Modules and Categories

Course manual 2018/2019

Course content

Modern algebra, number theory, topology, and geometry make extensive use of the language of modules and categories. In this course, a first introduction into these abstract theories is provided. 

We study modules over a ring (a common generalization of abelian groups and vector spaces), exact sequences (a powerful tool to work with generalizations of the 'isomorphism theorems'), tensor products, categories and functors. We make a start with homological algebra which combines techniques from 'modules' and 'categories'. 

Study materials

Syllabus

• L. Taelman, 'Modules and Categories'

Objectives

At the end of the course, the student

• can verify identities between modules through a 'diagram chasing' argument involving exact sequences
• recognizes categories and functors implicit in other mathematical subjects
• can prove elementary properties of functors by using the existence of adjoint functors
• can compute tensor products, quotients, presentations of modules in simple, explicit examples
• recognizes which tensor product and Hom operations between left-, right- and bi-modules are meaningful
• is able to compute with finitely generated modules over a principal ideal domain
• has an active vocabulary of examples of interesting modules, tensor products, categories, functors, (co-)limits, adjunctions
• can compute simple examples of free resolutions and Ext groups between explicitly given modules

Teaching methods

• Hoorcollege
• Werkcollege
• Self-study

Each week, there will a lecture on new materials and an exercise class given by the teaching assistant. Besides attending the lectures, the students are also expected to study on their own and work on the homework problems.

Learning activities

Academic skills

The student practice their skills by doing the homework problems and by attending exercise classes. These skills will be assessed in the exams.

Attendance

Programme's requirements concerning attendance (OER-B):

• Each student is expected to actively participate in the course for which he/she is registered.
• If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
• It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
• In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
• In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article 6.4, an other arrangement will be proposed in consultation with the study advisor.

Assessment

There will be weekly homework, leading to a homework grade.

Final grade = 60% Final + 20% Midterm + 20% Homework.

There is one retake covering both the midterm and final exam.

Inspection of assessed work

Contact the course coordinator to make an appointment for inspection.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Additional information

Prerequisites: Algebra 1, Algebra 2, Linear Algebra, Topology.

Occasionally we will use examples coming from other mathematical subjects such as Representation Theory or Galois Theory. These are not crucial to the course.

Contact information

Coordinator

• dr. M. Shen

Docenten

• Reinier Kramer