MAT9570: Algebraic K-theory

Rough plan for the lectures

1. Category theory [Mac Lane]
   1. Categories and functors
   2. Discrete and additive representations
   3. (De-)categorification
   4. Limits and colimits
   5. Adjoint pairs
   6. (ETC)
2. Homotopy theory [Hatcher, Goerss-Jardine, Waldhausen]
   1. Weak equivalences and quasi-fibrations
   2. Simplicial sets and spaces
   3. The gluing lemma
   4. The realization lemma
   5. A fibration criterion
3. Classifying spaces [Quillen]
   1. The nerve of a category
   2. Quillen's Theorem A
   3. Quillen's Theorem B
4. Waldhausen K-theory [Waldhausen, R.]
   1. Categories with cofibrations and weak equivalences
   2. The S.-construction
   3. K-theory of finite sets
   4. The additivity theorem
   5. The approximation theorem
   6. The fibration theorem
5. Quillen K-theory [Quillen]
   1. Exact categories
   2. Segal subdivision
   3. The Q-construction
   4. Dévissage
   5. Localization

Some foundational papers

• Daniel Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85--147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.
• Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293--312.
• Friedhelm Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), 318--419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
• Bob Thomason and Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990.

Some books on algebraic K-theory

• Jon Berrick, An approach to algebraic K-theory, Research Notes in Mathematics, 56, Pitman (Advanced Publishing Program), Boston, Mass.--London, 1982.
• Eric Friedlander and Dan Grayson (editors), Handbook of K-theory. Vol. 1, 2, Springer-Verlag, Berlin, 2005.
• Dan Grayson, Algebraic K-theory, http://www.math.uiuc.edu/~dan/Courses/2003/Spring/416/GraysonKtheory.pdf
• John Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, NJ, 1971.
• Jonathan Rosenberg, Algebraic K-theory and its applications, Graduate Texts in Mathematics, 147, Springer-Verlag, New York, 1994.
• Vasudevan Srinivas, Algebraic K-theory, Second edition. Progress in Mathematics, 90, Birkhäuser Boston, Inc., Boston, MA, 1996.
• Chuck Weibel, The K-book: An introduction to algebraic K-theory, http://www.math.rutgers.edu/~weibel/Kbook.html