## C.: I.M. HOMOLOGYIn presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented on my entire manuscript; their advice has led to many improvements. ANDERS KOCK and JACQUES RIGUET have read the entire galley proof and caught many slips and obscurities. Among the others whose sug gestions have served me well, I note FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED RICH KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE TER PUPPE, JOSEPH YAO, and a number of my current students at the University of Chicago - not to m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more versions of more chapters than she would like to count. Messrs. |

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### Contents

II | 8 |

III | 9 |

IV | 13 |

V | 15 |

VI | 19 |

VII | 21 |

VIII | 25 |

IX | 28 |

LXVI | 206 |

LXVII | 210 |

LXVIII | 211 |

LXIX | 215 |

LXX | 218 |

LXXI | 220 |

LXXII | 224 |

LXXIII | 227 |

X | 34 |

XI | 35 |

XII | 39 |

XIII | 42 |

XIV | 44 |

XV | 49 |

XVI | 51 |

XVII | 54 |

XVIII | 57 |

XIX | 61 |

XX | 63 |

XXII | 67 |

XXIII | 72 |

XXIV | 76 |

XXV | 82 |

XXVI | 87 |

XXVII | 92 |

XXVIII | 95 |

XXIX | 96 |

XXX | 99 |

XXXI | 102 |

XXXII | 103 |

XXXIII | 104 |

XXXIV | 105 |

XXXV | 108 |

XXXVI | 111 |

XXXVII | 114 |

XXXVIII | 120 |

XXXIX | 121 |

XL | 124 |

XLI | 129 |

XLII | 131 |

XLIII | 134 |

XLIV | 138 |

XLV | 141 |

XLVI | 142 |

XLVII | 146 |

XLVIII | 148 |

XLIX | 150 |

L | 154 |

LI | 159 |

LII | 163 |

LIII | 166 |

LIV | 170 |

LV | 173 |

LVI | 175 |

LVII | 177 |

LVIII | 181 |

LIX | 184 |

LX | 187 |

LXI | 189 |

LXII | 193 |

LXIII | 197 |

LXIV | 200 |

LXV | 204 |

LXXIV | 228 |

LXXV | 233 |

LXXVI | 236 |

LXXVII | 238 |

LXXVIII | 244 |

LXXIX | 248 |

LXXX | 249 |

LXXXI | 254 |

LXXXII | 257 |

LXXXIII | 260 |

LXXXIV | 262 |

LXXXV | 265 |

LXXXVI | 270 |

LXXXVII | 273 |

LXXXVIII | 278 |

LXXXIX | 280 |

XC | 283 |

XCI | 288 |

XCII | 290 |

XCIII | 293 |

XCIV | 295 |

XCV | 298 |

XCVI | 301 |

XCVII | 303 |

XCVIII | 308 |

XCIX | 311 |

C | 315 |

CI | 318 |

CIII | 322 |

CIV | 326 |

CV | 332 |

CVI | 336 |

CVII | 340 |

CVIII | 342 |

CIX | 345 |

CX | 347 |

CXI | 351 |

CXII | 355 |

CXIII | 358 |

CXIV | 359 |

CXV | 361 |

CXVI | 364 |

CXVII | 367 |

CXVIII | 371 |

CXIX | 375 |

CXX | 379 |

CXXI | 386 |

CXXII | 389 |

CXXIII | 394 |

CXXIV | 397 |

CXXV | 400 |

404 | |

CXXVII | 413 |

415 | |

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### Common terms and phrases

A-+A abelian category additive relation axioms bar resolution bifunctor bigraded bimodule boundary chain complex chain transformation cochain coefficients cohomology cokernel commutative diagram composite congruence connecting homomorphisms construction contracting homotopy contravariant corresponding couniversal covariant functor defined definition degree denote DG-algebra differential dimension direct sum dual elements epic epimorphism equivalent extension factor set filtration finite formula free abelian group free module function given gives graded algebra graded K-module hence homo homology groups Horn identity implies induced injective integers inverse K-algebra kernel left exact left module Lemma long exact sequence module homomorphism monic monomorphism morphism natural isomorphism object pair phism polynomial projective resolution Proof Prop proper projective proper short exact Proposition prove quotient right exact right modules ring short exact sequence simplicial singular spectral sequence splits subgroup submodule tensor product torsion product unique vector space yields zero

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