MATH 210A NOTES ARUN DEBRAY DECEMBER 16, 2013 These notes were taken in Stanford's Math 210a class in Fall 2013, taught by Professor Zhiwei Yun. I TEXed them up using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to
[email protected]. Contents Part 1. Basic Theory of Rings and Modules 2 1. Introduction to Rings and Modules: 9/23/132 2. More Kinds of Rings and Modules: 9/25/135 3. Tema con Variazioni, or Modules Over Principal Ideal Domains: 9/27/138 4. Proof of the Structure Theorem for Modules over a Principal Ideal Domain: 10/2/13 11 Part 2. Linear Algebra 14 5. Bilinear and Quadratic Forms: 10/4/13 14 Part 3. Category Theory 17 6. Categories, Functors, Limits, and Adjoints: 10/9/13 17 Part 4. Localization 20 7. Introduction to Localization: 10/11/13 20 8. Behavior of Ideals Under Localization: 10/16/13 23 9. Nakayama's Lemma: 10/18/13 26 10. The p-adic Integers: 10/23/13 28 Part 5. Tensor Algebra 30 11. Discrete Valuation Rings and Tensor Products: 10/25/13 30 12. More Tensor Products: 10/30/13 33 13. Tensor, Symmetric, and Skew-Symmetric Algebras: 11/1/13 35 14. Exterior Algebras: 11/6/13 38 Part 6. Homological Algebra 41 15. Complexes and Projective and Injective Modules: 11/8/13 41 16. The Derived Functor Exti(M; N): 11/13/13 44 17. More Derived Functors: 11/15/13 47 18. Flatness and the Tor Functors: 11/18/13 50 Part 7.