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^{-1}A, 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 |