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 m1 … mr is a minimal set of generators. Assume m1 lies in JM. Show that m1 lies in the module generated by the other mi, contradicting the minimality. 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 Am for all maximal ideals 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[x1, …, xd]. You may assume k to be algebraically closed. |