Section outline

►Homework Assignments

  • Time: Thursdays 9:15–11:00 till March 26th, then 10:15–12:00 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.
    • Thursday February 5th we skip due to an interesting seminar at KTH.

    Teachers:

    Material: 

    • Gelfand–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:

    1. Homework problems during the course (see below)
    2. A presentation (details announced during the course; 35 mins each, towards the end of the course, preliminary April 16–June 4)

    IMPORTANT: if you are a MASTER STUDENT and want to take the course for credit you have to have it registered as a selected topic in mathematics (write to Jennifer to do it). In addition, to the above 1 and 2 there will also be a short oral examination for you. If you don't want credits for the course, we could instead write a certificate that you have followed the course.

    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
    • axioms of triangulated categories
    • construction of the derived category of an abelian category
    • derived functors
    In the second part, we are going to present brief overviews of several applications, mainly to the field of algebraic geometry.
     
    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.

►Homework Assignments