# 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.