Author | Osborne, M. Scott. author |
---|---|
Title | Basic Homological Algebra [electronic resource] / by M. Scott Osborne |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1278-2 |
Descript | X, 398 p. online resource |
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 Localization -- 6 Derived Functors -- 6.1 Additive Functors -- 6.2 Derived Functors -- 6.3 Long Exact SequencesโI. Existence -- 6.4 Long Exact SequencesโII. Naturality -- 6.5 Long Exact SequencesโIII. Weirdness -- 6.6 Universality of Ext -- 7 Abstract Homologieal 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 Abelian Categories -- 7.7 Long Exact Sequences -- 7.8 An Alternative for Unbalanced Categories -- 8 Colimits and Tor -- 8.1 Limits and Colimits -- 8.2 Adjoint Functors -- 8.3 Directed Colimits, ?, and Tor -- 8.4 Lazardโs Theorem -- 8.5 Weak Dimension Revisited -- 9 Odds and Ends -- 9.1 Injective Envelopes -- 9.2 Universal Coefficients -- 9.3 The Kรผnneth Theorems -- 9.4 Do Connecting Homomorphisms Commute? -- 9.5 The Ext Product -- 9.6 The Jacobson Radical, Nakayamaโs Lemma, and Quasilocal Rings -- 9.7 Local Rings and Localization Revisited (Expository) -- A GCDs, LCMs, PIDs, and UFDs -- B The Ring of Entire Functions -- C The MitchellโFreyd Theorem and Cheating in Abelian Categories -- D Noether Correspondences in Abelian Categories -- Solution Outlines -- References -- Symbol Index