Noncommutative Algebra

Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 90 minutes / session

Prerequisites

This is an introductory graduate course, although advanced undergraduates are also welcome to attempt it. Close familiarity with undergraduate algebra (groups, rings, modules etc.) will be assumed. 

Course Description & Topics

The beginning of the course will cover fundamental concepts forming the basic vocabulary of the algebraic language in today’s math. 

• Concepts:
  • Rings, modules
  • (Semi)simple rings, division rings
  • Wedderburn theorem, Jordan-Hoelder and Krull-Schmidt theorem
  • Prime radical, Jacobson radical
  • Artinian rings
• Categories:
  • Yoneda lemma, adjoint functors
  • Additive and Abelian categories
  • Morita theory
• Basic homological algebra: 
  • δ-functors
  • Tor’s, Ext’s
  • group (co)homology
  • global dimension, etc.
• These will be followed by somewhat more specialized topics:
  • Koszul algebras
  • Ore localization and Goldie theorem
  • Central simple algebras and Brauer group
  • PI algebras
  • Growth of algebras
  • Some ideas of noncommutative geometry

Lecture Notes and Readings

There is no required textbook. A full set of lecture notes is provided. Possible other references include the following:

• Cohn, P. M. 2003. Further Algebra and Applications. London: Springer. ISBN: 9781852336677              
  See chapters 2, 4, 5, 7, 8.  An earlier edition of basically the same text as volumes 2, 3 of the series Algebra by the same  author.

  The author of the recommended textbook is Professor Paul Moritz Cohn (1924 -2006) of University College, London. He was born in Hamburg, Germany to a Jewish family which was later persecuted by the Nazis and perished in Holocaust. Paul Cohn himself was saved by Kindertransport, an organised rescue effort for children of persecuted Jewish families that helped them emigrate to the United Kingdom.

• Weibel, Charles A. 1994. An Introduction to Homological Algebra. Cambridge England: Cambridge University Press. ISBN: 9780521559874

• Notes by Prof. Michel Artin 

• Notes by Prof. Pavel Etingof 

Assignments and Grading

Biweekly homework assignments will be given. Grades will be based on the homework. Students are also invited to prepare an optional expository paper for extra credit.  There are no exams.