# Reading Seminar "Homological algebra II - Derived Categories"

## Content

The aim of this seminar is to learn some homological algebra beyond what the course MA3204 usually discusses. In particular we will try to get some understanding of what derived categories are, and why they are useful.

More precisely, I am thinking of covering the following aspects

- Abelian categories
- Triangulated categories
- Stable categories of selfinjective algebras
- Homotopy categories
- Derived categories
- Derived categories of hereditary categories
- equivalences D
^{b}(mod A) = K^{b,-}(proj A) = K^{b,+}(inj A) - derived functors, in particular RHom & \(\otimes^L\)
- connection to Ext & Tor

After that, we will learn about Rickard's Derived Morita theorem (following Keller's paper "On the construction of triangle equivalences").

## Schedule

We usually meet **Thursdays**, from **10:15** to **12:00** in room **656**. Sometimes the meetings are rescheduled to **Tuesdays** form **16:16** to **18:00** (see below).

All information of future speakers and subjects may shift if we take longer to discuss one subject than the schedule suggests.

Date | Time | Subject |
---|---|---|

Jan 13 | 10:15–12:00 | Abelian categories |

Jan 20 | 10:15–12:00 | Triangulated categories (definition and basic properties) |

Jan 27 | 10:15–12:00 | Stable categories of selfinjective algebras |

Feb 03 | 10:15–12:00 | Homotopy categories |

Feb 08 | 16:15–18:00 | Derived categories I |

Feb 17 | 10:15–12:00 | Derived Categories II |

Mar 01 | 16:15–18:00 | Derived categories III Derived Categories of Hereditary Categories I |

Mar 03 | 10:15–12:00 | Derived Categories of Hereditary Categories II D ^{b}(mod A) = K^{b,-}(proj A) = K^{b,+}(inj A) |

Mar 10 | 10:15–12:00 | D^{b}(mod A) = K^{b,-}(proj A) = K^{b,+}(inj A) |

Mar 17 | 10:15–12:00 | Derived functors |

Mar 24 | 10:15–12:00 | Derived functors Unbounded resolutions |

Mar 31 | 10:15–12:00 | DG algebras I |

Apr 05 | 16:15–18:00 | DG algebras II |

Apr 14 | 10:15–12:00 | Rickard's Morita theorem |

## Literature

We will not follow a specific book in order, but try to take the information necessary for the different talks from wherever we can find them. Our main sources will be

**Charles A. Weibel**, *An introduction to homological algebra* – the introduction to derived categories here is very dense, and we will have to make sure to read very slowly when using this book

**Masaki Kashiwara** and **Pierre Schapire**, *Categories and Sheaves* – this book contains a lot more details than we will focus on – this book is available from SpringerLink via NTNU library (i.e. you have to be on campus or surf via campus).

**Dieter Happel**, *Triangulated categories in the representation theory of finite-dimensional algebras*

**Thorsten Holm** and **Peter Jørgensen**, *Triangulated categories: Definitions, properties and examples* – version on Holm's homepage

**Bernhard Keller**, *On the construction of triangle equivalences* – Paper on SpringerLink

## Format

We will take turns presenting aspects of the subject to each other, and discuss what is being presented. It is important that the people not at the blackboard follow the talks and contribute to the discussion to make sure that we understand what is going on, rather than just accept it.

For people who wish to do so, it will be possible to have an exam at the end (as MA8001 - Doktorgradsseminar). If we decide to stop the seminar before enough content is covered people willing to take an exam will have to read further on their own.

## Guide

**Steffen Oppermann**, room 844, Sentralbygg II, Steffen [dot] Oppermann [at] math [dot] ntnu [dot] no