MATH 6330 - Noncommutative Algebra

Yuri Berest, spring 2015.

The course is intended to be a survey of noncommutative algebra. Topics will include module categories and Morita theory, Noetherian rings (Ore and Gabriel localization, classical quotient rings, Goldie Theorem), basic dimension theory (Krull dimension, global dimension and Gelfand-Kirillov dimension), filtered and graded rings, classical algebraic $K$-theory (projective modules, $K_0$, $K_1$ and $K_2$ of rings). Depending on the audience's interest, we may review some special (more advanced) topics, including $D$-modules, abstract homotopy theory (a.k.a. homotopical algebra), and derived algebraic geometry.

Our guiding principle will be to focus on topics and algebraic techniques that play a role in other areas of mathematics, mostly in algebraic geometry, algebraic topology and representation theory.