Algebraic K-theory and Chow groups

This mini-course is part of Algebraic Geometry and Moduli seminar at ETH, which runs on Zoom.

 

The lectures will take place on 28 October and 4, 11, 18 November at 3-4:30 pm. 

 

The notes will be uploaded here during the course and the references will be updated. 

Algebraic K-theory, introduced by Grothenideck, is an invariant of schemes, which is both of algebro-geometric and homotopic nature. Over the last sixty years, algebraic K-theory proved to be one of the richest invariants of schemes. While being hard to compute in practice, this invariant has numerous features and appears in important problems in various areas of mathematics, for example, in number theory and geometric topology.

 

In this mini-course, we will follow Grothendieck’s original motivation, coming from the perspective of algebraic cycles. We will introduce algebraic K-theory and its cousin G-theory, and focus on their connection with Chow groups.

 

Throughout the course, we will spot various instances of K-theory within the wildlife of algebraic geometry!

 

In the first lecture we will introduce the zeroth K-theory group of a scheme and discuss its basic properties, as well as consider various examples. 

 

In the second lecture we will introduce zeroth G-theory group, which is closely related to K-theory, and its coniveau filtration. We will discuss (a corollary of) the Grothendieck-Riemann-Roch theorem, which gives a rational isomorphism between zeroth G-theory and Chow groups. 

 

In the third lecture we will talk about less explicitly defined higher K-theory groups, and get an explicit model for the first K-theory group. We will mention a connection with higher Chow groups and motivate why they are an important instrument for computing classical Chow groups. 

 

In the fourth lecture we will learn about Suslin’s rigidity theorem, which leads to a full computation of K-theory with finite coefficients of algebraically closed fields. The proof of this theorem is a very beautiful combination of algebro-geometric and homotopic methods, so we will discuss its main ideas. Surprisingly, while the statement of the theorem is unrelated to algebraic cycles, the proof crucially uses cycle-theoretic techniques. 

references and helpful materials

  1. C. Weibel "The K-book: an introduction to algebraic K-theory", 2013
  2. W. Fulton "Intersection theory", 1998
  3. D. Quillen "Higher algebraic K-theory: I", 1973
  4. A. Suslin "On the K-theory of algebraically closed fields", 1983
  5. Marc Hoyois' lecture course on algebraic K-theory
  6. Binda-Khan's lecture course on the Grothendieck-Riemann-Roch theorem