# MA8202 Commutative Algebra (Fall 2010)

## Lecturer

**Steffen Oppermann**, room 844, Sentralbygg II, Steffen [dot] Oppermann [at] math [dot] ntnu [dot] no

## Schedule

Lectures will regularly take place Tuesdays and Wednesdays, **14:15-16:00**, in room **734**.

Exceptions | |
---|---|

Mo, Aug 23rd | extra lecture 10:15-12:00 in 734 |

Tu, Aug 24th | lecture moved to 656 |

Fr, Sep 10th | extra lecture 10:15-12:00 in 656 |

Tu, Sep 14th | no lecture |

We, Sep 15th | no lecture |

Tu, Sep 28th | lecture postponed to 15:15-17:00 |

## Book

The course will follow the book *Introduction to commutative algebra* by **M. F. Atiyah** and **I. G. Macdonald**.

The content of the course is mainly identical to the content of the book. However the course skips Chapter 4 and the first half of Chapter 5 (that is, Chapter 5 is reduced to the subjects "valuation rings" and "Hilbert's Nullstellensatz").

## Exercises

Date | Problems |
---|---|

Aug 25 – Aug 31 | Chapter 1, Exercises 1., 2. i) and ii), 5. i) iii), iv), and v), 10), 15) `Remarks:` 2. and 5. may be simplified by focusing on the case that A is a field; 15. gives a perspective of what will be important in algebraic geometry |

Sep 01 – Sep 07 | Prove the Nakayama Lemma: Let M be a finitely generated A-module, J the Jacobson radical of A. Assume that JM=M. Then M=0. `Hint:` Assume m is a minimal set of generators. Assume _{1} … m_{r}m lies in _{1}JM. Show that m lies in the module generated by the other _{1}m, contradicting the minimality. _{i}Chapter 2, Exercises 1., 3. + find counterexamples to the statement if A is not local or M is not finitely generated, 9, 13. |

Sep 10 – Sep 21 | Chapter 3, Exercises 5, 12 `Hint:` don't read the hint in the book, it is much more complicated than necessary, 13, 19. i) – vi) |

Sep 22 – Sep 28 | Consider the ring A = k[x,y]/(xy), where k is an algebraically closed field. 1. Determine the maximal ideals of A. Draw a picture. 2. Determine the local rings A for all maximal ideals _{m}m of A. |

Sep 29 – Oct 05 | Chapter 6, Exercises 1. and 4. |

Oct 06 – Oct 12 | Chapter 7, Exercises 8., 11., and 14. `Remark:` Exercise 14 is a version of Hilbert's Nullstellensatz which explains the relationship between ideals and geometry in more detail than the version we discussed before. Chapter 8, Exercise 3. |

Nov 10 – Nov 16 | Determine (independently) d(A), delta(A), and dim A for the local rings A = k[[x]] and A = k[[x,y]]/(xy). |

Nov 17 – Nov 23 | Determine dim Z and dim k[x_{1}, …, x_{d}]. You may assume k to be algebraically closed. |