基本同調代數(shù)

前言

　　Five years ago, I taught a one-quarter course in homological algebra. I discovered that there was no book which was really sultable as a text for such a short course, so I decided to write one. The point was to cover both Ext and Tot early, and still have enough material for a larger course （one semester or two quarters） going off in any of several possible directions. This book is "also intended to be readable enough for independent study.　　The core of the subject is covered in Chapters 1 through 3 and the first two sectionsof Chapter 4. At that point there are several options, Chapters 4 and 5 cover the more traditional aspects of dimension and ring changes. Chapters 6 and 7 cover derived functors in general. Chapter 8 focuses on a special property of Tor. These three groupings are.independent, as are various sections from Chapter 9, which is intended as a source of special topics. （The prerequisites for each section of Chapter 9 are stated at the beginning.）　　Some things have been included simply because they are hard to find else- where, and they naturally fit into the discussion. Lazard's theorem （Section 8.4）.is an example; Sections 4, 5, and 7 of Chapter 9 contain other examples, as do the appendices at the end.　　The idea of the book's plan is that subjects can be selected based on the needs of the class. When I taught the course, it was a prerequisite for a course on noncommutative algebraic geometry. It was also taken by several students interested in algebraic topology, who requested the material in Sections 9.2 and 9.3. （One student later said he wished he'd seen injective envelopes, so I put them in, too.） The ordering of the subjects in Chapter

內容概要

Five years ago, I taught a one-quarter course in homological algebra. I discovered that there was no book which was really suitable as a text for such a short course, so I decided to write one. The point was to cover both Ext and Tot early, and still have enough material for a larger course (one semester or two quarters) going off in any of several possible directions. This book is 'also intended to be readable enough for independent study.

書籍目錄

Preface 1 Categories 2 Modules   2.1 Generalities   2.2 Tensor Products   2.3 Exactness of Functors   2.4 Projectives， Injectives， and Flats 3 Ext and Tor   3.1 Complexes and Projective Resolutions   3.2 Long Exact Sequences   3.3 Flat Resolutions and Injective Resolutions   3.4 Consequences 4 Dimension Theory   4.1 Dimension Shifting   4.2 When Flats are Projective   4.3 Dimension Zero   4.4 An Example 5 Change of Rings   5.1 Computational Considerations   5.2 Matrix Rings   5.3 Polynomials  5.4 Quotients and Localization6 Derived Functors  6.1 Additive Functors  6.2 Derived Functors  6.3 Long Exact Sequences-Ⅰ.Existence  6.4 Long Exact Sequences-Ⅱ.Naturality  6.5 Long Exact Sequences-Ⅲ.Weirdness  6.6 Universality of Ext7 Abstract Homological Algebra  7.1 Living Without Elements  7.2 Additive Categories  7.3 Kernels and Cokernels  7.4 Cheating with Projectives  7.5 （Interlude）Arrow Categories  7.6 Homology in Abelin Categories  7.7 Long Exact Sequences  7.8 An Alternative for Unbalanced Categories8 Colimits and Tor9 Odds and EndsA GCDs，LCMs，PIDs，and UFDsB The Ring of Entire FunctionsC The Mitchell-Ereyd Theorem and Cheating in Abelian Cat-egoriesD Noether Correspondences in Abelian CategoriesSolution OutlinesReferencesSymbol IndexIndex

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