Mathematics 423/502, Topics in Algebra
TTh 14:00-15:20, MATX 1102

  • Textbook : Atiyah and Macdonald, Introduction to commutative algebra.
  • Course description : This is a course in Commutative Algebra, with some homological algebra mixed in. This material is of interest in its own right; it is also important for advanced work in algebraic geometry, algebraic topology and algebraic number theory. Specific topic include:
  • Prerequisites: The official prerequisites are Math 412 or Math 423/501. The most important pre-requisite not listed in the calendar is Math 323 or equivalent. In other words, I will expect a high comfort level with linear algebra, and some familiarity with rings and modules.
  • Homework will be assigned on a bi-weekly bases. Interaction and collaboration on homework is encouraged, but if you collaborate, please acknowledge this in writing.

    Problem Set 1. Due Tuesday, January 16. Chapter 1, Exercises 1, 2, 4, 5, 7, 8, 10, 12.
    In Exercise 5, ignore the question about the converse in (ii).

    Problem Set 2. Due Tuesday, January 30. Extended to Tuesday, February 6. Chapter 2, Exercises 2, 4, 5, 6, 7, 9, 10, and
    Problem A: Let f(x) be a polynomial in one variable with integer coefficients. Suppose there are finitely many primes p1, ..., pk such that for each integer n, f(n) is divisible by pi for some i between 1 and k. Use the Chinese Remainder Theorem to show that there exists an i between 1 and k, such that f(n) is divisible by the same prime pi for each integer n.

    Problem Set 3. Due Tuesday, February 13. Chapter 2, Exercises 1, 3, 12. In Exercise 1, prove the converse as well; the tensor product is 0 if and only if m and n are relatively prime.
    Chapter 6, Exercises 1, 2, 3, 4. In Exercise 4, answer only the first question (about the Noetherian property).

    Problem Set 4. Due Tuesday, March 6.

  • Problem Set 5. Due Tuesday, March 20.

    Solution to Problem 1(a) from Problem Set 5.

    Problem Set 6. Due Thursday, April 5

  • Evaluation : Course marks will be based on the homework and two midterm exams. The midterms will be given in class Thursday, February 15 and Thursday, March 22.

    Midterm 1 will cover the following topics from Chapters 1, 2 and 6 in the text.

    Chapter 1: Zorn's Lemma, rings and ideals, the nilradical and the Jacobson radical, the radical of an ideal, the Chinese Remainder Theorem.
    Chapter 2: Submodules and quotient modules, direct sums and products of modules, free modules, exact sequences, Nakayama's Lemma and its variants, operations on modules (Hom and tensor product), exactness properties of tensor products, flat modules.
    Chapter 6: Artinean and Noetherian rings and modules, connection between the Noetherian and finitely generated modules, composition series and modules of finite length, statement of the Hilbert Basis Theorem.

    Midterm 1 problems and solutions

    Midterm 2 will cover the following topics: the Hilbert Basis theorem from Chapter 7, Grobner bases, rings of fractions from Chapter 3, integral ring extensions from Chapter 5. Khovanskii bases (SAGBI) and Noether Normalization will not be covered.

    Midterm 2 problems and solutions