SES # | TOPICS | ASSIGNMENTS |
---|---|---|

Rings and ideals | ||

1 | Introduction, examples, prime ideals | R: Problems 1.5, 1.6, 1.12(a) SLK: [*] 1 |

2 | Maximal ideals, Zorn's lemma | R: Problems [*] 1.18, 1.19 SLK: 2 |

3 | Nilpotents, radical of an ideal, idempotents, local rings | R: Problems 1.4, 1.11, [*] 1.10 |

Modules | ||

4 | Homomorphisms, generators, Cayley-Hamilton theorem, determinant trick, Nakayama's lemma | R: Problems 2.1, [*] 2.6 (and prove that every minimal generating set of M has the same number of elements), 2.8 (and prove that the idempotent is unique) |

5 | Exact sequences, ascending chain condition, Noetherian rings | R: Problems [*] 2.9, 3.2, 3.5 (prove it, or give a counterexample) SLK: 3 |

6 | Hilbert basis theorem, Noetherian modules | R: Problems 3.3, 3.4, 3.7 (with p a prime), [*] 3.8 |

Integral dependence | ||

7 | Integral closure, Noether normalization | R: Problems 4.1(a), [*] 4.5 AM: Problem 5.6 |

8 | Proof of Noether normalization, weak Nullstellensatz | R: Problems 4.9, [*] 4.10 AM: Problem 5.9 |

Localization | ||

9 | Construction of S^{-1}A, basic properties | R: Problems: [*] 6.3(a), 6.1 (show that the ring A of problem 6.3(a) works), 6.5 (describe each subring A as a localization of Z) |

10 | Ideals in A and S^{-1}A, localization of modules | R: Problem [*] 6.13 SLK: 4, 5 |

11 | Exactness of localization | AM: Problem [*] 3.1 SLK: 6, 7, 8 |

12 | Support of a module SuppM, definition and properties of AssM | R: Problem [*] 7.2 (if true, prove it; if false, give a counterexample) SLK: 9 |

13 | Relation between Supp and Ass, disassembling a module | R: Problems 7.4, 7.6, [*] 7.7 |

Primary decomposition | ||

14 | Primary ideals, primary decomposition, uniqueness of primary decomposition | R: Problems 7.8, 7.10 SLK: [*] 10 |

Dedekind domains | ||

15 | Definition of a DVR | R: Problems 8.1, 8.2, 8.4 SLK: 11 |

16 | Main theorem on DVRs, general valuation rings | R: Problem [*] 8.6 SLK: 12, 13 |

17 | Serre's criterion of normality, Dedekind domains | R: Problem 8.7 SLK: [*] 14, 15 |

18 | Fractional ideals | AM: Problems 9.7, 9.8 SLK: [*] 16 |

19 | Finiteness of normalization | SLK: 17, 18, [*] 19, 20 |

Dimension theory | ||

20 | Going up, lying over, going down, dimension of affine rings | E: Problems 13.2, [*] 13.3 SLK: 21, 22, 23 |

21 | Artin rings | E: Problem 9.4 SLK: [*] 24, 25, 26 |

22 | Krull's principal ideal theorem, parameter ideals | SLK: 27, [*] 28, 29 |

Tensor product | ||

23 | Tensor product of modules, restriction and extension of scalars, flatness | SLK: [*] 30, 31, 32 |

Length | ||

24 | Modules of finite length | SLK: 33, [*] 34, 35 |

25 | Graded rings and modules, associated graded ring, Hilbert polynomials | |

26 | Filtrations, Artin-Reese lemma, dimension and Hilbert-Samuel polynomials |