtained projective Proof Proposition prove quotient R-module regular Remark resolution restriction result ring satisfies sends short exact sequence Show simplicial smooth space spectral sequence split subgroup Suppose taking Theorem topological triangle trivial unique universal write yields zero