## A First Course of Homological AlgebraBased on a series of lectures given at Sheffield during 1971-72, this text is designed to introduce the student to homological algebra avoiding the elaborate machinery usually associated with the subject. This book presents a number of important topics and develops the necessary tools to handle them on an ad hoc basis. The final chapter contains some previously unpublished material and will provide additional interest both for the keen student and his tutor. Some easily proven results and demonstrations are left as exercises for the reader and additional exercises are included to expand the main themes. Solutions are provided to all of these. A short bibliography provides references to other publications in which the reader may follow up the subjects treated in the book. Graduate students will find this an invaluable course text as will those undergraduates who come to this subject in their final year. |

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### Contents

The language of functors | 1 |

The Horn functor | 23 |

A derived functor | 53 |

Polynomial rings and matrix rings | 105 |

Duality | 135 |

Local homological algebra | 168 |

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### Common terms and phrases

A-injective A-projective A-submodule A[x]-module abelian groups Accordingly additive covariant functor annihilator Artinian assume belong Chapter Coker g commutative diagram composition series Consequently considered construct an exact covariant functor Deduce denote direct sum direct summand epimorphism essential extension exact rows exists a A-homomorphism ExtA finite base finitely generated left free module Hence HomA A/yA HomA(P Homz identity element identity functor identity map inclusion mapping induced injective envelope injective resolution integer isomorphism LA(A left A-module left exact left ideal left local ring left Noetherian Lemma maximal MjJM monomorphism morphism natural transformation naturally equivalent Noetherian ring notation polynomial projective cover projective module proof is complete proved quasi-local ring reflexive resp right A-module right ideal right Noetherian ring with radical S-algebra S-sequence semi-commutative semi-reflexive Show Solution split exact sequence statements are equivalent submodule Suppose surjective T)-bimodule Theorem 14 two-sided ideal Z.PdA zero zerodivisor