Winter 2009

The syllabus will be updated as the course progresses. Unless otherwise indicated, section numbers and exercises refer to the course text.

Date |
Sections/Notes |
Material
| Recommended Exercises |

Jan 7 | Introduction | Overview of the course, motivation | None |

Jan 12 | A.1 | Review of rings | Questions in main text of A.1.1, A.1.2; A.4 (Page 207): A.4.1a, A.4.1b, A.4.2a, A.4.2b |

Jan 14 | A.1 | Noetherian rings, factorial rings, basic topology | A.4.3a, A.4.3b (Page 208) |

Jan 19 | I.1 | Affine algebraic sets, Zariski topology | Page 24: Exercises 1, 2 |

Jan 21 | I.2, I.3 | Ideal of an affine algebraic set, irreducibility | Page 24: Exercise 3 |

Jan 26 | I.3, I.4 | Irreducibility (cont.), Hilbert's Nullstellensatz | Page 24: Exercise 4 |

Jan 28 | I.4 | Hilbert's Nullstellensatz (cont.) | Page 24: Exercise 5 |

Feb 2 | I.5, I.6 | Intersection of plane curves, morphisms | Page 24: Exercise 6 |

Feb 4 | I.6 | Relation between affine algebraic sets and algebras of finite type | Page 24: Exercises 7, 8 |

Feb 9 | II.1, II.2, II.3 | Projective space | Page 34: Exercises 1, 2 |

Feb 11 | II.3, II.4 | Conics in projective space, projective algebraic sets | Page 35: Exercise 3 |

Feb 23 | II.5, II.7 | Graded rings, ideals of projective algebraic sets | Page 35: Exercise 4, Page 36: Exercise 5 |

Feb 25 | N/A | Midterm Exam
| N/A |

Mar 2 | II.6, III.0, III.1 | Graded rings associated to projective algebraic sets, sheaves | Page 66: Exercise B.1 |

Mar 4 | III.1, III.2 | Sheaves, ringed spaces | Page 63: Exercise A.1 |

Mar 9 | III.2 | Structure sheaf of an affine algebraic set | Page 63: Exercise A.2, Page 67: Exercise B.2 |

Mar 11 | III.3, III.4 | Affine varieties, algebraic varieties | Page 63: A.3a |

Mar 16 | III.4, III.5 | Subvarieties, local rings | Page 64: A.4, A.5 |

Mar 18 | III.5, III.6 | Local rings, sheaves of modules | |

Mar 23 | III.6, III.8 | Sheaves of modules, projective varieties | |

Mar 25 | III.8, III.11 | Projective varieties, some more on morphisms | Page 64: A.6, Page 68: B.4 |

Mar 30 | IV.1 | Dimension | |

Apr 1 | IV.2, V.0, V.1 | Dimension, tangent spaces | IV.1 (this one will require a bit of work) |

Apr 6 | V.1 | Tangent spaces | |

Apr 8 | V.2, Appendix B | Singular points, schemes |