Basic Homological AlgebraSpringer 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 |
385 | |
391 | |
393 | |
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Häufige Begriffe und Wortgruppen
A₁ Abelian category additive category B₁ biproduct chain complex Chapter cokernel colim colimits commutative diagram coproduct Corollary covariant functor defined definition denote derived functors dimension theorem direct sum direct summand dn+1 epic essential extension exact rows example Exercise exists F-dim filler finitely flat resolution follows function functor from RM Furthermore global dimension Hence Hom(A Hom(B Hom(C Hom(Pn Homz i+j=n injective envelope injective resolution integral domain isomorphism kernel kernel-exact Krull dimension Künneth left exact left ideal left R-module Lemma LG-dim long exact sequence maximal ideal monic Mor(A morphism Noetherian nonzero Note nth homology one-to-one P-dim P₁ Pn+1 pre-Abelian category prime ideal projective dimension projective resolution proof of Proposition quasilocal quasiprojective R-module right exact ring short exact sequence Show submodule Suppose F Theory Torn unique W-dim zero object
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Verweise auf dieses Buch
Algebras, Rings and Modules, Band 1 Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko Eingeschränkte Leseprobe - 2004 |