2023/2024

Homological Methods in Commutative Algebra

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 1, 2 module

Open to: students of all HSE University campuses

Instructors: Alexander Pavlov

Language: English

ECTS credits: 6

Contact hours: 60

Course Syllabus

Abstract

The course is devoted to methods and results of commutative algebra that are beyond the basic commutative algebra course. The first part of the course covers more classical results. We start with regular sequences, depths, and Koszul complexes. We prove the main results about Cohen-Macaulaev, Gorenstein and complete intersection rings and look at examples of such rings in singularity theory and invariant theory. The second part of the course is devoted to local cohomology, local Grothendieck duality and geometric applications. The results of the first and second parts have a large number of applications in algebraic geometry. The last part of the course will be an introduction to Gorenstein's homological algebra. It turns out that some of the classical results (such as the Serre criterion for regularity, the Auslander-Buchsbaum formula) have an analogue for Gorenstein rings. We will discuss the concepts of G-class, Gorenstein projective\injective\flat modules, G-dimension (analogous to the projective module dimension for the Gorenstein case). Auslander-Bridger formula and Auslander-Buchweitz approximation.