# 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 Db(mod A) = Kb,-(proj A) = Kb,+(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
Db(mod A) = Kb,-(proj A) = Kb,+(inj A)
Mar 10 10:15–12:00 Db(mod A) = Kb,-(proj A) = Kb,+(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 examplesversion on Holm's homepage

Bernhard Keller, On the construction of triangle equivalencesPaper 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