Descent in algebraic K-theory


Time and place

Summer semester 2020
July 27–31, online (discussion on Discord)


In this lecture series, we will give a modern account of the landmark paper [TT] on the higher algebraic K-theory of schemes. Topics to be covered include:

  1. ∞-Categories: the basic theory, pre/stable ∞-categories, and animation
  2. Animated quasi-coherent sheaves on schemes
  3. Perfect complexes
  4. Waldhausen K-theory of stable ∞-categories
  5. Localization sequence and Zariski descent for algebraic K-theory

Prerequisites: Familiarity with algebraic geometry (scheme theory). The language of ∞-categories will be used, but we will give a quick review of the basics (optional background reading: HTT, Chap. 1, or HCHA, Chap. 1).


  • [TT] R. W. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in: The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math. 88, Birkhäuser (1990).


Lectures 10-12 and 14-16 every day, except Thursday.

Lect. 1: Monday 10:15–11:45
Lect. 2: Monday 14:15–15:45
Lect. 3: Tuesday 10:15–11:45
Lect. 4: Tuesday 14:15–15:45
Bonus: Tuesday 16:15–17:00(~)
Sebastian Wolf: Animated modules vs. connective complexes
Lect. 5: Wednesday 10:15–11:45
Lect. 6: Wednesday 14:15–15:45
Lect. 7: Thursday 14:15–15:45
Lect. 8: Thursday 16:15–17:45
Lect. 7: Friday 10:15–11:45
Lect. 8: Friday 14:15–15:45

Lecture notes


LiveTeX'd notes kindly shared by Milton Lin


Adeel Khan