### SoSe 20: Algebraic K-theory

**Lectures:** Wednesday 08-10 M104, Friday 08-10 M103

**Exercises:** Friday 10-12 M102

Algebraic K-theory was invented by Grothendieck in the 1950s in his proof of the Grothendieck–Riemann–Roch theorem. Nowadays algebraic K-theory plays an important role in various fields of mathematics, notably algebraic number theory, algebraic geometry, homotopy theory, and geometric topology. In particular it appears in the formulation of many deep conjectures. In this course, we will introduce algebraic K-theory in its various forms (K-theory of rings, of schemes, of exact categories, of Waldhausen categories) and prove some of the fundamental theorems of Quillen, Suslin, Waldhausen, etc.

### Exercises

- Sheet 1 (due 06.05)
- Sheet 2 (due 15.05)
- Sheet 3 (due 20.05)
- Sheet 4 (due 27.05)
- Sheet 5 (due 03.06)
- Sheet 6 (due 10.06)
- Sheet 7 (due 17.06)
- Sheet 8 (due 24.06)
- Sheet 9 (due 01.07)
- Sheet 10 (due 08.07)
- Sheet 11 (due 15.07)
- Sheet 12 (due 22.07)

### Lecture notes

- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5
- Lecture 6
- Lecture 7
- Lecture 8
- Lecture 9
- Lecture 10
- Lecture 11
- Lecture 12
- Lecture 13
- Lecture 14
- Lecture 15
- Lecture 16
- Lecture 17
- Lecture 18
- Lecture 19
- Lecture 20
- Lecture 21
- Lecture 22
- Lecture 23
- Lecture 24
- Lecture 25
- Lecture 26
- Lecture 27

### References

A good introductory reference is:- C. Weibel,
*The K-book: an introduction to algebraic K-theory*, 2013

*Handbook of K-theory*, 2004

- D. Quillen,
*Cohomology of groups*, ICM 1970 - D. Quillen,
*On the cohomology and K-theory of the general linear groups over a finite field*, 1972 - D. Quillen,
*Higher algebraic K-theory: I*, 1973 - D. Quillen,
*Higher algebraic K-theory*, ICM 1974 - D. Grayson,
*Higher algebraic K-theory: II*, 1976 - F. Waldhausen,
*Algebraic K-theory of spaces*, 1983 - A. Suslin,
*On the K-theory of algebraically closed fields*, 1983 - A. Suslin,
*On the K-theory of local fields*, 1984 - R. Thomason and T. Trobaugh,
*Higher algebraic K-theory of schemes and of derived categories*, 1990