| Chapter | Section(s) | Topic | Notes | |
|---|---|---|---|---|
| 1 | Appendix A, Category Theory Language, pgs. 417-429 |
A.1 Categories A.2 Functors A.3 Natural Transformations A.4 Abelian Categories; The Exact Imbedding Theorem A.5 Limits and Colimits | Note: The Exact Imbedding Theorem will only be stated; and, we'll show you how to use it. It is proved in "Imbedding of Abelian Categories", Transactions of the American Mathematical Society, 1960, (by myself). | |
| 0 | Chapter 1, pgs. 1-18, 25-29 Chain Complexes |
1.1 Complexes of R-Modules 1.2 Operations on Chain Complexes 1.3 Long Exact Sequences 1.4 Chain Homotopies 1.6 More on Abelian Categories | ||
| 1 | Chapter 2: Derived Functors pg. 30-51, 58-65 |
2.1 delta-Functors 2.2 Projective Resolutions 2.3 Injective Resolutions 2.4 Left Derived Functors 2.5 Right Derived Functors 2.7 Balancing Tor and Ext | Note: The proof of the results in section 2.7 will be delayed; they are much easier to prove once one is familiar with the specral sequence of a double complex. | |
| 2 | Chapter 3: Tor and Ext, pg. 66-76, 80-87 |
3.1 Tor for Abelian Groups 3.2 Tor Flatness 3.3 Ext for nice rings 3.5 Derived Functors of the Inverse Limit | Some proofs will be delayed -- they are easier using spectral sequences | |
| 3 | Chapter 4: Homological Dimension, pg. 102-103 |
4.3 Change of Rings Theorems |
We will only cover Nakayama's Lemma, General Version of Nakayama's Lemma, and some applications | |
| 4 | Chapter 5: Spectral Sequences, pg. 120-159 |
5.1 Introduction 5.2 Terminology 5.5 Convergence 5.9 Exact Couples 5.4 Spectral Sequence of a Filtered Cochain Complex 5.6 Spectral Sequences of a Double Complex 5.8 Grothendieck Spectral Sequence Additional Topics: Category of Sheaves on a Topological Space Cohomology of Sheaves Direct and Inverse Image of a Sheaf The Leray Spectral Sequence. | ||
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Labor Day: Sept 2
OCTOBER BREAK: Oct 14 - 15 | ||||
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Thanksgiving Recess: Nov 27 - Dec 1
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Last Day of Classes: Dec 9
Last updated Aug 18, 2024; may be modified during the term. Back to the Math 537 Home Page... | ||||