# Advanced topics in algebra: Hochschild (co)homology

## Description

The goal of the course is to give an introduction to Hochschild (co)homology, focussing on its applications in *deformation theory* of algebras (and schemes), and the role of the *Hochschild–Kostant–Rosenberg decomposition* in all this. There will be three parts:

- Hochschild (co)homology for algebras
- Hochschild (co)homology and derived categories
- The Hochschild–Kostant–Rosenberg decomposition for schemes

I will provide **lecture notes**. There will be an oral exam.

Lectures are on Tuesday, from 10:15 to 11:45, and Wednesday, from 14:15 to 15:45. First lecture will be on April 10.

## Preliminaries

For the first part some basic knowledge of homological algebra is required, e.g. on the level of the first chapters of Weibel. For the second and third part we will need standard algebraic geometry, e.g. on the level of Hartshorne. Familiarity with derived categories will be helpful, but is not required. Daniel Huybrechts will be teaching a course on derived categories, which might be useful.