The readings listed are taken from the three course textbooks:
R: Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts. Cambridge, UK: Cambridge University Press, April 26, 1996. ISBN: 9780521458894.
AM: Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1994. ISBN: 9780201407518.
E: Eisenbud, David. Commutative Algebra: With a View Toward Algebraic Geometry. New York, NY: Springer-Verlag, 1999. ISBN: 9780387942698.
| SES # | TOPICS | READINGS |
|---|---|---|
| Rings and ideals | ||
| 1 | Introduction, examples, prime ideals | R: Chapter 0 |
| 2 | Maximal ideals, Zorn’s lemma | R: Chapters 1.4-1.9 |
| 3 | Nilpotents, radical of an ideal, idempotents, local rings | R: Chapters 1.10-2.3 |
| Modules | ||
| 4 | Homomorphisms, generators, Cayley-Hamilton theorem, determinant trick, Nakayama’s lemma | R: Chapters 2.4-2.8 |
| 5 | Exact sequences, ascending chain condition, Noetherian rings | R: Chapters 2.9-3.3 |
| 6 | Hilbert basis theorem, Noetherian modules | R: Chapters 3.4-3.6 and chapters 4.1-4.3 |
| Integral dependence | ||
| 7 | Integral closure, Noether normalization | R: Chapters 4.4-4.8 |
| 8 | Proof of Noether normalization, weak Nullstellensatz |
R: Chapters 4.9-5.2 and chapter 6.1 Handout: Proof of the refined version of the Noether normalization lemma (PDF) |
| Localization | ||
| 9 | Construction of S^{-1}A, basic properties | R: Chapters 6.2-6.3 |
| 10 | Ideals in A and S⁻¹A, localization of modules | R: Chapters 6.4-6.8 |
| 11 | Exactness of localization | R: Chapters 7.1-7.2 |
| 12 | Support of a module SuppM, definition and properties of AssM | R: Chapters 7.3-7.5 |
| 13 | Relation between Supp and Ass, disassembling a module | R: Chapters 7.6-7.9 |
| Primary decomposition | ||
| 14 | Primary ideals, primary decomposition, uniqueness of primary decomposition | R: Chapters 7.10-7.12 |
| Dedekind domains | ||
| 15 | Definition of a DVR | R: Chapter 7.13 and chapters 8.1-8.3 |
| 16 | Main theorem on DVRs, general valuation rings | R: Chapters 8.4-8.6 |
| 17 | Serre’s criterion of normality, Dedekind domains | R: Chapters 8.7-8.9 and 9.3(e)-(f) |
| 18 | Fractional ideals | AM: Chapter 9 |
| 19 | Finiteness of normalization |
AM: Chapter 9, pp. 96-98 R: Chapters 8.11-8.13 |
| Dimension theory | ||
| 20 | Going up, lying over, going down, dimension of affine rings |
AM: pp. 61-62 R: Chapter s8.11-8.13 |
| 21 | Artin rings |
AM: pp. 62-64 and 78 E: Chapter 13 |
| 22 | Krull’s principal ideal theorem, parameter ideals |
AM: Chapter 8 E: Chapter 10 |
| Tensor product | ||
| 23 | Tensor product of modules, restriction and extension of scalars, flatness |
AM: pp. 24-27 E: Chapter 10 |
| Length | ||
| 24 | Modules of finite length | AM: pp. 24-31 and 39-40 |
| 25 | Graded rings and modules, associated graded ring, Hilbert polynomials |
AM: pp. 76-78 E: Chapter 2.4 |
| 26 | Filtrations, Artin-Reese lemma, dimension and Hilbert-Samuel polynomials |
AM: pp. 106-107, 111-112, and 116-121 E: Chapters 5.0-5.2 and chapter 12 |
