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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.
Course readings.
| 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
|