Charles Weibel: K-theory of algebraic varieties (Lecture 1)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields.

Charles Weibel: K theory of algebraic varieties

Abstract:
Lecture 1 will present definitions for the Waldhausen K-theory of rings, varieties, additive and exact categories, and dg categories. The "+=wS." theorem for rings and simplicial rings will be mentioned. Basic properties, such as the localization, approximation and additivity properties will be described, and used to construct several localization sequences.

Lecture 2 will continue with localization sequences, using Karoubi filtrations. This includes the construction of negative/non-connective K-theory, and the K-theory of bounded complexes.

Lecture 3 will cover computational issues. This includes Zariski descent, K-cohomology and Bloch's formula, as well as excision for K0 and K1 of rings. Excision for varieties in low dimensions will also be covered.

Lecture 4 will survey computations for regular rings and smooth varieties. This includes motivic-to-K-theory methods, étale cohomology and regulators.