Cornell Math - MATH 633, Fall 2006

MATH 633: Noncommutative Algebra (Fall 2006)

Instructor: Stephen Chase

Meeting Time & Room

Prerequisites: MATH 631. Some material from MATH 632 will also be used; e.g., Galois theory, basic properties of projective and injective modules. For such material, definitions and statements of results (but few or no proofs) will be given, and references will be provided.

The course will treat, to the extent that time permits, the following topics from ring theory and representation theory. All of these topics are part of the standard arsenal of the algebraist, and are also useful to those working in areas close to algebra, such as algebraic number theory, algebraic and non-commutative geometry, algebraic K-theory, and algebraic topology.

  1. Basic Concepts for Rings, Modules, and Algebras: Idempotents, annihilators of ideals and modules, endomorphism rings, chain and finiteness conditions, Artinian rings.
     
  2. Some Important Classes of Rings and Algebras: Monoid and group rings, filtered and graded rings and modules, universal enveloping algebras of Lie algebras, exterior and Clifford algebras, symmetric and Weyl algebras.
     
  3. Semi-simple Rings and Modules and Related Concepts: Jacobson's density theorem, Wedderburn structure theorem for simple and semi-simple Artinian rings, Jacobson-Bourbaki theorem, von Neumann regular rings.
     
  4. Division and Central Simple Algebras and the Brauer Group: Skolem-Noether theorem, double centralizer theorem, Wedderburn's theorem that finite division algebras are fields, Brauer group of a field, splitting fields of central simple algebras and the relative Brauer group, factor sets and connections with group extensions and group cohomology, cyclic and quaternion algebras.
     
  5. Radicals: Jacobson and nil radical of a ring, Nakayama's lemma, nilpotent ideals, lifting idempotents modulo a nil ideal.
     
  6. Introduction to Representation Theory of Finite Groups: Maschke's theorem, representation ring and Burnside ring of a finite group, characters, orthogonality relations, induced representations, Frobenius reciprocity law, examples and applications to group theory.
     
  7. Possible Additional Topics: Indecomposable modules and the Krull-Schmidt-Azumaya theorem, Morita equivalence of rings, coalgebras and Hopf algebras.

Text and References: There will be no official text for the course; instead, a number of texts relevant to the course will be put on reserve in the Mathematics Library. The following texts will be especially useful to students in the course: Dennis & Farb, Non-Commutative Algebra, Graduate Texts in Mathematics 144, Springer-Verlag, 1993; Dummit & Foote, Abstract Algebra (Edition 3) John Wiley & Sons, 2004.