SoSe 25: Algebraic K-theory

Lectures: Tuesday 08-10 M101, Thursday 08-10 M101
Exercises: Friday 12-14 M101

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 the algebraic K-theory of rings, starting with elementary definitions of K₀ and K₁ and making our way to Quillen’s construction of higher algebraic K-theory.

Exercises

Lecture notes

From SoSe 20:

• 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

An standard reference is:

• C. Weibel, The K-book: an introduction to algebraic K-theory, 2013

Some historical references in chronological order:

• 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