A Course on Derived Categories

2026-01-02T09:35:32+00:00

The objective of this notes is to present an introdutory material to the undergraduate and
graduate students that would like to know some ideas about the derived category.
These are the notes a one week series of introductory lectures which I gave in the XXIIIEscola
de ´Algebra, in Maring´a, Paran´a, Brazil. Firstly we introduced the concepts of additive
and abelian category to show the axioms of triangulated category that are our main objective.
The triangulated category obey four axioms. We first introduced the first three axioms and
their consequences on chapter one and then the octahedral axioms in various equivalent forms
in a separate section of the first chapter.
The objective of this section is to give a model capable of making this axiom more palatable
since, in general, the form that it is presented in the literature does not remind the reader of
any similar structure in other fields of mathematics. So, we make the necessary efforts here
to present another form of this axiom that is similar to other tools that could be seen in the
abelian categories.
We present in chapter one the main example of triangulated category, the homotopy category
of complexes. Secondly, to understand the morphisms in the derived category I introduced
the concept of localization in chapter two. To those that are starting to study localization, we
present the necessary background to understand the localization of non commutative ring. We
believe that with this model in mind the student will profit more from the study of localization
of categories.
On chapter two, the student will find the necessary information and exercises to begin
to manipulate morphisms in the derived category. So, on chapter three we introduce the
definition of derived category of an abelian category and we explain how one sees the original
abelian category as a subcategory of its derived category.

After having done all this work, it is natural to have many questions about the behavior
of the derived category or its applications. Therefore, we present here a bibliography in
portuguese and in english that will help the students to make further investigations.
The reader that whishes to know the history and the motivation of the begining of the
derived category with many details, should read the introduction of the book ”Sheaves on
Manifolds - M. Kashiwara and M. Schapira ([15]).
Acknowledgements: I am particularly grateful to Sˆonia Maria Fernandes-DMA-UFV,
Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IME-USP, who carefully
worked through the text and sent me detailed lists of corrections, questions and remarks.
These notes were writen for the first time in 2014 and were used in a minicourse which I tough
in the XXIII-Escola de ´Algebra in Maring´a, Paran´a, Brazil. The last version was written during
my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104−3.

Boletim da Sociedade Paranaense de Matemática

A Course on Derived Categories