HomologySpringer Science & Business Media, 6 déc. 2012 - 422 pages In 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. |
Table des matières
Ground Ring Extensions and Direct Products | 293 |
Homology of Tensor Products | 295 |
The Case of Graded Algebras | 298 |
Complexes of Complexes | 301 |
Resolutions and Constructions | 303 |
Twostage Cohomology of DGAAlgebras | 308 |
Cohomology of Commutative DGAAlgebras | 311 |
Homology of Algebraic Systems | 315 |
Homology of Complexes | 34 |
Differential Groups | 36 |
Complexes | 44 |
Cohomology | 47 |
The Exact Homology Sequence | 48 |
Some Diagram Lemmas | 49 |
Additive Relations | 51 |
Singular Homology | 54 |
Homotopy | 60 |
Axioms for Homology | 61 |
Extensions and Resolutions | 64 |
Extensions of Modules | 65 |
Addition of Extensions | 67 |
Obstructions to the Extension of Homomorphisms | 72 |
The Universal Coefficient Theorem for Cohomology | 76 |
Composition of Extensions | 82 |
Resolutions | 90 |
Injective Modules | 92 |
Injective Resolutions | 95 |
Two Exact Sequences for Ext | 96 |
Axiomatic Description of | 99 |
The Injective Envelope | 102 |
Cohomology of Groups | 104 |
The Group Ring 2 Crossed Homomorphisms | 106 |
Types of Algebras | 121 |
Group Extensions | 137 |
Modules over Algebras | 184 |
Identities on Hom | 193 |
Dimension | 200 |
Products | 220 |
Acyclic Models | 246 |
Categories of Diagrams | 257 |
Comparison of Allowable Resolutions | 260 |
Relative Abelian Categories | 262 |
Relative Resolutions | 265 |
The Categorical Bar Resolution | 270 |
Relative Torsion Products | 273 |
Direct Products of Rings | 278 |
Cohomology of Algebraic Systems | 280 |
The Cohomology of an Algebra | 283 |
The Homology of an Algebra | 288 |
Homology of Groups and Monoids | 290 |
Factor Sets | 316 |
Spectral Sequences | 318 |
Fiber Spaces | 322 |
Filtered Modules | 326 |
Transgression | 332 |
Exact Couples | 336 |
Bicomplexes | 340 |
The Spectral Sequence of a Covering | 342 |
The Bar Resolution | 343 |
The Characteristic Class of a Group Extension 8 9 | 345 |
Restriction Inflation and Connection | 347 |
The Lyndon Spectral Sequence | 351 |
The Comparison Theorem | 357 |
Derived Functors | 358 |
Squares | 359 |
Subobjects and Quotient Objects | 361 |
15 | 364 |
Proper Exact Sequences | 367 |
Ext without Projectives | 371 |
The Category of Short Exact Sequences | 375 |
Connected Pairs of Additive Functors | 379 |
Connected Sequences of Functors | 386 |
Derived Functors | 389 |
Products by Universality | 394 |
Proper Projective Complexes | 397 |
19 | 399 |
The Spectral KÜNNETH Formula | 400 |
404 | |
405 | |
406 | |
63 | 407 |
408 | |
409 | |
410 | |
List of Standard Symbols | 413 |
415 | |
416 | |
417 | |
418 | |
420 | |
Autres éditions - Tout afficher
Expressions et termes fréquents
abelian category additive relation axioms B₁ bar resolution bifunctor bigraded bimodule boundary chain transformation coefficients cohomology coker composite congruence connecting homomorphisms construction contracting homotopy contravariant corresponding couniversal covariant functor defined definition degree denote DG-algebra DG-module differential dimension direct sum elements epic epimorphism equivalent extension fiber finite formula free abelian group function given gives graded algebra graded K-module hence homo homology groups identity II-module implies induced injective integers inverse K-algebra K-split kernel left R-modules Lemma long exact sequence module homomorphism monic monomorphism morphism natural isomorphism object pair phism projective resolution Proof Prop proper projective proper short exact Proposition prove quotient R-module right exact right R-module ring short exact sequence simplicial spectral sequence split subgroup submodule tensor product torsion product unique yields Z(II zero