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 |

SUMMARY

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 elseยญ 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

CONTENT

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 Sequencesโ{128}{148}I. Existence -- 6.4 Long Exact Sequencesโ{128}{148}II. Naturality -- 6.5 Long Exact Sequencesโ{128}{148}III. 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 Lazardโ{128}{153}s 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, Nakayamaโ{128}{153}s 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 Mitchellโ{128}{148}Freyd Theorem and Cheating in Abelian Categories -- D Noether Correspondences in Abelian Categories -- Solution Outlines -- References -- Symbol Index

Mathematics
K-theory
Mathematics
K-Theory