This is the website for the course I'm co-teaching with Kaif Hilman on algebraic K-theory in the summer semester of 2024 at the Max Planck Institute for Mathematics in Bonn.
Course: Introduction to algebraic K-theory
When: Wednesday 14.15 - 16.00 and Friday 10.15 - 12.00.
Where: MPIM Seminar Room
Lecture notes: Link
The overall goal of this course is to cover the fundamental theorems of algebraic K-theory in the modern formalism of ∞-categories. We provisionally aim to cover the following topics:
Preliminaries on ∞-categories
Group completion K-theory and the group completion theorem
Quillen's +-construction
Algebraic K-theory of stable ∞-categories
Waldhausen's additivity theorem
The localization theorem
The theorems of the heart
K-theory of pullbacks after Land--Tamme
Application: Purity in telescopically localized algebraic K-theory
Schedule:
Week 1:
24/4: Adjunctions, (co)limits, Bousfield localisations, Yoneda lemma, Dwyer--Kan localisations, and straightening/unstraightening.
26/4: Semiadditivity and stability.
Week 2:
1/5: Holiday (Labour day)
3/5: Presentability, multiplicative matters and group completion K-theory.
Week 3:
8/5: The group completion theorem, interlude on group theory, cyclic invariance.
10/5: Cyclic invariance continued, the +-construction, and k(Fin).
Week 4:
15/5: The Q-construction, definition of algebraic K-theory of stable categories, and Verdier sequences.
17/5: Split Verdier sequences, additive functors and verdier localisations, Waldhausen's additivity theorem.
Week 5:
No lectures due to Pfingstferien.
Week 6:
29/5: Universal property of K as the initial grouplike and additive functor under the core. Waldhausen's generic fibration theorem.
31/5: K-theory as a Verdier localisation. Karoubi sequences.
Week 7:
5/6 (at 12.15-14.00 due to Hausdorff colloquium by Thomas Nikolaus): K-theory as a Karoubi localisation, Thomason--Neeman localisation theorem and Quillen's localisation sequences, non-connective K-theory.
7/6: Overview of various devissage results. Waldhausen categories, exact categories, and their K-theory via the S-construction. The +=S theorem.
Week 8:
12/6 (at 12.15-14.00 in MPIM Lecture hall): +=S theorem, continued. Quillen's devissage theorem and beginning of t-structures.
14/6: t-structures continued and beginning of the theorem of the heart for t-structures.
Week 9:
19/6: Theorem of the heart, almost perfectness, coherence, and regularity.
21/6: The Blumberg--Mandell localisation sequence and Burklund--Levy unipotent devissage.
Week 10:
26/6: No lecture due to MPIM evaluation.
28/6: Beginning of Land--Tamme circle-dot excision, oriented pullbacks, and the circle-dot construction for rings.
Week 11:
3/7: Identifying the circle-dot ring and the circle-dot construction for categories.
5/7: The main theorems of Land--Tamme and quantitative descent of algebraic K-theory.
Week 12:
10/7: Quantitative descent continued, example of the circle-dot ring, truncating localising invariants.
12/7: Finished all things circle-dot. Began overview on chromatic homotopy theory.
Week 13:
17/7: Finished overview of chromatic homotopy theory and began discussion of the Land--Mathew--Meier--Tamme purity theorem.
19/7: Finished the discussion of Land--Mathew--Meier--Tamme purity theorem.