Mathematics 423/502, Topics in Algebra
January-April 2013, MWF 13:00-13:50, Room MATH 204
J.J. Rotman, Advanced Modern Algebra,
Course description :
This course is a sequel to
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,
Office: 1105 Math Annex
E-mail: reichst at math.ubc.ca
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
Project 4: Effective Nullstellensatz. Possible starting point:
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
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.