Avsnittsöversikt
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Time: Thursdays 9-11 till February 21st, then 10-12 for the remainder of the semester.
Place: Cramér room (SU Campus Albano, house 1, floor 3 — entrance near the Sonya Kovalevsky display)Exceptions to the schedule:
- Thursday January 22nd 2026 the course will be 10:15-12:00 in Cramer.
Teachers:- David Rydh dary@math.kth.se
- Sofia Tirabassi tirabassi@math.su.se
Material:
- Geland–Manin, Methods of homological algebra (available for free from SU/KTH library website) [GM]
- D. Huybrechts, Fourier–Mukai transforms in algebraic geometry [Huy]
- A. Yekutieli Derived Categories (free arXiv-version) [Yek]
- Various scientific papers (more information below)
Examination: Consists of two parts:
- Homework problems during the course (see below)
- A presentation (details announced during the course; 20–25 mins each, towards the end of the course)
IMPORTANT: if you are a MASTER STUDENT and want to take the course for credit you have to have it registered has a selected topic in mathics (write to Jennifer to do it). In addition, to the above 1 and 2 there will be also a short oral examination for you.
About:
In the past three decades, triangulated categories in general and derived categories in particular have been useful tools in many areas of mathematics. In this course we are going to introduce triangulated categories but focus on the main example: the (bounded) derived category of an abelian category.The course is roughly divided in two parts. The first is purely algebraic, more or less covering- additive categories and functors
- abelian categories and exact functors
- derived functors
- construction of the derived category of an abelian category
- axioms of triangulated categories
Below (see modules) you will be able to find a more detailed lecture by lecture plan which will be constantly updated during the course.Prerequisites: For the first part some basic homological algebra: the tensor product is required and preferably you have seen Tor and Ext. The latter are motivations for the general constructions that otherwise could be difficult to understand and motivate. For the second part some knowledge about sheaves and algebraic geometry will be beneficial, however this is not necessary to complete the course. In particular, most/all homework will be on the first part.
Homework problems
If you are taking the course for credit, you should solve the exercises and submit your solutions (on Moodle). If you cannot make it by the date indicated for some reason, that is probably fine – just write to us ahead of time to let us know. If you have discussed the problems with others (which we encourage you to do), please indicate this in your hand-in. Also, if you have used other sources (forums, Stacks project, AI, ...) please indicate this as well. -
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Öppnades: måndag, 12 januari 2026, 00:00Senaste inlämningsdatum: torsdag, 5 februari 2026, 23:59
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After a brief introduction with motivations we will start to review a bit of category theory. We will introduce addittive categories and functors and we will start delve into abelian categories.
Literature
[Yek] We do as much as possible of chapter 2, we will continue into the next lecture.
One can find some of the material in [GM] and [Huy] but treated much more superficially.
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We keep going over [Yek] Chapter 2.
Important notions
- addititve categories and functors
- abelian categories
- exactness properties for functors
- projective and injective objects
- there are enough of them in
-mod.