Course content

The course deals with homological algebra for abelian categories in general, and modules over a ring in particular.

First category theory is introduced, both in the setup of categories in general and abelian categories in particular, and some basic properties are discussed (functors, natural transformations, limits and colimits, in particular kernels, cokernels, pullbacks, pushouts).

The main part of the course focuses on the study of derived functors, in particular the derived functors Ext and Tor. To this end, the concepts of complexes, homotopy, homology, projective and injective resolutions are introduced and studied. The discussion of the first Ext also involves comparison to short exact sequences (Yoneda-Ext).

Finally triangulated, and in particular derived categories are introduced, and Ext is interpreted as morphism set in the derived category.

Learning outcome

1. Knowledge.
The student knows the fundamental concepts of categories and functors, and in particular of the Hom and tensor functors. The student can construct derived functors, and is familiar with the derived functors Ext and Tor. Further the student knows how the derived category is constructed, and how to interpret Ext in terms of this category.

2. Skills.
The student can read, discuss, and write arguments using categorical language.
Given a right (or left) exact functor between abelian categories with enough projectives (injectives), the student can construct the left (right) derived functors, and interpret what their values mean for the exactness of the original functor.

Learning methods and activities

Lectures. The lectures will be given in English if they are attended by students from the Master's Programme in Mathematics for International students.

The lecturer may give and discuss exercises (which are not obligatory but recommended) in order to practice the concepts introduced.

Recommended previous knowledge

Though no specific results from these courses will be used, it is useful to have participated in (or to participate simultaneously in) one or more other algebra or (algebraic) topology courses, such as MA3202 Galois Theory, MA3203 Ring theory, MA3403 Algebraic topology I.

Required previous knowledge

Participants should have some experience working with modules over rings, in particular know what a module and a homomorphism of modules is, and preferably what kernel, cokernel, and image of such a homomorphism are.

For instance this knowledge could have been obtained by participating in the course MA3201 Rings and Modules.

Course materials

Will be announced at the start of the course.

Credit reductions