Modules and Categories

6 EC

Semester 1, period 1, 2

5122MOCA6Y

Owner Bachelor Wiskunde
Coordinator dr. M. Shen
Part of Minor Logic and Computation, year 1Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Minor Mathematical Themes, year 1Bachelor Mathematics, year 3Double bachelor Mathematics and Physics, year 3Double bachelor Mathematics and Computer Science, year 3
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Course manual 2025/2026

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

Objectives

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

Teaching methods

  • Self-study
  • Lecture

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

Activiteit

Aantal uur

Hoorcollege

28

Tentamen

3

Tussentoets

3

Werkcollege

28

Zelfstudie

106

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 cannot 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, it can be decided to deny the positive effects of homework to be counted towards the final grade of the course. In case of personal circumstances, as described in OER-A Article A-6.4, a different arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight Details

Final grade

1 (100%)

Deeltoets

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

Final grade = 70% Final + 20% Midterm + 10% Homework. 

Inspection of assessed work

Contact the course coordinator to make an appointment for inspection.

Assignments

See canvas page for this course.

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

Weeknummer Onderwerpen
1 Modules, examples, homomorphisms, kernels and cokernels, sums and products
2 Generators, free modules, exact sequences, five lemma
3 Split short exact sequences, finitely generated modules over principal ideal domains
4

Jordan normal form. Categories: definition, small and big examples, isomorphism in a category
5 Mono- and epimorphisms. Final and co-final objects. Functors: definition and examples
6 Contravariant functors. Morphisms of functors, equivalences of categories
7 Tensor products: universal property, examples, bimodules, functoriality
8 Tensor product is right exact. Tensor-hom adjunction
9 Adjunction of functors
10 Limits and colimits: definitions and examples
11 Limits and colimits in Set; Yoneda; adjoint functors and limits 
12 Chain complexes, homology functors, long exact sequence, homotopy
13 Free resolutions: definition, functoriality, uniqueness up to homotopy
14 The functors Ext^n, interpretation of Ext^0 and Ext^1
15  
16  

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

  • dr. M Shen

Staff

  • W. Huang