Algebraic K-Theory

Contents

Algebraic K-theory, which to any ring R associates a sequence of groups K₀R, K₁R, K₂R, etc., can be viewed as a theory of linear algebra over an arbitrary ring. We will study in detail the first three of these groups. The higher K-groups, as defined by Quillen, will be the subject of the course “Higher algebraic K-theory” in the fall semester of 2011.

Applications of algebraic K-theory to number theory, algebraic topology, algebraic geometry, representation theory and functional analysis will be sketched as well.

Schedule

• Lectures: Fridays, 8:15 to 10:00
• Exercices: Fridays, 10:15 to 12:00
• Room: MA 12

Syllabus

1. Elementary category theory and module theory
2. K₀ : Grothendieck groups, stability, tensor products, change of rings
3. K₁ : elementary matrices, commutators and determinants
4. K₂: Steinberg symbols, exact sequences, Matsumoto’s theorem

Bibliography

Bruce A. Magurn, An Algebraic Introduction to K-Theory, Cambridge, 2002.

John Rognes, Lecture Notes on Algebraic K-theory, University of Oslo, 2010.

Jonathan Rosenberg, Algebraic K-theory and its Applications, Springer, 2004.

Charles Weibel, The K-book: An Introduction to Algebraic K-theory, (in progress).

Homework sets

• Exercise set 1
• Exercise set 2
• Exercise set 3
• Exercise set 4
• Exercise set 5
• Exercise set 6
• Exercise set 7
• Exercise set 8
• Exercise set 9

Various files to download

• Syllabus
• Schedule of remaining lectures