Masters Seminar in Algebraic Geometry: Introduction to algebraic K-theory
This seminar will give an introduction to algebraic K-theory, starting with the Grothendieck group of algebraic vector bundles. Next, we look at the early constructions of the functors K_1 and K_2 from the general linear group over a ring, as discussed in Milnor's book. This includes examples and computations, such as Matsumoto's theorem on K_2 of a field. We will then turn to Quillen's construction of higher algebraic K-theory, first for rings via the plus construction then for exact categories, leading to the definition of the K-theory and G-theory of schemes. We will discuss the fundamental properties of higher algebraic K-theory, and recall Quillen's proof of Bloch's conjecture relating the cohomology of the K-sheaves to the Chow groups. Other aspects of higher algebraic K-theory, such as the Merkujev-Suslin theorem and its applications, or the K-theory of dg categories and its universal property in that setting, will be discussed if time permits. For additional information, please see the seminar program.