MA9N9-15 Representation theory of Algebras (with an algebraic geometric viewpoint)
Introductory description
This course will be about learning the modern use of derived categories in the study of the representation theory of finite-dimensional algebras. Derived categories are ubiquitous across representation theory, algebraic geometry, commutative algebra, homotopy theory, and other areas. Still, usually modules on derived categories tend to focus more on their use or prevalence in only one particular area. In this course, we will see how several of the tools and techniques that one learns regarding tackling these categories, and various other related questions in representation theory, can be useful more generally in areas such as algebraic geometry. All of the material that will be covered in this course are absolutely standard. Quite a lot of the focus will be placed on developments that have occurred in the last two decades or so.
Module aims
The main aim of this module is to familiarize students with some modern techniques in using derived categories arising in both representation theory and algebraic geometry – that will involve learning about (a) several important and useful generation properties of various kinds of derived categories, (b) Rickard’s Morita theory and its modern generalizations to algebraic geometry, (c) several longstanding open conjectures in finite-dimensional algebras and the use of derived categories in attacking them, etc.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Derived categories of rings, algebras, and schemes; Compact generation of triangulated categories in representation theory and algebraic geometry; Fundamentals about modules over finite-dimensional algebras; Homological conjectures in representation theory
Learning outcomes
By the end of the module, students should be able to:
- develop a good understanding of the fundamentals of the representation theory of finite-dimensional algebras
- be well-equipped to use a range of categorical tools in their own research.
Subject specific skills
Students taking this module will develop some fine skills in dealing with derived categories arising in algebra and algebraic geometry. Being skilled in derived categories is always a great asset for students doing research in these areas.
Because of the general approach that I will take in introducing all the concepts and the ideas in this course, despite the course being primarily based on finite-dimensional algebras, the students will become skilled in detecting how even in areas far from finite-dimensional algebras, certain objects or categories will have some key properties that will let them use their knowledge derived from this course.
Transferable skills
- sourcing research material
- prioritising and summarising relevant information
- absorbing and organizing information
- presentation skills (both oral and written)
Study time
| Type | Required |
|---|---|
| Lectures | 30 sessions of 1 hour (20%) |
| Private study | 120 hours (80%) |
| Total | 150 hours |
Private study description
Review lectured material.
Work on suplementary reading material.
Source, organise and prioritise material for additional reading.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group B
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
Assessment component |
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| Oral Exam | 100% | No | |
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An oral exam involving a presentation by the student, followed by questions from the panel (2 members of the department) |
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Reassessment component is the same |
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Feedback on assessment
Students will receive feedback from the course instructor after the oral exam, to cover also areas like presentation skills and use of technologies (or blackboard)
There is currently no information about the courses for which this module is core or optional.