## Introduction to Hochschild and cyclic (co)homology

### MA 697- Fall 2021, Prof. Manuel Rivera, Purdue University

**Course description:** In this course we will carefully develop the foundations and basics of Hochschild and cyclic homology of associative algebras assuming minimal requirements: we will assume basic homological algebra as discussed in an algebraic topology course on homology and cohomology such as 572. More specifically, we will assume familiarity with chain complexes, free/projective/injective resolutions, derived functors, Tor/Ext, etc… Knowledge of algebraic geometry will be helpful but not required. We will focus on applications to the algebraic topology of loop spaces. The course will be roughly divided into four parts:

- The basics of Hochschild and cyclic (co)homology and their relationship to differential forms via the Hochschild-Kostant-Rosenberg Theorem
- Variations of Hochschild and cyclic (co)homology
- Hochschild and cyclic (co)homology in algebraic topology: relationship to loop spaces
- Operations on Hochschild and cyclic complexes

**Instructor:** Prof. Manuel Rivera (manuelr at purdue dot edu), Office: Mathematics Building 708

**Textbook: **We will not follow any particular text. However, my exposition will be highly influenced by the book *Cyclic homology by J.L. **Loday*.

Other useful notes and papers (more will be added throughout the semester):

- P. Belman’s lecture notes
- D. Kaledin’s Tokyo lecture notes and Seoul lecture notes
- K. Cieliebak and E. Volkov, Eight flavors of cyclic homology

**Course schedule: **Monday, Wednesday, Friday 1:30-2:20pm at Mathematics Building 215

**Office hours: **TBD

**Homework: **Homework exercises will be posted every two weeks.

**Grading scale: **80% homework, 20% presentation

**Special accommodations**: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.

In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php

**Course notes** (updated weekly)