2020 – 2021, V, VII, IX semester

The seminar serves as an introduction to classical algebraic K-theory with a stress on lower K-functors K_0, K_1, and K_2 defined by Grothendieck, Bass, and Milnor. Algebraic K-theory can be considered as a generalized cohomology theory of algebraic varieties. In particular, the lower K-functors are important invariants of commutative rings and, more generally, of algebraic varieties, and their study is closely related to classification of vector bundles over those. Algebraic K_2-functors are also connected to reciprocity laws in algebraic number theory. Compared with higher K-theory, lower K-functors allow very explicit definitions in terms of projective modules and the general linear group. On the other hand, their properties are already quite non-trivial to prove.

**References**

- Weibel, The K-book: an introduction to algebraic K-theory, Graduate Studies in Math. vol. 145, AMS, 2013.
- J. Milnor, Introduction to Algebraic K-theory, Princeton Univ. Press, 1971.
- H. Bass, Algebraic K-theory, Benjamin, 1968.
- J. Rosenberg, Algebraic K-Theory and its applications, 1994.