The goal of the seminar is a quick introduction to algebraic $K$-theory with applications in algebraic geometry and algebra. We plan to consider several approaches to algebraic $K$-groups, mainly in the context of exact categories, and also to analyze concepts and constructions in algebraic geometry related to algebraic $K$-theory. The apotheosis of the seminar should be a detailed study of a proof of the famous Merkuryev-Suslin theorem which relates Milnor $K$-groups and Brauer groups of fields.

Talks will be given mainly by undergraduate and graduate students participating in the seminar. As for preliminaries, familiarity with algebra and algebraic geometry, homotopy theory, category theory, and a little homological algebra will be required.

Program

1. Grothendieck groups, explicit definition of $K_0, K_1, K_2$ for rings.
2. Milnor $K$-groups, Moore-Matsumoto theorem.
3. Plus construction, $Q$-construction, Waldhausen $S$-construction.
4. Comparison of various definitions of higher $K$-groups.
5. General properties of algebraic $K$-groups.
6. $K$-theory of schemes, $K$-cohomology.
7. Brown-Gersten spectral sequence, Gersten conjecture.
8. The Merkuryev-Suslin theorem.

Seminar organizers
Fonarev Anton Vyacheslavovich
Gorchinskiy Sergey Olegovich

Financial support
The seminar is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center