241022 Exact categories (V) (WiSe 2023/2024)
Contents, comment
-
Exact categories are a generalization of abelian categories (we use the definition of Quillen). They are a minimal set of axioms which is required for homological algebra - and as a structure they are much more flexible than abelian categories. They are ubiquitous in nearly every part of pure mathematics but rarely studied for their own sake. A good first source to start can be found in the arxiv (Theo Buehler: Exact categories) -Here one can find famous commutative diagram constructions such as "The snake lemma" in exact categories. Apart from the basic definitions, I want to first give an idea on how one can find and construct exact categories to understand the objects that we are studying. Then we look at homological invariants (numerical one such as projective dimension, global dimension and then later on more complicated ones like derived categories, singularity categories and Hall algebras). We may also consider invariants which have been studied in the past as K-theory and Hochschild homology of exact categories. Webpage comes in September.
Requirements for participation, required level
-
This course is in english. Basic concepts from homological algebra are necessary (such as categories, functors, cohomology of complexes).
It is recommendable that one has encountered a categories of modules over a ring (apart fom vector spaces of a field...).
Bibliography
-
Theo Buehler: exact categories (arxiv)
Teaching staff
Dates ( Calendar view )
| Frequency | Weekday | Time | Format / Place | Period |
|---|
Subject assignments
| Module | Course | Requirements | |
|---|---|---|---|
| 24-M-P1 Profilierung 1 | Profilierungsvorlesung (mit Übung) - Typ 2 | Study requirement
|
Student information |
| 24-M-P1a Profilierung 1 Teil A | Profilierungsvorlesung (mit Übung) - Typ 2 | Study requirement
|
Student information |
| - | Graded examination | Student information | |
| 24-M-P1b Profilierung 1 Teil B | Profilierungsvorlesung (mit Übung) - Typ 2 | Study requirement
|
Student information |
| - | Graded examination | Student information | |
| 24-M-P2 Profilierung 2 | Profilierungsvorlesung (mit Übungen) - Typ 2 | Study requirement
|
Student information |
| 24-M-PWM Profilierung Wirtschaftsmathematik | Profilierungsvorlesung (mit Übung) -Typ 2 | Study requirement
|
Student information |
| - | Graded examination | Student information |
The binding module descriptions contain further information, including specifications on the "types of assignments" students need to complete. In cases where a module description mentions more than one kind of assignment, the respective member of the teaching staff will decide which task(s) they assign the students.
To understand the notion of an exact category (following Quillen) one needs to know what is an additive category and what are the kernels and cokernels in a category. In examples it may also be helpful if the notion of a Krull-Schmidt category is known but it can also be explained in the lectures. At a later stage of the lecture: Derived categories of exact categories are considered - this will require triangulated categories and Verdier quotients of them. According to the wishes of the audience we will include more or less background material for it.
E-Learning Space
A corresponding course offer for this course already exists in the e-learning system. Teaching staff can store materials relating to teaching courses there:
- Registered number: 7
- This is the number of students having stored the course in their timetable. In brackets, you see the number of users registered via guest accounts.
- Address:
- WS2023_241022@ekvv.uni-bielefeld.de
- This address can be used by teaching staff, their secretary's offices as well as the individuals in charge of course data maintenance to send emails to the course participants. IMPORTANT: All sent emails must be activated. Wait for the activation email and follow the instructions given there.
- If the reference number is used for several courses in the course of the semester, use the following alternative address to reach the participants of exactly this: VST_428514488@ekvv.uni-bielefeld.de
- Coverage:
- 7 Students to be reached directly via email
- Notes:
- Additional notes on the electronic mailing lists
- Email archive
- Number of entries 1
- Open email archive
- Last update basic details/teaching staff:
- Wednesday, June 28, 2023
- Last update times:
- Monday, August 21, 2023
- Last update rooms:
- Monday, August 21, 2023
- Type(s) / SWS (hours per week per semester)
- V / 2
- Language
- This lecture is taught in english
- Department
- Faculty of Mathematics
- Questions or corrections?
- Questions or correction requests for this course?
- Planning support
- Clashing dates for this course
- Links to this course
- If you want to set links to this course page, please use one of the following links. Do not use the link shown in your browser!
- The following link includes the course ID and is always unique:
- https://ekvv.uni-bielefeld.de/kvv_publ/publ/vd?id=428514488
- Send page to mobile
-
Click to open QR code
Scan QR code:
- ID
- 428514488