Mathematics 423/502, Topics in Algebra
January-April 2013, MWF 13:00-13:50, Room MATH 204

  • Textbook : J.J. Rotman, Advanced Modern Algebra, 2nd edition.

  • Course description : This course is a sequel to Math. 422/501. It will to cover a range of topics in commutative and homological algebra, including some of the algebraic prerequisites for advanced work in number theory, algebraic geometry and algebraic topology. Topics will include the structure theorem for modules over principal ideal domains, Hilbert Basis Theorem, Noether Normalization Lemma, Hilbert's Nullstellensatz and an introduction to affine algebraic geometry, Groebner bases, tensor products, group cohomology.

  • Class projects : Each student will be required to choose a project and submit a paper on it during the term. The paper should include complete statements of the relevant results and complete proofs. Depending on the topic, I will ask some students to present their projects in class.

    Project 1: Give an example of a principal ideal domain which is not a Euclidean domain.

    Project 2: Tsen-Lang theory (C_n fields).

    Project 3: Wedderburn's Little Theorem (about associative division rings of finite order) and Artin-Zorn Theorem (generalization to alternative division rings).

    Project 4: Effective Nullstellensatz. Possible starting point: MR0944576 (89h:12008).

    Project 5: The fibre dimension theorem. (One possible source for this is Basic Algebraic Geometry by Shafarevich.)

    Project 6: Hilbert's theorem on the finite generation of the ring of invariants.

    Project 7: Krull dimension.

    Project 8: Efficient implementations of Buchsberger's algorithm Possible sources: Becker and Weispfenning MR1213453, Cox, Little and O'Shea MR1189133.

    Project 9: Application of Grobner bases. Start with section 2.8 of Cox, Little and O'Shea (see the reference above).

    Project 10: SAGBI = Sabalgebra analogue of Grobner bases for ideals. Possible sources: my paper MR1966757 and the references there.

    Completed projects

    Homework Assignments