Basic Homological Algebra

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Springer Science & Business Media, 19.05.2000 - 398 Seiten
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 Tor 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 sections ofChapter 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; Sections4,5, and 7ofChapter 9 containother examples, as do the appendices at the end.
 

Inhalt

Categories
3
Modules
13
22 Tensor Products
16
23 Exactness of Functors
24
24 Projectives Injectives and Flats
30
Ext and Tor
41
32 Long Exact Sequences
49
33 Flat Resolutions and Injective Resolutions
57
74 Cheating with Projectives
188
75 Interlude Arrow Categories
204
76 Homology in Abelian Categories
215
77 Long Exact Sequences
227
78 An Alternative for Unbalanced Categories
241
Colimits and Tor
259
82 Adjoint Functors
266
83 Directed Colimits 0 and Tor
272

34 Consequences
68
Dimension Theory
75
42 When Flats are Projective
81
43 Dimension Zero
84
44 An Example
93
Change of Rings
101
52 Matrix Rings
106
53 Polynomials
108
54 Quotients and Localization
112
Derived Functors
125
62 Derived Functors
128
63 Long Exact SequencesI Existence
132
64 Long Exact SequencesII Naturality
142
65 Long Exact SequencesIII Weirdness
149
66 Universality of Ext
153
Abstract Homological Algebra
167
72 Additive Categories
171
73 Kernels and Cokemels
175
84 Lazards Theorem
276
85 Weak Dimension Revisited
282
Odds and Ends
287
92 Universal Coefficients
292
93 The Runneth Theorems
298
94 Do Connecting Homomorphisms Commute?
311
95 The Ext Product
320
96 The Jacobson Radical Nakayamas Lemma and Quasilocal Rings
326
97 Local Rings and Localization Revisited Expository
333
GCDs LCMs PIDs and UFDs
339
The Ring of Entire Functions
347
The MitchellFreyd Theorem and Cheating in Abelian Categories
361
Noether Correspondences in Abelian Categories
365
Solution Outlines for Selected Exercises
375
References
385
Symbol Index
391
Index
393
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